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1. th t 1 tteps 3 par eps 001 g y y 1 y7n v gamma e v beta0 g f0 int max v th t t 0 xi t t p gamma 10 n 3 f0 8 beta0 600 tmax 1 625 e 1 6 init v 9 TOTAL 20 DT 0 01 XLO 0 XHI 20 YLO 30 YHI 10 VMAXPTS 400 done 5 2 Stochastic equations XPPAUT allows you to simulate a variety of random processes including con tinuous Markov processes and Brownian motion XPPAUT has two functions that return random values normal a b which returns a normally distributed random variable with mean a and standard deviation b The function ran c produces a uniformly distributed random variable from the interval 0 c XPPAUT uses the general purpose random number generator described in Numerical Recipes in C ran1 You can always start the random number generator at the same value by picking the same seed with the nUmerics stocHastic Seed U H S com mand Suppose you want to simulate the system dx f x dt de 5 2 Stochastic equations 103 Then here is the correct Euler discretization a t At z t f At At d where is normally distributed with zero mean and unit variance XPPAUT saves you the trouble of doing this by introducing wiener parameters These are nor mally distributed numbers with mean zero and variance vdt where dt is the nu merical time step Thus the ODE file would contain these statements code fragment wiener w x f x sigma w Thi
2. 0 6 F 04 F 0 2 F 0 d 1 n n 1 0 20 40 60 80 100 Figure 5 5 The probability of existence of the mutant species as a function of time stocHastic Mean to load the mean values of all the variables Escape to the main menu and plot the mean of z as a function of time You should see something like figure 5 5 5 2 2 Gillespie s method The above method of simulating a Markov process is good for small numbers of such events However if you want to simulate thousands of molecules using the markov declaration is out of the question Furthermore this method of simulating the Markov process is only an approximation to the real continuous time process Events can only occur at values of the time step dt Dan Gillespie 1977 devised a method which is exact and can be used to simulate thousands of random processes The method is basically to use the ideas of mass action and thus simulate each reaction at the single molecule level This method has been used to study channels in active membrane models as well as a number of chemical processes We illustrate it here on a simple chemical reaction and then with the Brusselator a classic chemical oscillator Consider the reaction X Z for the degradation of the molecule X at rate r Suppose that there are N molecules of X Since Markov processes are exponentially distributed the time of the next reaction taking place is a random variable with probability exp at where a
3. W2u The way that we avoided the trivial solution u x 0 was to set the derivative to 1 at x 0 A more standard way of normalizing the solution to the linear 136 Chapter 6 Spatial problems PDEs and boundary value problems e A 5 KS we PS ofl 3536 Oot of lt lt c Figure 6 4 Establishing shooting sets for the bistable reaction diffusion equation problem is to force it s L norm to be 1 uo dz 1 So how can we do this in XPPAUT Consider the differential equation dz 2 ae U along with the boundary conditions z 0 0 and 2 1 1 Clearly u 0 is not a solution Thus we can write our boundary value problem as this four dimensional system 1 u v v wu yl 2 w 0 with the four conditions u 0 u 1 z 0 0 and z 1 1 Try solving this with XPPAUT using different initial guesses for w 2 Solve the nonlinear problem 6 1 Boundary value problems 137 with the boundary conditions u 0 0 and u 1 1 2 Dendrites with active currents have been the subject of much recent interest in the computational neuroscience community Consider the model with a nonlinear calcium current 0 Vrz V g V E V with g V g 1 exp v 40 5 E 200 and the following boundary conditions V 0 Vhora and Vz 5 0 The two parameters of interest are g the strength of the calcium conductance and the holding potential
4. 0 free parameter sig 0 xe ye are the fixed points at the ends f xe ye g xe ye project off the fixed point from unstable manifold hom_bcs 0 project onto the stable manifold hom_bcs 1 par per 8 1 a 0 init x 1 y 1 total 1 01 meth 8 dt 001 oF OH FO OO H 188 Chapter 7 Using AUTO bifurcation and continuation xlo 2 xhi 1 6 ylo 1 yhi 1 xp x yp y done The only new feature is the projection conditions XPPAUT s boundary value solver will not work here since there are more equations than conditions and it doesn t know about the integral condition I have set the total integration time to 1 and have added the additional parameter per corresponding to the parameter P in the differential equation I use the Dormand Prince order 8 integrator as it is pretty accurate I have also set the view to be the x y plane Note that this is a pretty rough approximation of the true homoclinic We will use AUTO to improve this before continuing in the parameter a Run XPPAUT with this ODE file and integrate the equations You will get a rough homoclinic pretty far from the fixed point Click on File Auto tothe the AUTO window Now click on Axes Hi Choose xmin 0 xmax 50 ymin 6 ymax 6 and also select sig as the variable in the y axis Click on OK and bring up the AUTO window Numerics dialog Change Ntst 35 Dsmin 1e 4 Dsmax 5 Par Max 50 EPSL EPSU EPSS 1e 7 and click OK Now before you run the program click on
5. Consider the simple linear differential equation dx aos ax by dy y TETY 2 1 where a b c d are parameters We will explore the behavior of this two dimensional system using XPPAUT even though it is easy to obtain a closed form solution To analyze a differential equation using XPPAUT you must create an input file that tells the program the names of the variables parameters and the equations By convention these files have the file extension ode and I will call them ODE files Here is an ODE file for 2 1 10 Chapter 2 A very brief tour of XPPAUT linear2d ode right hand sides x a x bx y y cx x dx y parameters par a 0 b 1 c 1 d 0 some initial conditions init x 1 y 0 we are done done I have included some comments denoted by lines starting with these are not necessary but can make the file easier to understand The rest of the file is fairly straightforward I hope The values given to the parameters are optional by default they are set to zero The init statement is also optional The minimal file for this system has 4 lines x a x bx y y c x dx y par a b c d done Of course in this minimal file all parameters are set to zero as are the initial conditions Use a text editor to type in the first file exactly as it is shown or download the files Name the file linear2d ode and save it That s it you have written an ODE file I give a number of practice examples at the e
6. Run this in silent mode and you will get 11 data files that contain the outputs from the solutions to z x with x 0 1 j and j 0 10 Parameter sets can contain any of the declarations used in option files as well as initial data and parameter declarations Take advantage of the one parameter range integration as well If all you are varying is one parameter or initial condition then you can set the ODE file for a range integration in silent mode Here are two examples In the first one file is produced which contains all the runs ex1 ode x y y x rangeover y rangestep 10 rangelow 1 rangehigh 1 range 1 rangereset no done Here the initial value of y varies between 1 and 1 If you type xpp ex1 ode silent one output file will be produced that has all the integrations together There are occasions when you want to do this I doubt this is such an occasion In the second example 11 different files are produced one for each run ex1 ode x y y x rangeover y rangestep 10 rangelow 1 rangehigh 1 range 1 rangereset yes done 248 Chapter 9 Tricks and advanced methods The only difference is rangereset yes which tells XPPAUT to reset the storage at every run In silent mode XPPAUT produces files with names output dat 0 output dat 1 etc If you have many sets of many parameters and want to run these in a single run in batch mode keeping each run in a separate file then here
7. XPPAUT uses the numbers from 0 10 to describe colors of plots These are trans lated into linestyles for black and white postscript files Here are the color numbers and their approximate names Figure A shows the corresponding linestyles for black and white postscript Color 0 Black on a white screen and white on a black screen Color 1 Red Color 2 Red orange Color 3 Orange Color 4 Yellow orange Color 5 Yellow Color 6 Yellow green Color 7 Green Color 8 Blue green Color 9 Blue Color 10 Purple Keep these in mind for hardcopy For example to change the linetype of the null clines change the color that they are plotted in the program using the options shown in Appendix B and then when hardcopy is made the linetype will be as shown in the figure Negative linetypes in XPPAUT do not join the points together and instead draw either single points LT 0 or circles or varying diameter LT 1 2 no 255 256 Appendix A Colors and linestyles SOMIAKAWNHO I Figure A 1 Available postscript linestyles Appendix B The options XPPAUT enables the user to incorporate a large number of options within an ODE file These have the form opti value opt2 value2 where opti is the name of an option and value is the assigned value Almost every option can also be set within XPPAUT from the menus Similarly most items that you want to set in XPPAUT can be set in the ODE file The follow
8. to the Morris Lecar equations 7 5 Importing the diagram into XPPAUT In this section I will describe a nice geometric idea that underlies the complex firing patterns of some neurons Many neurons exhibit what is known as bursting behavior which consists of bouts of rapid spiking followed by period of relative quiescence A typical voltage trace is shown in figure 7 9 The model used to produce this is the Morris Lecar model previously described but such that the applied current satisfies a differential equation depending on the potential d CL T g1 V Ve gcatoc v V Voa gxw V Vx dw _ Wolv w dt Tw v dI gt e Vo v 196 Chapter 7 Using AUTO bifurcation and continuation with parameters given below in the associated ODE file mlsqr ode dv dt I gca minf V V Vca gk w V VK gl V V1 c dw dt phix winf V w tauw V dI dt eps v0 v v 0 41 84 w 0 0 002 minf v 5x x 1 tanh v v1 v2 winf v 5x 1 tanh v v3 v4 tauw v 1 cosh v v3 2 v4 param eps 001 v0 26 vk 84 v1 60 vca 120 param gk 8 g1 2 c 20 param vi 1 2 v2 18 param v3 12 v4 17 4 phi 23 gca 4 dt 1 total 4000 meth cvode xp 1 yp v x10 25 xhi 45 yl0 60 yhi 20 done This is set up in the J v plane to illustrate how the current J increases and then decreases as the voltage either moves through an apparent steady state or oscillates Plot I against t to see that the oscillati
9. y where y e x Note that there is no closed form inverse of y e special name FUN defines a one dimensional array name that represents a special summation involving an array of variables and a table FUN can be one of the following conv type n m w u name j 3 w m Dshift u 7 1 j 0 n 1 l m where w is a table of length 2m 1 and type even 0 periodic de termines how u j l is defined for j 1 lt 0 and j l gt n fconv type n m w u v f m name j Y wm Dfu i 0 v 3 0 n 1 l m 273 sparse n m w l u evaluates to m 1 name j Y wljmt dul c jm 1 j 0 n 1 1 0 where w is a table of length mn and cis a table of indices of length mn fsparse n m w l u v f evaluates to m 1 name j D w jm 1 f u c im 1 v 7 1 0 mmult m n w u evaluates to m 1 name j Y w l mj u 1 j 0 n 1 1 0 fmmult m n w u v f evaluates to m 1 name j Y w l mj f u 1 v 3 j 0 n 1 1 0 These can then be used in right hand sides of ODEs For example table w 21 10 10 exp abs t special k conv even 101 21 w u0 u o 100 u j1 x 31 Here k evaluates as the discrete convolution of exp x with the array u set name x1 z1 x2 z2 defines a named set of declarations including pa rameter values initial data and options which can be invoked while running XPPAUT with the File Get par set command For example set hopf a 3 x 1 2 b 9
10. 0 n 1 l m is represented by the statement special r fconv type n m w u v f This is similar to the conv operator but has additional arguments The first 5 are exactly like in the conv operator type is either zero even or periodic Additionally there is another array v and the name of a function of two variables f XPPAUT applies the function to u j l and v j before multiplying by the weight w 1 and adding up Thus to solve 6 5 we would define a weight a function of two variables and then the special convolution table w 11 5 5 exp abs t gam u v ggamf u v special r fconv zero 40 5 w theta0 theta0 gam Note that I use the same array for u and v in the fifth and sixth arguments of fconv they need not be different Thus 5 r j Y e MP Oj41 9 7 0 39 l 5 Since I am interested in the relative phases that evolve I will subtract off d0o dt which is proportional to r 0 Thus the ODE file for 6 5 is phschn ode a chain of 40 phase oscillators in relative phase coordinates 6 2 Partial differential equations and arrays 153 table w 11 5 5 exp abs t gam u v gamf u v gamf u a0 al cos u b1 sin u b2 sin 2 u special r fconv zero 40 5 w theta0 theta0 gam theta 0 39 r 31 r 0 par a0 25 a1 4 b1 1 b2 2 total 200 dt 25 meth cvode nplot 5 yp theta5 yp2 thetal0 yp3 thetal15 yp4 theta20 xhi 200 y10 0 yhi 10 done NOTES The first argument in fconv
11. 11 11 12 12 16 17 19 19 20 21 21 24 24 25 27 27 28 29 vi 3 3 3 4 3 5 Contents FunctioDs s a e e di a EA aaa 3 3 1 User defined functions Auxiliary and temporary quantities 3 4 1 Fixed variables ooo cc a A 3 4 2 IEXGTCISESS ca A AS it a eh ea cee Discontinuous differential equations 3 5 1 Integrate and fire models 3 5 2 Clocks regular and not o o 3 5 3 The dripping faucet o 3 5 4 Exercises ui a seg ad a a e de A XPPAUT in the classroom 4 1 4 2 4 3 4 4 4 5 4 6 5 1 5 2 Plotting funetions lt nm a kg AA a A a E tay E a Discrete dynamics in one dimension 4 2 1 Bifurcation diagrams 0 4 2 2 Periodic points ici ah aed Aa a e 4 2 3 Liapunov exponents for l d maps 4 2 4 The Devil s staircase 2 o a 4 2 5 Complex maps in one dimension 4 2 6 Iterated function systems 0 One dimensional ordinary differential equations 4 3 1 Non autonomous 1D systems Planar dynamical systems o Nonlinear systems ee ee ee 4 5 1 Conservative dynamical systems in the plane 4 5 2 Homework cmo Bae Se ee A Three and higher dimensions 0 0 4 6 1 Poincare maps FFTs and chaos 4 6 2 Poineareamaps 4 e aa a A de da aS 4 6 3 Hom
12. Appendix D Structure of ODE files All ODE files are just text files which consist of a series of definitions and directives to the XPPAUT parser The order in which the definitions are given is usually unimportant with one major exception So called fixed or temporary variables are evaluated in the order in which they are defined Thus you should never use a named fix variable before it is defined as this will lead to some rather bizarre results when you attempt to solve an ODE ODE files are all line oriented with a limit of 1024 characters per line You can use the standard line continuation symbol Here are all the possible ODE file directives AF comment line This is ignored by XPPAUT A comment that is extracted and can be displayed in a separate window a 1 b 2 A comment with some action associated with it When these comments are displayed in the comment window an appears next to them When clicked by the user various initial conditions parameters and XPP internal options can be set Iname formula This defines a derived parameter name whose values could depend on other parameters These do not appear in the parameter list since they are presumably tied to the other parameters Each time you change a parameter they also are updated options filename Insert a file name with common options This is include only for backward compatibility and is generally obsolete name t 1 formula or dname dt f
13. Parrondo s paradox game p e if ran 1 lt p e then 1 else 1 par eps 005 p2 1 p3 75 m 3 pa 5 gamma 5 probability for game 2 if it is played pg2 if mod ca m 0 then p2 else p3 probability for game 1 is it is played pgi pa probability for this game p if ran 1 gt gamma then pg2 else pg1 now play the game and increment or decrement our cash ca catgame p eps total 100 meth discrete dt 1 done As in the above two games run this simulation 1000 times You will find that you net on average about 1 2 after 100 games Figure 5 8 illustrates the effects of playing all three games 10000 times Here is the combined simulation parrondo ode this plays the combined game with probability 1 2 each Parrondos paradox game p e if ran 1 lt p e then 1 else 1 par eps 005 p2 1 p3 75 m 3 pa 5 gamma 5 game 1 alone ca_1 ca_1 game pa eps game 2 alone pg2p if mod ca_2 m 0 then p2 else p3 ca_2 ca_2 game pg2p eps game 3 pg2 if mod ca_c m 0 then p2 else p3 pgi pa p if ran 1 gt gamma then pg2 else pg1 ca_c ca_ctgame p eps total 100 meth discrete dt 1 done 5 2 Stochastic equations 115 cash 1 5 T 0 5 0 5 played Figure 5 8 Illustration of Parrondo s paradox games 1 and 2 are losers but played together become winners 5 2 4 Spike time statistics In many neurobiological applications we are interested in t
14. fP x We need to find roots of g x Newton s method 52 Chapter 4 XPPAUT in the classroom provides an iterative scheme Tn 1 Pn glrn g rn This rapidly converges if we have a close by starting guess But by sweeping over many values of r we are almost assured of catching all of them Since we are solving for the roots numerically we want to make sure that convergence to a true zero has occurred Thus if g r is less than a specified tolerance we will assume that we have found a root Finally we compute g r numerically to simplify the calculation With these preliminaries here isan XPPAUT file that will let you compute periodic points of any specified period logper ode finds the periodic points of the logistic map f x a x 1 x recursively define pth iterate ff x p if p lt 0 then x else ff f x p 1 find zeros of g g x x ff x p q g 1 Newtons method with numerical derivative r r eps q g rteps q a a if within tol of root then OK aux err tol abs q par p 3 eps 000001 tol 1e 7 meth discrete total 20 maxstor 50000 xp a yp r xlo 3 xhi 4 0001 ylo 0 yhi 1 00001 done NOTES I have defined the p iterate of f recursively The main iteration is the implementation of Newton s method I define a fixed variable q since I need to refer to it three times and thus the computation is sped up The parameter tol gives the tolerance that I require for a root
15. rane Wain Parm a 2nd Paras a Simin 15 Ym Xmax 75 and click Ok We are plotting X versus the parameter a in the window 75 75 x 3 3 Click on Numerics and set Par min 0 75 and Par max 0 75 then click on OK Now we are ready to run But have your mouse poised over the ABORT button since AUTO will run in circles until you stop it This is because the periodic solution is an isola an isolated circle of solutions and AUTO doesn t stop on such branches When you are ready to run click on Run Periodic Click on ABORT if it looks like AUTO is running in circles Another way to prevent this is to set the total number of computed points Npts to a low value from the Numerics menu However there is a danger that you won t compute all the points on the diagram A pair of ellipses should appear and you should see something like the Figure 7 3 The filled dots represent the maximum and minimum values of the stable periodic solution and the unfilled dots are the max min values of the unstable periodic branch A better way to look at this is to view the norm Click on Axes Norm and change Ymin to 0 Then click on OK and reDraw Use the Grab command and the arrow keys to move around the loop Note that at a 0 5 there is a limit point of oscillations 7 1 Standard examples 173 0 6 0 4 0 2 o 0 2 04 06 0 6 04 02 o 02 04 06 Figure 7 3 The diagram for the isola exam
16. the unstable manifolds are drawn in yellow and the stable in blue If you don t like this choice you can change it see Appendix B The second yellow trajectory is the one we are interested in it is the branch of the unstable manifold that joins with the fixed point 0 0 Change c to 1 and repeat the calculation Note that the solution trajectory crosses U 0 and spirals into 0 0 Change c to 3 and repeat again This is a perfectly valid solution Sometimes the initial data are quite far from the fixed point This means that the approximation to the invariant manifold is probably pretty bad To make the initial data closer make dt smaller within the nUmerics menu One point to note is that XPPAUT does not keep these trajectories in memory so that you cannot get hard copy of them However XPPAUT is aware of the initial conditions so you can repeat the calculations using the appropriate conditions For example we want the second trajectory computed This is an unstable manifold so if we compute forward in time we will go away from the fixed point as desired Click on Initialconds s H oot I H and choose 2 when given the choice of which since we want the second trajectory that XPPAUT computed You will get about two thirds of the trajectory You should extend the amount of time to integrate to say 40 to get the full trajectory For now just click on Continue C and continue to 40 You should have the traveling front To see the profile o
17. u ut u o 99 u j f cp k j cm delay u j tau dt 05 total 100 delay 10 autoeval 0 done 156 Chapter 6 Spatial problems PDEs and boundary value problems It is similar to the file for a regular network with the exception of the definitions of the the weights and the connectivity Note again we have added the statement autoeval 0 in the penultimate line to tell XPPAUT that the tables should not be recomputed automatically each time a parameter is chosen Before running this file there is one more point In many neural network models self connections are not allowed So how could we modify the connectivity function to avoid this Given an index we can add to this a randomly chosen number from 1 to 99 and mod out by 100 Thus would never be chosen The following fragment of code would replace line 4 in the above ODE file table con 1000 O 999 mod f1r t 10 1 f1r 99 ran 1 100 The flr t 10 is just the index i since the first 10 entries are connections to 0 the next 10 to 1 and so on The term 1 f1r 99 ran 1 generates a random integer from 1 to 99 Run this file and look at the space time evolution It appears that there is no pattern formation other than the small randomness Essentially the whole thing appears like a mean field model This suggests a theorem which is that the sparse excitatorily coupled oscillating network leads to synchronous oscillations as N m get large Compare the solutions to th
18. yp 0 export x y a b c d t xp yp par a 1 b 1 c 1 d 1 init x 1 y 2 done I have exported both the state variables as well as all four parameters and the independent variable t so the C program will set the fixed variables xp yp to the desired values depending on the state variables etc These are then the true right hand sides of the system Next you must write some C code to communicate with XPPAUT Here is the three right hand example include lt math h gt some example functions lv double in double out int nin int nout double var double con double x in 0 y in 1 double a in 2 b in 3 c in 4 d in 5 double t in 6 out 0 a x b y out 1 c y d x 9 8 Dynamic linking with external C routines 251 vdp double in double out int nin int nout double var double con double x in 0 y in 1 double a in 2 b in 3 c in 4 d in 5 double t in 6 out 0 y out 1 xta y 1 x x duff double in double out int nin int nout double var double con double x in 0 y in 1 double a in 2 b in 3 c in 4 d in 5 double t in 6 out 0 y out 1 x 1 x x ax sin b t c y Each right hand side has the form rhs double in double out int nin int nout double var double con The double array in contains all the exported variables etc in the order you exported them So in 0 contains the value of x The double array out shou
19. 1 You can edit animation files and load them into XPPAUT without having to leave XPPAUT 2 If you click on the little box in the upper right hand side of the animation window the animation will run simultaneously with the simulation This slows down the simulation 4 3 One dimensional ordinary differential equations 67 Exercises 1 Try animating the following equation x sinrx 1 05 To do this in the Main Window click on File Edit Functions F E F and change the first function to sin pi x 1 05 and then click on Ok Then run the ode with x 0 0 5 as an initial condition Rescale the phase variable axes for the animation by setting the parameters x0 5 and x1 10 Then run the animation 2 Try to create an ODE file and an animation file for a one dimensional non autonomous equation Note that since the function is non autonomous we must also redraw the function curve at every frame This is not the best way to look at nonautonomous 1d systems The next section suggests a better way Here are the two files phase line2 ode one dimensional nonautonomous system f x t x x 25 1 x a0 al sin w t par a0 0 a1 0 w 0 x f x t init x 26 transformation information for the animator par x0 5 x1 1 5 y0 1 y1 1 tr x x0 x 100 x1 x0 ff x t 4 tr x t y0 y1 yo ylo 25 yhi 1 25 total 40 xhi 40 done animation file for a phase line2 ode has time dependence permanent
20. 1 x fp x a 1 2 x init x 1 a 2 z 0 x f x a a z ztheav t 100 1n abs fp x 1e 8 aux liap z max t 100 1 xlo 2 xhi 4 ylo 2 yhi 1 total 2000 trans 2000 meth disc bound 1000000 xp a yp liap done NOTES The statement z ztheav t 100 1n abs fp x 1e 8 serves two pur poses First if f 0 the addition of the small amount le 8 prevents a singularity in the log Secondly premultiplying by heav t 100 only adds the numbers after 100 iterations The quantity liap is just the running average of z and it the quan tity of interest total 2000 trans 2000 tells XPPAUT to iterate for 2000 and only keep the last point Run this and integrate using the range option Use the same parameters in this as you did in computing the bifurcation diagram There is an easier way to compute Liapunov exponents over a range of parame ters XPPAUT has a built in algorithm for the computation of Liapunov exponents for differential equations and maps It works similarly to the idea described above Load up the logistic equation logistmap2 ode x t 1 a x 1 x par a 3 init x 54321 total 1000 trans 500 meth disc done I have set the total iterates to 1000 and thrown away the first 500 Once you have XPPAUT up and running click on nUmerics stocHastic Liapunov and choose 1 to look at a range Choose a as the parameter to range over choose 500 as the number of steps and choose 3 and 4 as the star
21. 10 Chapter 6 Spatial problems PDEs and boundary value problems u o 100 u 3 x 31 total 100 delay 10 meth euler done Write an ODE file for Nick Swindale s model for occular dominance columns in one dimension On z t Ot nle nle We yyt dy v Choose 101 points u 0 5 v 1 3 W x e71 0 3567 0252 Approximate the continuous convolution by a sum from 15 to 15 on a periodic domain Use random initial data between 0 and 1 Here is part of the ODE file table wa 31 15 15 exp a t t c exp b t t special k conv periodic 101 15 wa n0 Solve the globally coupled system u u d uj_1 2u 541 kuj 1 8 where 1 N 120 Take N 50 d 1 and vary k between 1 and 4 Use random initial data and reflecting boundary conditions Does this suggest any kind of spontaneous pattern formation Can you analytically determine the stability of the origin for this Chapter 7 Using AUTO bifurcation and continuation For many people the main reason to use XPPAUT is because it provides a fairly simple interface to the continuation package AUTO Doedel 1986 AUTO now has a GUI interface in the distributed version but still requires that you write FORTRAN code to drive it XPPAUT allows you to use most of the common features of AUTO including following fixed points periodic orbits boundary value problems homoclinic orbits and two parameter continuations This chapter
22. 5 5 be the bifurcation parameter Plot the prey x in a window 0 lt a lt 6 and finally set Dsmax 0 1 You will get a nice corkscrew 7 3 Boundary value problems XPPAUT is able to solve boundary values problems as we saw earlier in the book However the method used by XPPAUT depends on shooting which can be wildly difficult when the desired evolution equation has large positive eigenvalues Fur thermore XPPAUT cannot track solutions through turning points nor does it find branch points which are often indicators of non trivial solutions Thus if you can write your problem as an autonomous boundary value problem on the interval 0 1 and find a simple starting solution then AUTO is probably your best bet for track ing a solution We will examine a few cases EXAMPLE 1 Here is an example form the original AUTO manual that involves branch switching 1 u v v rr u u u 0 u 1 0 Here is an ODE file that has all the required AUTO parameters set for you bvp1 ode using AUTO to find the branches of a BVP 7 3 Boundary value problems 183 par r 0 u vy v rx pi 2 u u 2 bu b vu total 1 dt 01 dsmax 2 ds 2 autoxmin 0 autoxmax 6 autoymin 0 autoymax 100 ntst 5 parmax 5 meth cvode bound 10000 xhi 1 ylo 50 yhi 50 done Run XPPAUT and integrate this once to get the trivial solution loaded Click on File Auto to get the AUTO window up In the AUTO window click on Run Bndry Value an
23. 5 then f x else y note that t is the iteration number 0 1 2 if t is even evaluate f otherwise keep the old y y t 1 8 x y x t 1 if t 0 then x else y note that x t 2 f x t so every other point is the map f x a x 1 x par a 3 95 these are just useful for plotting f x x par xlo 0 xhi 1 nit 25 init y 0 x 25 always start y 0 here I create scaled x values to plot f x xx xlo xhi xlo t nit aux map f xx aux st xx some convenient settings for the graphics xlo 0 ylo 0 xhi 1 001 yhi 1 001 O xp x yp y nplot 3 add the plots y x and y f x O xp2 st yp2 st xp3 st yp3 map tell xpp that it is discrete and iterate 25 times meth discrete total 25 done This may seem a bit involved but it works and it is very easy to adapt it to other functions Simply edit the function f x and perhaps the ranges of the map xlo xhi Run this for a variety of values of a lt 4 Try the map f x mod x 3 4 asin 27 1 for a lt 1 2 50 Chapter 4 XPPAUT in the classroom x n 1 0 a 1 1 f 1 02 04 0 6 08 x n Figure 4 2 A cobweb plot of the logistic map 4 2 1 Bifurcation diagrams Another classic picture that is shown in many books is the bifurcation diagram for the logistic map This is done by plotting the parameter a along the x axis and iterates after transients of the map alo
24. If all the eigenvalues of A have negative real parts then this iteration scheme always contracts initial data to the origin for any positive value of h A numerical method is stable if it has the same property as backward Euler XPPAUT implements backward Euler for the general nonlinear equation This means that at each step it must solve the nonlinear equation Unt Tp hF t h n41 Newton s method is used to solve this XPPAUT requires two parameters MaxIt the maximum number of iterates and Tolerance the tolerance for convergence See above for a description of Newton s method XPPAUT has one more standard fixed step integrator Adams Bashforth which is a fourth order predictor corrector I will not describe this method except to say that it requires fewer function evaluations than Runge Kutta while maintaining fourth order accuracy It is an explicit method and quite unstable In fact try Adams on the logistic equation and figure out at which value of a the method becomes unreliable with a time step of 0 05 I don t recommend this method as it is quite unstable and the extra speed up you get over Runge Kutta is not worth the risk XPPAUT also has a symplectic integrator These integrators should only be used on systems of the form amp fj 01 Tn C 2 Integrators 265 written as follows Tjp Vj Ui fj Symplectic integrators obey the discrete equivalent of a conservation law so that they will preserve energy for H
25. Postscript aXes opts expOrt Colormap Freeze Freeze Delete Edit Remove all Key Bif Diag Clr BD On ff freeze nUmerics Total Start time tRansient Dt Ncline ctrl sIng pt ctrl nOutput Bounds Method dElay rUelle plot lookup bndVal ESC exit 275 Appendix E Complete command list stocHastic New seed Compute Data Mean Variance Histogram Old hist Fourier Power fIt data Liapunov Stat Poincare map None Section Max min Period Averaging New adjoint Make H Adjoint Orbit Hfun File Prt info Write set Read set Auto Calculator Save info Bell off c Hints Quit Transpose tIps Get par set Edit RHS s Functions Save as Load DLL Parameters Erase Make window Create Kill all Destroy Bottom Auto Manual SimPlot On Off Text etc Text Arrow Pointer Marker Delete all markerS Edit Change Move Delete Sing pts Go Mouse Range monteCarlo View axes 2D 3D Array Toon Xi vs T Restore 3D params Bndryval Range No show Show Periodic AUTO Parameter Axes Hi Norm hI lo Period Two par Zoom last 1 par last 2 par Fit fRequency Average Numerics Run Grab Usr period Clear reDraw File Import orbit Save diagram Load diagram Postscript Reset diagram Clear grab Write pts All info E 3 Browser commands E 3 Browser commands Find Get Replace Load Write Unreplace Table Add col Del col 277 278 Append
26. The AUTO window you will vary for the session Choose the names of any of your parameters the default is the first 5 that you have defined in the ODE file e Use the Axes command to tell XPPAUT what parameters are varied and what is to be plotted as well as the range of the graphs e Use the Numerics command to define all the AUTO numerical parameters such as the direction to go output options step size etc e Use the Run command to run the bifurcation Once you have run a bifurcation use the Grab command and the arrow keys to move around the diagram and the File command to save things or reset AUTO The commands Clear and reDraw just erase and redraw the screen respectively The Usr period item allows you to tell AUTO to save certain points such as oscillations of a certain period or points when a parameter takes a particular value AUTO is essentially independent of XPPAUT but there is some communica tion back and forth For example if you grab a point in the AUTO window then the state variables are loaded as initial data into XPPAUT and the parameters are changed as well Similarly when you start fresh in AUTO the initial fixed point or orbit is computed in XPPAUT Finally you can import a bifurcation diagram from AUTO into XPPAUT and plot this We will use this feature later to show bursting in a biological model With these preliminaries let s compute the bifurcation diagram for our cusp model The main parameter is a and
27. The theory of Hopf bifurcations 5 1 Functional Equations 93 Delay Ics x En al bo 18 Figure 5 1 The initial condition window for delay equations implies that the periodic orbit that arises from the destabilization of the fixed point has a period of roughly 27 1 88 3 34 where the denominator is the imaginary part of the unstable eigenvalue Verify that the actual period is around 3 6 which is pretty close Unlike ordinary differential equations delay equations require an entire inter val of initial data XPPAUT lets you input this by editing the Delay ICs window shown in Figure 5 1 To get this window click on the button labeled Delay on the top line of the Main Window Click in any variable and put a formula of t and then click on Ok or Go This will initialize the variable for t between 0 and r where r is the maximal delay You can also set initial conditions in the ODE file by adding a line of the form x 0 f t where x is the variable The delays do not need to be constant they simply must be non negative and smaller that the maximum delay Here is a cool example of a problem due to Ken Cooke The equation is x Z ra t 1 2 where ft is the integer part of t This is half way between an ordinary differential equation and a map In fact integrating it from t t to t t 1 produces a map that can be easily analyzed Nevertheless with a little thought we can write an ODE file for t
28. This is an example of 3 1 phaselocking Open another graphics window by clicking on Makewindow New and in this window graph both x1 and x2 against t Window this for 80 lt t lt 100 so you can see it more clearly e Gradually decrease w2 toward 1 and see what other kinds of mode locking are possible w1 1 9 is interesting 9 6 Arcana Here I describe a number of useful tricks that belong to no special category 9 6 1 Iterating with fixed variables Because fixed variables in XPPAUT are evaluated in succession one can actually compute iterative procedures with them which will be used in the right hand sides of the equations In chapter 4 we saw an example of this in the Taylor series plots Here I present a more sophisticated example of this trick I want to create and analyze the Morse Thule sequence Mn This is a sequence of 0 s and 1 s The nt number in the sequence is the number of 1 s in the binary expression of n modulo 2 For example the 23 element is found by writing 23 10111 This has four 1 s 9 6 Arcana 237 and 4 0 modulo 2 so M23 0 Suppose that n lt 2 Then the number of 1 s in the binary expansion of n is found from the iteration s 0 x n for i 0 i lt p it s s mod x 2 x flr x 2 Here flr x is the integer part of x Thus I just keep dividing x by 2 and checking if the result is odd This can be done much more quickly in C than I have indicated here but this is l
29. a x 1 x72 fx v cxcos omega t par a 1 f 2 c 3 omega 1 init x 1 v 1 done Try running this equation Change the total integration time to 200 by clicking on nUmerics Total and entering 200 Then click on Esc and integrate the equations with Initialconds Go Click on Window Fit to fit the entire time series into the window Create a new window clicking on Makewindow Create Stretch it out to make it bigger Then make a plot of the phase plane by clicking the little boxes next to the variables x and v in the Initial Data Window and then clicking on the xvsy button in the Initial Data Window You should see something like Figure 3 1 In a similar vein we can write nonautonomous maps in the same manner For example the logistic equation with slowly increasing carrying capacity Tn l an n41 Y2Lp 1 becomes time dependent carrying capacity par a 01 r 1 8 init x 1 x t 1 r x 1 x 1 a t meth discrete total 200 done 3 3 Functions 31 ECON TON CIC TO mG as cos 8 osno eos 7 a paa ero pea 7 Fabs Pa Jan inte tog ine loge sat ve max Gy max min x y sign a a rest fe gt 0 thon Toke me ertto ote moae y emoduloy Tena x amp y ANDy bessely d Yat Iy e esto ossa O Table 3 1 Names of standard XPPAUT functions 3 3 Functions XPPAUT comes with all of the usual functions you expect with the standard names In addition there
30. and choose y as the variable to transform In the Data Viewer the two columns formerly occupied by x and y are now filled with the power and the phase That is the FFT returns the cosine and sine coefficients of the spectrum c s the power p Vc s is the magnitude and the phase arctan s c is the angle The column occupied by t is replaced by the frequency You can plot the spectrum by graphing x against t Note the broadband frequency There are solitary peaks at about 318 636 and so on corresponding to the unstable periodic orbit You can get back the original orbit by clicking on the nUmerics Stochastic Data entry and then tapping Esc to get back to the main menu There are a number of other interesting ways to analyze chaotic data We will return to these at a later time 3 5 3 The dripping faucet One of the classic examples of chaos in a physical system is the dripping faucet Bob Shaw 1985 collected data on the interval between drips of the faucet He found through a variety of analyses that the process was chaotic and simulated it on an analog computer The model consists of a mass on a spring The mass grows linearly and when the spring extends beyond a certain limit a fraction of the mass drops off The fraction is proportional to the velocity The equations are d dx dx dm an k gt with the condition that if x t crosses 1 then the mass is decreased by an amount that depends on the velocity
31. back to the main menu Click on Xi vs t and plot N You will see a very strange power spectrum which is self similar That is if you blow up a region it looks like the full spectrum Try this with 64000 iterations and it looks essentially the same 9 6 2 Timers There are often situations where you may want a switch to turn on after a proscribed amount of time For example if you have a model of a synapse that has the form ds dt A t 1 s s r where A t Ao is T lt t lt T h and zero otherwise T is the firing time for the cell that corresponds to the synapse s Here is a way to do this init tf 1000 s a0 heav tfth t 1 s s tau tf 0 global 1 v vt tf t I have used the global flag to determine when the cell s voltage v crosses some threshold vt At this point the variable tf is set to the current time t The Heaviside step function switches to 1 and remains as long as t lt tf h and thus turns off h time units later Coupled phase response curves Related to this issue of tracking time is keeping a record of interspike intervals and or relative firing times in a system of neural oscillators For simplicity consider two coupled neural oscillators say coupled by a phase response curve see section rh W1 6 a2 Par 11 x W2 x Pi2 z2 We assume that whenever x crosses 1 it is reset to zero and k k P x F a We also assume that P is 1 periodic and vanishes at x 0 Then
32. coupling between oscillators occurs only through the voltage and thus the integral 9 1 involves only one component of the adjoint the voltage component For whatever reason the numerical algorithm for computing the adjoint for a given component converges best if you start the oscillation at the peak of that component Thus since we will only couple through the potential in this example we should start the oscillation at the peak of the voltage Here is a good trick for finding that maximum In the Data Viewer click on Home which makes sure that the first entry of the data is at the top of the Data Viewer Click on Find and in the dialog box choose the variable v and for a value choose 1000 and then 0k XPPAUT will find the closest value of v to 1000 which is obviously the maximum value of v Now click on Get in the Data Viewer which grabs this as initial conditions In the Main Window click on Initialconds Go to get a new solution Plot the voltage versus time Use the mouse to find the time of the next peak With the mouse in the graphics window click the button and move the mouse around At the bottom of the windows read off the values The next peak is at about 22 2 In the Data Viewer scroll down to this time and find exactly where v reaches its next maximum Note that it is as we suspected at t 22 2 In the Main Window click on the nUmerics menu and then Total to set the total integration time Choose 22 21 it is always best to
33. differential equation dX lt F X One of the common types of solution that are sought for this are trajectories joining fixed points That is one is interested in a solution that satisfies lim X t X t 00 where F X 0 are fixed points If Xt X the orbit is called a homoclinic trajectory otherwise it is called a heteroclinic trajectory Given a fixed point X to a system of ODEs one computes the linear stability for it Suppose none of the eigenvalues lie on the imaginary axis k eigenvalues have positive real parts and n k have negative real parts Then under fairly general circumstances there are sets A with dimensions k and n k respectively such that every point on A tends to the fixed point as t oo and every point on A tends to the fixed point as t oo They are called respectively the unstable and stable manifolds for the fixed point X Thus a heteroclinic trajectory going from X to X can be regarded as an intersection of the unstable manifold of X and the stable manifold of X Now when can we expect such an intersection to occur If the sum of the dimensions exceeds the dimension of the total system then it the intersection of these is generic that is it is not surprising For example suppose that n 1 and there are two fixed points stable and unstable Then we expect that there is a solution joining them Suppose that n 2 one fixed point is a saddle point and the
34. g a y 4 3 As with linear systems direction fields can lead to a good qualitative picture of the behavior Another qualitative method related to the computation of direction field is the use of nullclines Nullclines break the phase plane into regions where dx dt dy dy are of constant sign Thus they serve to give a sense of the direction of the flow The z nullcline resp y nullcline is the curve defined by f x y 0 resp g x y 0 Furthermore the points where the x and y nullclines intersect are fixed points XPPAUT easily computes the nullclines it can even animate them as a function of a parameter Suppose that a nonlinear system has a fixed point and that the linearization has no eigenvalues with zero real part Then locally the behavior of the nonlinear system is identical to that of the associated linear system In particular i the stability is unchanged and ii the stable and unstable manifolds of saddle points are tangent to those of the corresponding linearized system This means that XPPAUT can compute both the stability and the stable unstable manifolds for nonlinear systems With these tools it is easy to put together a complete picture of the dynam ics of any two dimensional system Let s consider the following example from a textbook d S etyt ey 0 d Y Z y 2ry 2 2 y dy 4 5 Nonlinear systems 73 Figure 4 8 Phase plane showing trajectories and nullclines for the two dimensional n
35. is flexible and with a little thought by the instructor it becomes relatively easy to write ODE files illustrating all the standard figures in most textbooks I have used XPPAUT with many differential equations and modeling texts at the undergrad uate and graduate levels Some of these are specialty courses like Computational Neuroscience or Quantitative Cardiac Physiology However I will not delve into such specialized applications in this chapter Rather you can go to the WWW and see some examples from various advanced courses I will start with some simple examples of using XPPAUT to create plots in two and three dimensions without reference to differential equations at all I will then consider a variety of topics in first order discrete dynamical systems I will illustrate how to make cobweb diagrams draw bifurcation diagrams compute rotation numbers and diagrams of the maximal Liapunov exponent I will then demonstrate how to use XPPAUT to make Julia sets and the famous Mandelbrot set In the next subsection I will discuss one dimensional non autonomous differential equations and finally planar dynamical systems 4 1 Plotting functions Suppose that you want to plot the function of one variable f x from xo to x Then the following ODE file will do the trick plot1 ode plot f x par xlo 2 xhi 2 f x x 1 x 2 s 1 45 46 Chapter 4 XPPAUT in the classroom x_ xlo s xhi xlo aux y f x_ aux X X_ O xp x yp
36. know about the Gillespie method so we must approximate the integration by a discrete time iteration The obvious choice would be to use Euler s method to advance the voltage and the potassium gating variable w However Euler s method is fraught with instabilities if the time step is too big Thus we introduce another first order method that is often used to solve membrane models exponential Euler Consider the differential equation y a by where a and b may be complicated functions of other variables and time For the period of time At we assume that a b are approximately constant Then we can exactly integrate this equation for an amount of time At y At a b y 0 a b exp bAt This trick is called the exponential Euler method The advantage of it is that if b is very large the algorithm remains stable without At being very small See also Duman and Mascagni in Koch amp Segev 1989 It is just as inaccurate as Euler s method but is not unstable Since both the voltage equation and the potassium gating variable can be written as y a by where b gt 0 we can use this method to advance both variables Thus the method for integrating the equations is to first figure out the time of the next reaction giving At and update the channels using the Gillespie method and update the deterministic equations using the exponential Euler method Yn 1 a b yn a b exp bAt Thus the XPPAUT implementation o
37. r rmax 1 s0 s define a two step Markov process with rate r markov z 2 0 r 1 0 define a flag so that when z jumps past 1 2 reset it to 0 turn on refractory period and increment the count global 1 z 5 s 1 z 0 count count 1 exponential decay of the refractory period s s tau count 0 par rmax 05 tau 10 s0 0 set it up so only the value at the end of an expt is computed meth euler dt 1 total 1000 bound 1000000 nout 500 trans 1000 done I have added a number of comments that explain what is going on We will simulate this over many trials and look at the spike count distribution Time is in milliseconds so that with a rate 0 05 we expect about 50 spikes per trial We will simulate 500 trials Click on Initconds Range and range over s from 0 to 0 in 500 steps and set Reset Storage to N This will take a few minutes When done you will have the standard Poisson process with no refractory period The Fano factor is the variance of the count divided by the mean Click on nUmerics stocHast Stat Mean Dec and choose count You will get the mean spike count of about 50 and a standard deviation of around 7 The Fano factor is about 77 50 which is close to 118 Chapter 5 More advanced differential equations the theoretical value of 1 for a Poisson process Make a histogram of the counts Click on stocHast Histogram with 40 bins of the variable count with a range between 30 and 70 Plot this and it looks
38. that they appear to be true damped oscillations Set 7 0 75 and solve again It goes to a complex zig zag pattern Increase T or change the parameter cm to see some other cool patterns You may have to increase the amount of time to see the asymptotic state Figure 6 9 shows a typical simulation Another animation Let s create an animation of these patterns which treats the spatial array as a rope that evolves in time If you want to understand animation files completely consult Chapter 8 For now type in the file nnetdelay2 ani use with nnetdelay ode a one liner 152 Chapter 6 Spatial problems PDEs and boundary value problems line 05 9 0 99 101 05 8 u j 05 9 j 1 101 05 8 u j 1 BLACK 2 done Click on Viewaxes Toon The animation window pops up Click on File in the animation window and load the above file nnetdelay2 ani Then press Go Rerun the simulation with different parameters and look at the animation More spatial operators There are other spatial operators which can be used in conjunction with tables to create fairly complex models A chain of weakly coupled oscillators provides a good example of the fconv operator Phase models arise from weak coupling and often have the form i T 6 5 where I is a 27 periodic function This function arises from the theory of averaging which we describe in detail in Chapter 9 In XPPAUT the expression m RG Y wf D vG
39. va fl 0 6 ye 04 oe A 0 2 Y A 0 H 0 0 2 0 4 0 6 08 1 0 35 T 0 3 F E 0 25 F Ei 0 2 F ye f 0 15 a 0 2 0 22 0 24 0 26 0 28 0 3 0 32 0 34 b Figure 4 5 Rotation number for the standard map and a expanded view over a small range of the parameter b Let z u iv and consider the complex map Zn 1 oe or in real coordinates Unt1 Un 1 Up Vas 2UnUn Suppose that we restrict z to have a magnitude of 1 so that u v 1 Then u v 1 2u and we see that the logistic map with a 4 is equivalent to the complex map z 2 restricted to the unit circle The dynamics on the unit circle can best be studied by writing z exp 2770 so that the map is now On41 20n mod 1 This means that the dynamics is chaotic in almost every sense of the word If we express a number between 0 and 1 in base 2 then the map just shifts to the left and 58 Chapter 4 XPPAUT in the classroom drops the lowest bit Every possible periodic solution is found e g 0 100100100 is a period three point Furthermore the Liapunov exponent of this is In2 since the map expands all points by a factor of 2 Looking again at the complex map z z it is clear that any initial condition that starts inside the unit circle will tend to the fixed point 0 while any initial condition which starts outside of the unit circle will tend to infinity Thus the unit circle serves to separate points in the complex plane which a
40. we can write the ODE file for this as map2 ode pair of coupled maps x1 w1 X2 W2 p x a sin 2 pi x f x mod x p x 1 9 6 Arcana 239 par a 05 par w1 1 w2 9 global 1 x1 1 x1 0 x2 f x2 global 1 x2 1 x2 0 x1 f x1 dt 0101 done The strange value of dt is due to a problem with the global command which occasionally fails if the test condition is satisfied exactly i e x1 1 at a particular time step Now this particular ODE file gives the values of the state variables but not the timing difference between the two cells nor the firing time interval between them Let be the time difference between the time that cell 1 fires and cell 2 fires Let Tj be the interval between successive firings of the respective cells Then we can track these keeping in mind that the global command only allows you to modify state variables We introduce 5 new variables t1f t2f t1 t2 phi all of which satisfy u 0 Here is the new ODE file map2 ode pair of coupled maps and timing info x1 w1 x2 w2 tif 0 t2f 0 t1 0 t2 0 phi 0 p x a sin 2 pi x f x mod x p x 1 par a 05 par wi 1 w2 9 global 1 x1 1 x1 0 x2 f x2 t1 t tif tif t global 1 x2 1 x2 0 x1 f x1 phi t t1f t2 t t2f t2f t dt 0101 total 50 bound 10000 done Each time z fires I update the interval between the last firing t1 t t1if and then update the firing time of 1 t1f t Each time that
41. 10 wei u0 special sii mmult 10 10 wii v0 fu x 1 1 exp x uth fv x 1 1 exp x vth u 0 29 ul j fu cee see j ciexsie j v 0 9 gt v j fv ceixsei j cii xsii j tau par cee 10 cie 8 uth 1 55 cei 13 cii 8 vth 2 tau 4 aux uu sum 0 29 of shift u0 i 30 aux vv sum 0 9 of shift v0 i 10 autoeval 0 total 50 done I have added the line autoeval 0 which tells XPPAUT not to re evaluate the tables every time the parameters are changed Thus the weights are fixed as we vary the other parameters The default for XPPAUT is to re evaluate tables whenever parameters are changed Since the tables here are random we don t want to alter them I have also added the population averages of the excitatory and inhibitory populations uu vv as auxiliary variables Curiously enough this network seems to synchronize and produces rhythmic output quite robustly As an 6 2 Partial differential equations and arrays 155 extra credit problem simulate the mean field model for this equation and compare it to the full model The mean field equations are uw u fu Ceet Ciev TU 0 fulCeiu Cuv Some tricks using the network functions There are other network functions which are more flexible than the convolutions The sparse operator allows you to create networks of length n in which each element j is connected to m other elements 7 in the network with weight wji This operator requires that you p
42. 2 17 2 wx v v v 1 u7 2 17 2 wx u init u 1 x 043 par c 1 a 25 w 1 dt 01 total 6 281 x10 0 xhi 6 5 dsmax 05 parmin 1 parmax 1 O autoxmax 1 autoxmin 1 autoymin 5 autoymax 2 done I have set it up for AUTO Run XPPAUT with this file Integrate the equations once Now click on File Auto to get AUTO started The diagram is set up so that the forcing strength c is the main parameter Click on Run Periodic to pick up the periodic solution Click on ABORT to stop it from infinitely looping You should see a little four leafed clover pattern Remember that this is showing the maximum and minimum of each periodic solution There are two such solutions one is stable and one is unstable Click on Grab to load the first point Click on Enter right afterwards Now in the AUTO window click on File Clear grab This makes AUTO forget the point it just grabbed However the point is still loaded into XPPAUT In the Initial Data Window of XPPAUT change the initial value of x to 1 In the Main Window of XPPAUT click on Initialconds Go and then when that is done Initialconds Last We now have another periodic solution where x is large and not near 0 Back in the AUTO window click on Run Periodic and a new branch of solutions will appear The point of clearing the AUTO grab point is that AUTO can start fresh from a new point Grab the starting point of this most recent computation this may take many taps on the Tab key until t
43. 3 Simulate the Chua circuit whose XPPAUT file is given below chuas scroll chaos x ax y if x gt 1 then m1 x m0 m1 else if x 1 gt 0 then m0 x else m1 x m0 m1 y x ytz zZ b y par a 9 b 14 28 par m0 0 1428 m1 0 2856 init x 1 y 1 z 1 total 200 done Plot it in 3 dimensions and see why it is called scroll chaos Compute the spectrum and the Liapunov exponent Take a few different Poincare sections see if you can find one which reveals a one dimensional map lurking beneath 4 Chaos occurs in two dimensional systems as well if they are periodically forced The classic example is the forced Duffing equation which we will write as a x 1 f acost We can rewrite this as a system w v fu acos t a x 1 Write an ODE file Set a 3 and f 25 Integrate this for a total of 200 and look at it in the x v plane Window the plane 1 5 1 5 x 1 1 88 Chapter 4 XPPAUT in the classroom Now we will take a Poincare map with respect to t plotting a point every time t hits multiples of 27 If you choose your section variable to be t then XPPAUT assumes it is periodic and plots a point every time t is a multiple of the section value Thus to get a map choose the Poincare map option choose Section choose T as the variable and type in 6 283185307 for the section Change the total time to integrate to 10000 Then in the main menu change the linetype to 0 so dots are poi
44. B t Y forward in time it will converge to a periodic orbit proportional to X t as a con sequence of the stability of the limit cycle and the translation invariance Similarly the solution to Z B TZ will converge to a periodic orbit if we integrate it backward in time As you can see this is the desired adjoint solution and is how XPPAUT computes it Once X t is computed it is trivial to compute the integral and thus compute the interaction function 9 5 1 Computing a limit cycle and the adjoint To use XPPAUT to compute the adjoint a necessary first step to computing the interaction function H we first have to compute exactly one full cycle of the oscil lation Let s use as our example the Morris Lecar equation v I gile v grwlek w IcaMolU eca v w wa v w Aw v 9 5 Oscillators phase models and averaging 229 We will look at vi f v1 1 v2 v wi g v1 w1 vy f v2 w2 elvi v2 Wy g v2 w2 Since the method of averaging depends on the two oscillators being identical except possibly for the coupling all you need to do is integrate the isolated oscillator Use the file ml ode Change the parameters I 09 phi 0 5 Integrate the equations and click on Initialconds Last a few times to make sure you have gotten rid of all transients Now you are pretty much on the limit cycle To compute the adjoint you need one full period In many neural systems
45. F T and fill in the resulting dialog box as follows Colum 1 UL iets 100 colskip os 1 0 mos i Rovsrip 100 Row 0 will be moved to the column of the first variable U1 Since Rowskip 100 Row 100 will be moved to the second data column Row 200 to the third data column up to Row 1000 in the eleventh data column The values in the Time column will be replaced by the numbers 1 to 100 Thus if we plot U1 against t we will see the spatial profile of the initial data Plotting U2 against t will plot the spatial profile of the Row 100 which represents the spatial profile when t 0 5 To get rows 0 100 500 and 1000 we thus would plot U1 U2 U6 U11 respectively You can do 6 2 Partial differential equations and arrays 143 03 FS 06 b 04 0 2 F 0 2 F 0 4 F 0 6 F 0 8 F 10 20 30 40 50 60 70 80 90 100 Figure 6 7 Solutions to the cable equation at 4 different times this manually using the Graphics Add curve G A command Alternatively here is a shortcut In the Initial Data Window check the boxes corresponding to U1 U2 U6 U11 and in the Initial Data Window click the button xvst All four curves will appear on the plot Two of the curves are very close corresponding to t 2 5 and t 5 confirming what we saw in the space time plot see Figure 6 7 6 2 1 An animation file One more way to look at the evolution of space time activity is to animate it XPPAUT has some animation
46. I have also set delay 20 so up to 20 year cicadas can be modeled e The initial condition statement n 0 100 0 t sets the number of underground grubs from earlier generations to be 100 The apparently useless 0 t is there to tell XPPAUT that this is a formula so that this must be a delay initial condition Run this and then try different values of k like 9 11 13 and 17 Which ones lead to synchronous emergence Fabry Perot cavity Here is a rather strange functional equation that combines a map with a continuous dynamical system This problem arises in the analysis of something in laser physics called a pendular Fabry Perot cavity Juan M Aguir regabiria provides this model as an example of his excellent dynamical systems program Dynamics Solver for Windows Here are the equations z x Q z xo Asin 0 f f t 1 cosb expliz t rT f t 7 T r x t a t 7 22 c where zo zs Asin 0 1 cos 0 2 cos 0 cos xs f is complex and the delay 7 is implicitly defined To avoid this difficulty Aguirre gabiria differentiates the 7 equation to obtain T v t c v t 7 where u t x t We need to write f fr ifi in terms of its real and imaginary parts Equations such as f t G f t 7 present a challenge to XPPAUT since they are not ordinary differential equations We will fool XPPAUT into treating them as Volterra equations without the in tegrals Howev
47. N e Ro R R w v Try e 01 Ro 0 3 c 0 01 and integrate for a total of 1000 with dt 0 25 using qualrk as the method Plot the A u projection in a window 0 4 lt x lt 0 4 and 1 2 lt y lt 1 2 Next write an ODE file for the u v system with A as a parameter Start with A 4 and find a steady state Then track the fixed point as increases using AUTO Hint set Dsmax 0 1 and Par Min 0 5 in the AUTO numerics dialog Find the Hopf bifurcation and obtain the periodic solution Write the points to a file Then superimpose them on the elliptic burster A U plane 198 Chapter 7 Using AUTO bifurcation and continuation v v 20 10 10 I L ji 0 dl 10 i h N 10 fi I r Y A ao b i E 20 IN VE j 30 pe p af 30 Es d Q 40 40 E 50 L 50 E 60 0 500 1000 1500 2000 2500 3000 3500 4000 26 2 30 32 34 36 38 40 V 4 t I y 20 10 ol 10 Homoclinie nN 20 S 30 40 _ gt 50 L 60 2 28 30 32 34 36 38 40 42 4 Figure 7 9 Bursting in a modified membrane model as a function of time during a burst top right Top left the voltage the slow parameter I and the voltage bottom superimposition of the bifurcation diagram in the I v plane Chapter 8 Animation 8 1 Introduction XPPAUT has a built in animator which allows you to create cartoons of the systems that you are simulating The animation fil
48. Usr Period and choose 3 for the number We want AUTO to output at particular values of the parameter per corresponding to P When the dialog comes up fill the first three entries in as per 20 per 35 per 50 respectively and click OK This forces AUTO to output when P reaches these three values Now click on Run and choose Homoclinic A little dialog box appears Fill it in as follows and click OK heft Ea Xe Right Eq Xe Minstable 1 stare 1 You must tell AUTO the dimension of the stable and unstable manifolds as well as the fixed point to which the orbit is homoclinic Note that if you ever fill this in wrong or need to change it you can access it from the Main Window menu under Bndry Value Homoclinic Once you click on OK you should see a straight line across the screen as the homoclinic approximation gets better Click on Grab and grab the third point corresponding to the point Per 35 For fun in the Main Window click on Initial Conds Go and you will see a much better homoclinic orbit Now that we have a much improved homoclinic orbit we will continue in the parameter a as desired First let s make sure we get the orbits when a 6 4 2 2 4 6 We will click on Usr Period and choose 6 Type in the following in the first six entries a 6 a 4 a 2 a 2 a 4 a 6and click Ok Click on Axes Hi to change the axes and the continuation parameter Change the Main Parm to a Xmin 7 Xmax 7 and click OK Click on Numer
49. Vhora Set g to zero Vhola 100 and solve the boundary value problem Hint start with initial data V 0 100 V 0 100 which is close to the true solution Now you should have a nice solution to the linear problem We will use the Bndryval Range command to compute the solution to the nonlinear BVP as g varies between 0 and 0 8 Click on Bndryval Range Choose gbar as the parameter and range between 0 and 0 8 with 40 steps A whole lot of solutions will emerge The Data Viewer is filled with the following information The first column has the parameter and the remaining columns contain the value of V 0 and V 0 Write a few down noticing that there is an abrupt transition in behavior at about g 0 55 to see this plot v x versus t which contains the parameter Figure 6 5 shows a plot of one solution and the behavior conductance varies A simplified model for excitability in a nerve is the Morris Lecar system In this homework exercise we first compute the velocity of the front in the absence of a recovery variable and then in the next exercise compute the traveling wave for the full system The equations for the full PDE are 2 a 2y I gL V Vz gxw V Vx JCaMo V V Vea L OPV Dye T CV w w OE PAw V woo V w g V w The functions wa V mao V Aw V involve exponentials of the voltage and are given in the ODE file for the traveling wave below In absence of diffusion there is a
50. a 02 representing a pair of sinusoidally coupled oscillators with different uncoupled frequencies w y w2 and coupling strength a The ODE file for this is straightforward phase2 ode phase model for two coupled oscillators th1 w1 a sin th2 th1 th2 w2 a sin th1 th2 par wi 1 w2 1 2 a 15 total 100 done Fire this up and integrate the equations You will get the Out of bounds message at t 90 or so That is because the variables 9 are actually defined only modulo 217 and are not really going out of bounds but instead just wrapping around the circle Thus it is more proper to look at 6 modulo 27 If define an auxiliary variable which is 9mod 27 and then plot it you will get a series of ugly lines that cross from one part of the screen to the other This is because XPPAUT does not know that this particular variable is defined on the circle There is a simple way to tell XPPAUT which of the state variables lie on the circle Click on phAsespace Choose A C and when prompted for the period choose the default which is approximately 27 A little window will appear with all of your variables included Move the cursor to the left of each of them and click the mouse to see a little X next to the variable Put this mark next to both variables and click on done This tells XPPAUT that these are both considered folded variables and will be folded mod 27 Now reintegrate the equations This time you get no such out of bounds messag
51. a straight line along y 0 line 0 y0 y1 y0 1 y0 y1 y0 BLACK transient this code draws a function ff x t line 0 991 01 f j1 t j 11 01 3 1 1 t RED 2 following the dancing ball fcircle x x0 x1 x0 y0 y1 y0 02 BLUE end 68 Chapter 4 XPPAUT in the classroom 4 3 1 Non autonomous 1D systems Non autonomous systems are somewhat more difficult to study The exercise above gives you one way to look at them but this is strictly numerical Consider dx Z flet 4 1 One way to look at such a system is in the t x plane Direction fields are very helpful in understanding the dynamics At a regular array of t x values in the plane we draw a little arrow whose direction is that of the vector 1 f x t This tells us the tangent point to the solution to 4 1 through the point t x thus we can sketch an approximate solution without ever having to actually solve the differential equation XPPAUT has tools for drawing direction fields However it only works for two dimensional dynamical systems This is no problem Consider the differential equation ds dx g q 7S This is identical to 4 1 but is now sitting in two dimensions With these con siderations we introduce the general one dimensional nonautonomous differential equation file oned ode generic one dimensional ODE file f x t x 1 x s 1 x f x s xp s yp x xlo 5 xhi 15 ylo 2 yhi 2 done Run XPPAUT with
52. across your screen toward the right If nothing happens you probably forgot to set a good initial guess for the starting point You should make sure that you are on a fixed point in XPPAUT for a 1 b 1 the fixed point is about 1 3247 Assuming that you have a curve you can see that it is pretty boring and that the curve was produced for a increasing Click on Grab to move around on the bifurcation diagram You should see a cross on the screen If you don t see a cross on the screen then next time you run XPPAUT run it with the xorfix option which should fix the lack of a cross By default it runs without requiring this option on Linux and Windows Use the left and right arrow keys to move the cross around The stability circle shows the eigenvalue status The fact that the drawn curve is solid implies the fixed point is stable unstable fixed points are drawn with thin lines AUTO specially marks certain points with small crosses and numbers You can jump to these by tapping Tab As you move around the diagram the text area beneath the diagram gives you a summary of information about the current location such as the value of the parameter the state variable the period and the point type for special points There are many pointtypes in AUTO In this example you will see only EP which means endpoint The one you never want to see is MX since this means AUTO has had trouble and maxed out on some sort of numerical iteration Sometimes this c
53. adjusted global sign condition name1 form1 defines a global flag allowing XP PAUT to implement delta functions If sign 1 and condition changes from negative to positive or if sign 1 and condition changes from positive to negative or if sign 0 and condition vanishes identically then each of the variables name is immediately changed to the value of formula For exam ple global 1 x 1 x 0 y yta sin y init name1 vali1 name2 val2 sets the initial data of name to the number val 272 Appendix D Structure of ODE files name 0 expression sets the initial value of name to the expression If name is a delay differential equation variable then if expression is a function of time t then name t is set to the function for M lt t lt 0 where M is the maximum delay For example x delay x 2 x 0 sin t bdry expression defines a boundary condition at the start or the end of an interval If the variable name is primed then it refers to the end of the interval otherwise to the start of the interval For example bdry u v 1 forces the boundary condition u L v 0 1 0 where 0 L is the interval for the ODE 0 expression defines an algebraic condition for differential algebraic equations solve name expression tells XPPAUT to solve the algebraic conditions for the variable name with initial guess expression For example x y init x 0 O y exp y x solv y 56715 Solves the DAE aw
54. allow you to change the dimension of the dynamical system so we trick the program by introducing both equations but allowing no changes until the mutation occurs We assume that the rate of the mutation is ex1 that is it is proportional to the population of species 1 Here is the XPPAUT file Kepler model kepler ode init x1 1 e 4 x2 0 z 0 par eps 1 a 1 xi x1 1 x1 z x2 x2 z a x2 1 x1 x2 eps x1 the markov variable and its transition matrix markov z 2 0 eps x1 0 0 total 100 xhi 100 ylo 0 yhi 1 2 nplot 2 yp2 x2 done Here are some notes on the ODE file i I multiply the right hand side of x2 and also x2 in the x1 equation by z which is zero until the mutation comes alive ii the mutation never turns off so the second row of the transition matrix is all 0 s iii I have added the line nplot 2 yp2 2 so that XPPAUT also plots the second population against time Run this Here is a question What is the expected value of z as a function of t We can answer this through a Monte Carlo simulation As with the Brownian motion problem we can perform a repeated simulation and take the mean and variance Use the nUmerics stocHast Compute option to run this simulation 100 times choose z as the quantity and set the start and end to 0 It will take several seconds for this computation While still in the nUmerics menu type 106 Chapter 5 More advanced differential equations 08 F
55. and choosing x at the prompt Freeze this curve G F F and choose color 1 red Now in the Initial Data Window change the initial value of z from 30 to 30 001 and then click on the Go button The trajectories are pretty close up to about t 10 where after they diverge This gives a ballpark estimate of the rate of divergence The exact rate of divergence is governed by the maximal Liapunov exponent This is a numerically determined quantity which is found by integrating a linear differential equation obtained by linearizing the original equation around the trajectory By definition the divergence d t away from the trajectory is approximately d t d 0 exp At where A is the maximal exponent Taking t 10 d 0 0 001 and d 10 1 we get that A In 1000 10 0 69 which is close the actual published value of about 0 9 You can do better than this by clicking on nUmerics Stochastic Liapunov U S L choosing O to not get a range of values and then clicking on Esc to get back to the main menu after the exponent is presented The number you get here is also a little low You can improve the estimate by integrating away all the transients and integrating for a long time If you use the Initialconds Last I L and extend the time of integration to 99 you will get a value closer to 0 9 The main point is that the maximal Liapunov exponent is positive which implies that there is divergence of nearby trajectories Note XPPAUT computes the
56. appears to be at 0 0 Figure 2 10 shows a printout along with a representative trajectory when the parameter J 0 3 The stability of fixed points is determined by linearizing about them and finding the eigenvalues of the resulting linear matrix XPPAUT will do this for you quite easily XPPAUT uses Newton s method to find the fixed points and then numerically linearizes about them to determine stability To use Newton s method a decent guess needs to be provided For planar systems this is easy to do it is just the intersection of the nullclines In XPPAUT fixed points and their stability are found using the Sing pts command as singular points is a term sometimes used for fixed points or equilibrium points Click on Sing pts Mouse S M and move the mouse to near the intersection of the nullclines Click the button and a message box will appear on the screen Click on No since we don t need the eigenvalues A new window will appear that contains information about the fixed points The stability is shown at the top of the window The nature of the eigenvalues follows c denotes the number of complex eigenvalues with positive real part c is the number of complex eigenvalues with negative real part im is the number of purely imaginary eigenvalues r is the number of positive real eigenvalues and r is the number of negative real eigenvalues Recall that a fixed point is linearly stable if all of the eigenvalues have negative real
57. are a number of less standard functions that you can use in the program The following logical functions return 1 or 0 depending on the truth of the in equality x gt y x lt y x y x lt y x gt y x y The last means NOT EQUAL y For all these logical operations it is best to surround the right and left sides by parentheses Here are the remaining simple functions e ran x produces a uniformly distributed random number between 0 and z e normal x s produces a normally distributed random number with mean x and standard deviation y e if x then y else z returns y if x is true otherwise it returns z AN e sum x y of z produces _ z where z is some expression involving the index i For example sum 0 10 of i evaluates to 55 There are other complicated functions that act directly on variables we will describe them shortly 3 3 1 User defined functions It is easy to define functions of up to 9 variables in XPPAUT A function is defined with the statement such as f x y x x y Functions can depend on other functions but be careful of recursive definitions as XPPAUT does not check for this and will unceremoniously crash if such a function is called A legitimate recursive function will not cause a crash e g one that stops such as fac x if x gt 0 then x f x 1 else 1 For example here is an ODE file wc ode for the Wilson Cowan equations 32 Chapter 3 Writing ODE files for differential equa
58. as part of a Mathematical Biology program During the evenings I ported the DOS program to X Windows on a UNIX environment while enjoying the sun set and listening to the same cassette tape over and over I forget what it was The program has evolved a great deal from those early years and is available at no cost to anyone who wishes to download it I have also successfully compiled the X version to run under various 32 bit flavors of Windows and also the new Mac OS X I have added lots of integrators and tools as well as my own idiosyncratic interface to the amazing continuation package AUTO Most things that you might want to do that concern dynamics either discrete or continuous can probably be done with XPPAUT if you know a few of the tricks That is the point of this book I suspect many users do not take full advantage of the features of the program This is mostly my fault as the user manual that is distributed with the program while comprehensive in its description of all the features is hopelessly baroque in its organization Why should anyone want to use XPPAUT There are plenty of packages that will integrate differential equations for you Many people use MATLAB MAPLE or MATHEMATICA to study and analyze dynamical systems These are all general purpose packages that have the capability to do most everything that is described in this book However both of the symbolic packages are extremely slow when it comes to numerically so
59. below morris lecar PDE par d 1 h 1 note that I have to change the names of the parameters so that I can use v1 v2 etc as the voltage grid points params vai 01 va2 0 15 va3 0 1 va4 0 145 gca 1 33 phi 333 params vk 7 vl 5 iapp 04 gk 2 0 g1 5 om 1 minf v 5 1 tanh v val va2 ninf v 5 1 tanh v va3 va4 lamn v phi cosh v va3 2 va4 f v w g1 vl v gk w vk v gca minf v 1 v tiapp g v w 1amn v ninf vw w vO v1 v 1 100 7 v j w j d v j 1 2 v j v j 1 h h v101 v100 w 1 100 g v j w j init vi 0 v2 0 v3 0 v4 0 init v 5 100 4 total 50 done In this exercise we will use a classic reaction diffusion model to illustrate pat tern formation Turing 1955 proposed that spatial instabilities could arise if coupled chemical systems had differing diffusion constants The equations for a general two component reaction diffusion equation are Ou du ae f u v Duna Ov Ou z glu v Dogz 162 Chapter 6 Spatial problems PDEs and boundary value problems with boundary conditions such as Neumann or periodic or Dirichlet The Brusselator is an example of such a system where f g are f u v a b 1 utu7v g u v bu uv The spatially homogeneous equilibrium is u v a b a and it is stable to spatially homogeneous perturbations as long as b lt 1 a However if this condition holds and D D is sufficiently large then t
60. c to 2 and repeat the calculation Notice that the branch again goes to infinity but via a different path Try c 1 and c 1 5 Try to get a value of c that is good to two decimal places Plot two guesses on the same graph that are on either side of the correct value Hint since this is a stable manifold we must integrate backwards to compute it The second branch is the desired one to compute using the Initialconds sHoot command Finally choose c 1 08121222 and shoot Then set the total time to 13 and integrate the equations backwards with the appropriate initial condition and plot v w versus t the independent variable You will see a good approxi mation of the pulse over that time scale 6 2 Partial differential equations and arrays XPPAUT has many operators that make it relatively straightforward to solve spa tially discretized partial differential equations and a variety of convolution equa tions The only downside to this is the somewhat limited size of the arrays that can be solved with XPPAUT The number of equations XPPAUT can integrate in recent versions is about 800 The task of the user is to determine the appropriate discretization and then code it up I will start with a simple PDE a cable of length L with mixed linear boundary conditions at each end The equations are Ou Oru z Dra u 6 2 co aou 0 t bouz 0 t cy aLu L t brLus L t The first step is to spatially discretize the PDE to convert it to a
61. called lorenz50 ani and has the following form animation for lorenz50 ode fcircle x 1 50 20 40 z j 50 02 j 50 done What this does is plot little colored balls in the x z plane where the coordinates have been suitably scaled If you want to know exactly what is going on in this animation file see the section on animation In the animation window click on Go to start the animation You will see a single little ball that becomes a comet as the individual trajectories diverge and eventually the attractor is filled out by the 50 little balls Exit from the file lorenz50 ode and now we will get back to work on the Lorenz equation If you closed the original Lorenz file you should start it up again and remove the projection graphs Integrate the equation and plot x against time Now we will look at the power spectrum Click on nUmerics Stochastic Power and choose X at the prompt Then click on Esc to get back to the main menu The power is in the first second data column x and the mode number is in the first data column t so click on Xvst and choose X to plot the spectrum You may want to zoom into the lowest 200 or so modes Notice the spectrum has no real peaks indicative of chaos 4 6 2 Poincare maps As a final look at the Lorenz attractor we will compute a Poincare map For our purposes a Poincare map consists of a discrete set of values picked whenever one of the variables passes through some prescribed value In Lo
62. can be seen by rewriting the system in polar coordinates in which case it becomes fart Ss 1 7 1 Standard examples 171 e lo Figure 7 2 One and two parameter bifurcation diagrams for the cusp ODE x a bx x The one parameter diagram has b 1 The two parameter diagram also shows the artifactual line at b 1 from the one parameter continua tion Thus when f A 0 there is a limit cycle with magnitude VA Starting with the parameter value a 0 we see that A 1 5 so that a t V1 5cost y t V1 5sint is a solution I will leave the verification of this as an exercise to the reader as well as the stability of the cycle The reader should also show that for a 0 another solution is A 0 5 leading to an unstable limit cycle Finally the reader should prove that x y 0 is an asymptotically stable fixed point for all a Here is how to start AUTO from a limit cycle Note first that this does not always work AUTO has much more trouble starting from a periodic solution than from a fixed point 1 Integrate the equations until a nice limit cycle is formed 2 Determine the period either by looking at the data browser or by using the mouse to measure peak to peak distance in the oscillation Hints In the Data Viewer click on the button Find choose a variable and then pick a real big number XPPAUT will then find the value of the variable closest to this big number in other wor
63. columns r r2 73 You can also import data to XPPAUT using the Read button in the Data Viewer Each column goes into the associated column in the Data Viewer If there are more columns in the data file than in the Data Viewer the first k columns will be read in where k is the number of columns in the Data Viewer I often read in data from PDE simulations so that I can use the animation features of XPPAUT Text etc XPPAUT has some limited capabilities for putting text lines and symbols in your graph These are found under the menu item Text etc You can put text of five different sizes and two different fonts in the figure The fonts are Times Roman and Symbol The latter allows you to put Greek letter but you have to know the corresponding Latin letter E g a a is obvious but q 0 is not so obvious You can mix fonts and have sub and superscripts in the text that you write To do this use the five escape sequences VO Use the normal Times Roman font 1 Use the Symbol font s Subscript S Superscript n Normal no sub or superscripts expr evaluate the expression expr before rendering the text Thus if you type in 1q 0 sj n at b S2 n 1b then the output will be 0j a b 8 since q becomes 0 in the Symbol font and b becomes If you want a text expression to be evaluated and permanently set so that when the quantity expr changes the text will not then preface the string with the symbol For exa
64. else i2 x shift al i x shift b1 i y shift e1 i y shift c1 i x shift d1 i y shift f1 i meth discrete total 20000 maxstor 100000 xp x yp y xlo 3 xhi 3 ylo 0 yhi 10 1t 0 done Notes The first six lines give values for the affine maps ve art byt e y cr dy f The next lines give the four probabilities of each of the maps Notice that these probabilities are not uniform The numbers s1 s2 s3 are the cumulative probabil ities Next a random number z is selected The next three lines of code determine which map to use based on z by testing whether or not z is larger than each of the cumulative probabilities This sequence of code is similar to that used in our implementation of the Gillespie algorithm see Chapter 5 Finally the map is de fined by using the shift operator on each of the parameters for the map Thus if i 0 then the first map is used and so on For a much larger set of parameters you would likely use look up tables see for example the cellular automata example in Chapter 6 If you run this ODE file you will see the famous fractal fern 4 3 One dimensional ordinary differential equations 65 4 3 One dimensional ordinary differential equations Many courses in differential equations start off by describing a variety of exactly solvable first order differential equations However recent texts such as those by Blanchard et al Strogatz and Borelli and Coleman take a more qualitative approach and emphasi
65. equations ODE s or maps You type these in almost exactly in the way you would write them on paper There are two different ways to declare a differential equation Consider the ODE dx _ dt Then the corresponding line in an ODE file will be either x x or dx dt x I will generally use the former since it is less typing The spaces are not required here and are just for readability There are two different ways to define a map Consider Be Caj 2i You could write this in two ways x x 2 or x t 1 x 2 Both are interpreted the same way Again I will use the former as it is less typing Higher order differential equations and maps must be rewritten as systems of first order equations Thus the classic damped spring dex dx would be written as d d L U ae Y ge LU kl and the corresponding ODE file is written as the spring x v v f v k x 1 par f 0 1 1 k 1 done 3 2 Ordinary differential equations and maps 29 where I have assigned some arbitrary values to the parameters as well as a comment Consider the delayed logistic map Tn 1 TEn 1 Ln 1 which is a second order map To solve this in XPPAUT we rewrite it as a system of two first order maps Un 1 Yn Yny Tn42 TEn41 1 Ln TYn 1 Tn Thus the ODE file is delayed logistic x t 1 y y t 1 r y 1 x par r 2 1 init y 25 total 200 done I have added an initial condition statement for y I have also set the n
66. few time to get rid of transients For limit cycles get a good estimate of the period and integrate one full period and no more or less e To navigate the diagram quickly click on Tab as this just jumps to special point if you get lost in the diagram use the Axes Fit reDraw sequence to put the entire diagram on the page Or if there is a really complex part use the Axes Zoom feature to magnify the given area e Follow all branch points AUTO will generally go back to all branches of fixed points and try to follow these automatically However for limit cycles which have branch points such as a pitchfork bifurcation AUTO does not automatically compute them Use Grab to move to such points labeled BP and then click on Run Change the sign of Ds in the numerics dialog the branch off in a different direction e Grab all Hopf points and try to find the periodic solutions that arises from them e For two parameter bifurcations also change the sign of Ds to go in both di rections e If AUTO fails from the beginning as indicated by the MX label then you haven t given it a proper starting value Make sure you are on a good periodic orbit or at a true fixed point or have a true solution to the boundary value problem e If AUTO fails after partially completing the diagram and you would like to change numerical parameters and have another go destroy the diagram first before continuing Grab the initial starting point and File Reset diagram
67. fill in the resulting dialog box as follows praxis Z imax Y axis Z Ynax 45 Pain 30 kabel Yain 30 Yrabe1 and click on 0k You will see a diagonal line from one corner to the other Click on Graphics Edit and choose 0 Change the Linetype to 1 and click Ok The diagonal line becomes a series of points Now how can we see the map First lets freeze this diagonal line of points since with maps it is always nice to plot the line y x Click on Graphics Freeze Freeze and then click on Ok The diagonal dots become a solid line Finally we can plot the curve zn41 against Zn XPPAUT has a command called rUelle named after the famous mathematician David Ruelle who proved that you could reconstruct the dynamics of arbitrary systems from the time series of one variable x t by looking the points x t z t T1 t Tn Thus we will look at zn_1 Zn We thus want to shift the values on the x axis up by one Click on nUmerics rUelle U U and change the X axis shift to 1 Escape to exit the numerics menu and click on Restore to restore the plot You should see a cusp like curve that is almost one dimensional It intersects the diagonal at around z 38 5 and this corresponds to a periodic point If you are reasonably good at guessing formulas for graphs from their shape you might try to approximate this curve by some function of z Here is my guess florenz Z 27 18 exp z x 37 3 5 4
68. fixed point 0 0 is a saddle point with a positive eigenvalue Au and a negative one of A whose sum is the trace of the linearized matrix a The sum of the eigenvalues is called the saddle quantity and if it is positive for us a lt 0 then the homoclinic is unstable Figure 7 7 shows the continuation and some representative orbits Heteroclinics We now describe how to find heteroclinic orbits The methods are the same except that we must track two different fixed points Thus we need an additional n equations for the other fixed point As with homoclinic orbits we go from the unstable manifold to the stable manifold In this case the left fixed point is the one emerging from the unstable manifold and the right fixed point is the one going into the stable manifold Thus the dynamical system is dx SSD ds Flo a dLleft _ ds i dC right z ds f tef 0 0 190 Chapter 7 Using AUTO bifurcation and continuation f rignt 1 0 Ls x 0 teft 0 0 Lu x 1 trigni 1 0 The only difference is that we have the additional n equations for the right fixed point and the n additional boundary conditions It is important that you give good values for the initial conditions for the two fixed points since they are different and you need to converge to them The classic bistable reaction diffusion equation provides a nice example of a heteroclinic We examined this equation 6 1 in the previous chapter bo
69. follows from the uniqueness theorem for ODEs Since u 0 0 if v1 0 va 0 are two different branches then the solutions are identical Thus we introduce a dummy differential equation q 0 with a boundary condition q 0 v 0 Then the solution to this dummy equation satisfying the additional boundary condition is just q t v 0 for all t Add the following two lines to the above ODE file q 0 b q v The dimension of the system is a bit larger but this is a triviality Figure 7 6 shows the result of this trick 184 Chapter 7 Using AUTO bifurcation and continuation 600 400 H J he 2 J J J os E 200 F A j o coe l 1 200 N o2 L a0o f eA 600 0 Figure 7 6 Solution to the boundary value problems from example 1 left and example 2 right EXAMPLE 2 Here we revisit the example studied in chapter 4 which arises in the analysis of a reaction diffusion equation on a disk Recall that the boundary value problem is 0 A 1 A d A A r Ak 0 qA d k k r 2kA A with boundary conditions A 1 k 1 0 and A k ns IS r gt 0 r r gt 0 r We handled the behavior as r 0 by forming a Taylor series near the origin We let ro be close to zero and start the problem at r rg This problem is nonautonomous so we introduce the differential equation r 1 with b
70. format and a standard GIF file called twodl_0 gif will be created Kinescope Capture can be used to capture a sequence of frames and these can be saved in sequence or even animated Single frames can be viewed with a standard browser or most any graphics software e g xv on Linux machines On Windows machines you can capture the window to the clipboard and dump it into a Paint or Word file This is done by clicking on the window and pressing the Alt PrtScrn key Erase the screen and use the mouse to start some initial data at roughly 2 2 You will get a nice spiral Let s color code this trajectory according to the absolute velocity y y Click on nUmerics Colorcode Velocity U C V and choose Optimize Escape back to the main menu Your trajectory is nicely colored with the minimum velocity blue and the maximum red Let s color the whole phaseplane according to this color scheme Click on Dir field flow Colorize and change the grid to 100 You can colorize according to many different quantities such as energy for a conservative system as we shall see later While XPPAUT does not create a direct postscript version of your picture you can save the screen image as a GIF file as described above Individual color coded trajectories are rendered in the postscript file Erase the screen E and go back into the numerics menu and turn off the colorizing nUmerics Colorize No color and Esc back to the main menu XPPAUT can tell you th
71. get tired of repeating this try Init Conds mIce I I which being mice 14 Chapter 2 A very brief tour of XPPAUT an sor bid f I gt gt E Y Figure 2 4 The initial conditions window is many mouses Keep clicking in the window When you are bored with this click either outside the window or tap the escape key Esc Click on Erase and then Restore E R Note that all the trajectories are gone except the latest one XPPAUT only stores the latest one There is a way to store many of them but we will not explore that for now Printing the picture XPPAUT does not directly send a picture to your printer Rather it creates a postscript file which you can send to your printer If you don t have postscript capa bilities then you probably will have to use the alternate method of getting hardcopy Note that Word supports the import of PostScript and Encapsulated PostScript but can only print such pictures to a postscript printer You can download a rather large program for Windows called GhostView which enables you to view and print postscript on non postscript printers Linux and other UNIX distributions usually have a PostScript viewer included Here is how to make a PostScript file Click on Graphics Postscript G P and a dialog box will appear asking for three things i Black and White or Color ii Landscape or Portrait iii and the Fontsize for the axes Accept all the defaults for now by just clicking O
72. go a little bit over but just a little Escape to the main menu and click on Initialconds Go Now we have one full cycle of the oscillation The rest is easy Computing the adjoint Click onnUmerics Averaging New adjoint and after a brief moment XPPAUT will beep Sometimes when computing the adjoint you will encounter the Out of Bounds message In this case just increase the bounds and it recompute the adjoint Click on Escape and plot v versus time This time the adjoint of the voltage is in the Data Viewer under the voltage component Note that the adjoint is almost strictly positive and looks nearly like 1 cost This is no accident and has been explained theoretically 230 Chapter 9 Tricks and advanced methods 9 5 2 Averaging Now we can compute the average Recall that the integral depends on the adjoint the original limit cycles and a phase shifted version of the limit cycle X t G Xo t Xo t In XPPAUT you will be asked for each component of the function G For unshifted parts use the original variable names e g x y z and for the shifted parts use primed versions x y z The coupling vector in our example is V t 06 V t 0 Thus for our model the two components for the coupling are v v 0 This says that we take the phase shifted version of the variable v called v by XPPAUT and subtract from it the unshifted version of v With these preliminaries it is a snap to comp
73. if u iv is a complex number then r u v is its magnitude and 6 tan v u is its argument We then write the result of this as a two dimensional real map Here is the XPPAUT file julia ode julia ode julia set for z gt z 2 c par cx 0 cy 0 r sqrt sqrt x cx 72 y cy 2 th atan2 y cy x cx 2 s sign ran 1 5 init x 1 x s r cos th y s r sin th total 2000 meth disc trans 100 O xp x yp y xlo 2 xhi 2 ylo 2 yhi 2 1t 0 done NOTES The function atan2 v u takes into account the signs of u v to return the correct argument 0 for the number u iv I have thrown away the first 100 4 2 Discrete dynamics in one dimension 59 points and iterated it for 2000 times The last line 1t 0 tells XPPAUT that it should not connect the points with lines Start up XPPAUT with this file Before iterating turn off the axes so you don t confuse them with the Julia set click Graphics aXes opts G X and change the three entries X org 1 on etc from 1 to 0 thus turning off the axes Now integrate the equation and you will see an circular Julia set It looks elliptical since the graphics view is rectangular Change cy to 0 5 and integrate the equations again Try cy 0 5 Can you say something about the symmetry of the Julia set Try cy 0 cx 5 Try cy 1 5 cx 0 Do these look connected Try cy 1 cx 0 and iterate for 4000 times This may or may not be connected It is hard to tell Animated Julia s
74. integral if there is a function I x y such that dI dt 0 along trajectories A easy way to test if a system is integrable in two dimensions is to show that Of g Be Oy n In this case an integral is easy to find and I leave it as an exercise For example the system d yY y y 2 2 1 2 76 Chapter 4 XPPAUT in the classroom is integrable and the function I x y zy 2 yt 4 24 4 2 3 is constant along solutions Let s first explore this system and then we will turn to some more standard mechanical systems Here is the XPPAUT file for this system integrable ode an integrable system with its integral x x y y 3 y y72 2 x 2x 1 x here is the integral aux i x y 2 2 y 4 4 x 4 4 x73 3 here is the log of the integral aux li log x y 2 2 y 4 4 x 4 4 x 3 3 O xp x yp y x10 2 y10 2 xhi 2 yhi 2 total 100 done NOTE To improve scaling I have also computed the log of the integral taking the log of a quantity can be used to better delineate large orders of magnitude I have set up the plot as a phase plane Explore this system by setting some initial conditions in the plane Note that almost every solution that you compute is a periodic This is a common feature of conservative systems Check in the Data Viewer that the quantity J remains constant along every trajectory Turn the freeze option on Graphic stuff Freeze On freeze and then use the mouse to choose a bunch of nice trajectories Ini
75. interval is infinite which confounds things even more In general BVPs arise as steady states to certain partial differential equations The boundary conditions for the PDEs lead to boundary conditions for the ODEs that are derived from them It is not always the case that a solution even exists Indeed unlike the situation with initial value problems even the smoothest BVPs may not have a solution Sometimes they have a solution only when there are some particular parameters chosen correctly Take the classic problem of a vibrating string Utt Ura with u 0 t u 1 t 0 One looks for periodic solutions of the form u x t exp iwt u a Clearly wv Vas 123 124 Chapter 6 Spatial problems PDEs and boundary value problems This is an eigenvalue problem where w is the unknown parameter We know that the solution to this is u x sin nrx and w wn nr How could we solve this numerically We note that v 0 is always a solution for any w so we want to avoid this Since the problem is linear and second order we can assume that Vg 0 is not zero if it were then v 0 by the uniqueness theorem for ODEs So let s set vz 0 1 Then we have a second order equation with three conditions v 0 0 vz 0 1 and v 1 0 There are three free parameters the two initial conditions and the eigenvalue Thus there is hope that we can solve this equation XPPAUT solves boundary value problems using a very simple algorith
76. is how to do it I will use an example with three parameters a b c taking on 4 sets of values 1 2 3 1 2 4 1 0 2 and 2 2 6 I first make a table with 12 entries the first 3 are for set 1 and so on Here is the transposed table 12011123124 1022 26 called pars tab I make an ODE file with a parameter for the run number use range integration and defined parameters to complete the file Here is my example par ode table pp pars tab par n a pp 3 n b pp 3 n 1 c pp 3 n 2 x x ta O Gg z z C aux my_a a aux my_b b aux my_c c total 10 range 1 rangeover n rangelow 0 rangehigh 3 rangestep 3 rangereset yes done I have added auxiliary variables as a hardcopy of the actual parameter values used The n parameter runs from 0 to 3 in increments of 1 The table pp has a domain of 0 to 11 with the first three integers corresponding to the first three parameter values and so on That is pp 2 3 1 is the value of b for the third run Run this with the line xpp par ode silent and you will get four files one for each run You can make little inline tutorials by using quoted comments that have an action associated with them The user invokes these by clicking on File PrtInfo which creates a window with the ODE source code Clicking on Action inthis window produces special comments from the ODE file and cer tain associated actions For example here is a linear planar system planar ode with
77. is some font available to your X server This sets the small font which is used in the Data Browser and in some other windows BIG fontname sets the font for all the menus and popups SMC 0 10 sets the stable manifold color UMC 0 10 sets the unstable manifold color XNC 0 10 sets the X nullcline color YNC 0 10 sets the Y nullcline color 284 Appendix F Error messages The remaining options can be set from within the program They are LT int sets the linetype It should be less than 2 and greater than 6 e SEED int sets the random number generator seed XP name sets the name of the variable to plot on the x axis The default is T the time variable YP name sets the name of the variable on the y axis ZP name sets the name of the variable on the z axis if the plot is 3D NPLOT int tells XPP how many plots will be in the opening screen XP2 name YP2 name ZP2 name tells XPP the variables on the axes of the second curve XP8 etc are for the 8th plot Up to 8 total plots can be specified on opening They will be given different colors e AXES 2 3 determine whether a 2D or 3D plot will be displayed e TOTAL value sets the total amount of time to integrate the equations de fault is 20 DT value sets the time step for the integrator default is 0 05 e NJMP integer tells XPP how frequently to output the solution to the ODE The default is 1 which means at each integration step e T0 va
78. is the velocity c as a function of the threshold a Click on Grab and looking at the bottom of the screen wait until you see per 40 and then click on Enter Now open the Numerics dialog box and change Ds 02 to go the other direction Click on Run and you should see the rest of the line drawn across the screen Click on Grab and move to the point labeled a 0 25 Click on File Import Orbit and plot this in the AUTO window Amazingly there is essentially no difference in the two plots Can you explain why Notice the value of c at the point a 25 It is very close to the value we computed by shooting in chapter 6 192 Chapter 7 Using AUTO bifurcation and continuation 7 3 2 Exercises 1 A nonlinear PDE that arises in the study of pattern formation has the form Ut teeta Age 7 u rtanh u with boundary conditions u 0 t ee 0 t u 1 t wee 1 t 0 One steady state solution to this is u 0 If a is large enough the zero state loses stability as r increases and the resulting solution is a stationary spatial pattern Since this solution branches fro the origin we can use AUTO to try to find a non trivial branch of solutions The steady state solution is a fourth order nonlinear problem ul nf rtanh u u au m with u 0 u 0 u 1 u 1 0 We can rewrite this as a four dimensional system and then write an ODE file for it Here is a possible ODE file where I have set up AUTO for yo
79. it to the desired directory The binaries are missing things like the example files and the documentation Download the source code to get these 1 1 3 Additional UNIX setup In some systems the zooming and cursor movement does not always work properly In these systems you want to call XPPAUT with an additional command line argument e g xppaut xorfix file ode This will usually fix these problems By default XPPAUT comes up with only the main window visible You can always make the other windows visible by clicking on the top row of buttons on the main window You may want to have XPPAUT come up with all the windows visible To do this add the command line argument 1 2 Native MS Windows NT 95 98 2000 3 allwin You can use the alias command in your shell to call XPPAUT with these command line arguments Alternatively create a text file called e g myxpp with the following line in it usr local bin xppaut 1 2 allwin xorfix Save the file and make it executable by typing chmod x myxpp Now if you call myxpp it will have the two command line options enabled 1 2 Native MS Windows NT 95 98 2000 Just download the program winpp zip into a folder say wpp and then use Winzip or a similar program to unzip the file Create a shortcut to winpp This version does not have all the features of the full version Furthermore the interface is quite different Most of the equation files will work for this version and most of the
80. j j 1 20 wn j 1 RED 3 TRANSIENT fcircle v 0 9 6 1 1 wLlj 25 02 j 1 10 fcircle vtot 6 1 1 wtot 25 03 BLACK end The nullclines are drawn along with the values of the variables 8 3 My favorites In this section I will describe a number of my favorite toys and mechanical objects I will start with some variants of the pendulum Then I will create a roller coaster model and a model of a great ride the Zipper I will finish with the discrete analogue of the Lorenz equations following Strogatz 8 3 1 More pendula The plain old pendulum is rather boring However if you do things like periodically drive the pendulum or glue two of them together they produce pretty complex behavior They are also real fun to watch The first example I want to do is the parametrically driven pendulum As you know the down position is asymptotically stable for a damped pendulum and the straight up state is unstable However if you put the pendulum on an oscillating platform you can stabilize the up state This is how jugglers keep the broomstick standing up balanced on the palm of their 210 Chapter 8 Animation hand they jiggle it The differential equations are not hard to derive See for example Guckenheimer and Holmes You write down the Lagrangian see the next section and from this derive the equations of motion The result is the following ml 6 f6 mgsin amlw sin wt sin 8 where a is the amplitude of
81. left This is a curve of fold points along this curve in a b space the original system has a fold Once again invoke the Grab item and click on Tab until you have selected the limit point 3 again Look at the bottom of the AUTO window to see which points you move through We now want to change the direction of bifurcation to get the rest of the curve Click on Numerics and change Ds from 0 02 to 0 02 thus forcing AUTO to make the main parameter increase Click on Run and you will get the rest of the curve it is a nice cusp For each a b inside the cusp there are three fixed points and for a b outside the cusp there is only one fixed point Click on Axes Last 1Par and then on reDraw You will get the one parameter diagram again Click on Axes Last 2 par reDraw and you will see the cusp This is a shortcut to go between the most recently defined one and two parameter views Click on File Postscript and you can make a postscript version of the diagram Figure 7 2 shows the one parameter and two parameter diagrams 7 1 1 A limit cycle We will start on a limit cycle in this example The limit cycle is essentially what is called an isola since it is not connected to a main branch of fixed points Here are the equations v axf R y y yf R 2 where f R 0 25 a R 1 and R x y There is one parameter in this problem a and for 5 lt a lt 5 a pair of limit cycles one stable and one unstable This
82. lt name2 gt number lt namei gt lt valuel gt lt name2 gt lt value2 gt lt name gt lt x1 gt lt x2 gt lt xn gt lt formula gt table lt name gt lt filename gt table lt name gt lt npts gt lt xlo gt lt xhi gt lt function t gt global sign condition name1l form1 init lt name gt lt value gt delay initial conditions lt name gt 0 lt value gt or lt expr gt bdry lt expression gt O lt expression gt lt For DAEs solv lt name gt lt expression gt lt For DAEs special lt name gt conv type npts ncon wgt rootname fconv type npts ncon wgt rootname root2 function sparse npts ncon wgt index rootname fsparse npts ncon wgt index rootname root2 function comments lt name gt lt value gt set lt name gt x1l z1 x2 z2 options lt filename gt 281 282 Appendix F Error messages Active comments 7 3 b 3 Some nice text done INTEGRAL EQUATIONS The general integral equation u t f t f K t s u s ds becomes u f t int K t t u The convolution equation v t exp t f e t u s ds would be written as v t exp t int exp t 2 v If one wants to solve say u t exp t f t t K t t u t det the form is u t exp t int mu K t t u and for convolutions use the form u t exp t int mu w t u NETWORKS special zip conv type npts ncon wget root where root i
83. maximal Liapunov exponent along a computed trajectory by linearizing about each point in the trajectory advancing one time step using a normalized vector computing the expansion and summing the log of the expansion The average of this over the trajectory is a rough approximation of the maximal Liapunov exponent We can see this sensitivity in a rather dramatic fashion by solving the Lorenz equations over 50 different initial conditions that are close to each other and then looking at solutions in a two dimensional projection We will use the animator for this and create an ODE file with 50 independent versions of the Lorenz equation The file is called lorenz50 ode and here it is 50 lorenz equations x 1 50 s y j x j y 1 50 r x j1 y 3 x 3 z 3 84 Chapter 4 XPPAUT in the classroom z 1 50 bx z 3 x j x y1 3 par r 27 s 10 b 2 66 init x 1 50 7 5 y j 3 6 init z 1 50 30 00 j 1 total 50 dt 02 done Notes I have used an idiosyncratic aspect of the XPPAUT file reader to initialize z j The expression j is expanded to the number value of j so that if j 15 the expression 30 00 j 1 is expanded to 30 00151 Thus the initial data are all close but differ slightly The trailing 1 is included to distinguish 30 0041 j 4 from 30 00401 j 40 Run this program and integrate the equations I G Now click on View axes Toon to bring up the animation window The animation file for this illustration is
84. o 9 cp j total change in mass mdot sum 0 9 of shift cp0 i total mass m sum 0 9 of shift c0 i mc the waterwheel equations using 10 pendulums with mass of each cup theta thetap thetap nu thetap 1 1 mdot thetap 1 sum 0 9 of shift m0 i sin theta 2 pi i n m 1 1 main parameter to change is flow the spigot rate par flow 5 mu 1 n 10 1 15 mc 2 nu 1 init theta 05 some stuff for animation x 0 9 3 sin theta 2 pi j n 4 ylO 9 3 cos theta 2 pix j n 4 yc O 9 3x cos theta 2 pix j n 4 1 c 3 2 total 200 dt 05 meth cvode tol 1e 5 atol 1e 4 yp thetap xhi 200 y10 2 5 yhi 2 5 Notes I need to get the total change in mass as well as the total mass for this I use the sum command I have defined a bunch of variables x y yc which are used in the animation file They represent the spokes of the waterwheel and the height to the water in each cup Run this ODE and then load the animation file waterwheel ani waterwheel ani 218 Chapter 8 Animation waterwheel animation file line 4 4 x 0 9 y j j1 10 line x 0 91 y j1 x jl yc j BLUE 4 done The first part of the file draws the spokes of the waterwheel The second part draws little thick blue lines to represent the height of the water in the cups 8 4 The animator scripting language Animation files consist of nothing more than a series of commands to the anima tor These commands are usually draw
85. of the lower mass nUmerics stocHastic Power Now here is the animation file doubpend ani animation of double pendulum here is the base fcircle 5 5 025 BLACK first rod line 5 5 5 2 sin th1 5 2 cos th1 second rod line 5 2 sin th1 5 2 cos thi 5 2 sin th1 2 sin th2 5 2xcos th1 2 cos th2 bobs fcircle 5 2 sin th1 5 2 cos th1 04 RED fcircle 5 2 sin th1 2 sin th2 5 2 cos th1 2 cos th2 04 GREEN end This should be self explanatory Use this to look at the chaotic behavior There are many other related equations and I urge you to look at them 212 Chapter 8 Animation 8 3 2 A roller coaster How does one simulate a roller coaster Again we appeal to classical mechanics We will parameterize the coaster by a variable s which you can think of as an arclength We will also restrict the coaster to a plane although it will be allowed to loop Let x s y s define the coaster and assume that s 0 1 Let m be the mass of the car The kinetic energy is K 849 E w 8 y 38 The potential energy is just P mgy s The Lagrangian is L K P The equations of motion are given by the Euler Lagrange equations d OLY L dt s as mR s 3 mgy s A s 8 This leads to where R s x 8 y s A s x s s y s y 8 This is a general theory of roller coasters Now we need to design one I designed a very simp
86. opti val1 opt2 val2 sets various internal XPPAUT options For exam ple maxstor 100000 total 1000 dt 02 anything jl j2 expr j is expanded by XPPAUT into a series of ja j 1 y 81Jl J prLj p y J J statements e g x 1 4 x 31 3 274 Appendix D Structure of ODE files is expanded into x1 x1 1 x2 x2 2 x3 x3 3 x4 x4 4 j1 j2 expands all following statements until another is encountered as above For example C1 4 x j y j x j 1 y j x j is expanded into x1 y1 x0 y1 x1 x2 y2 x1 y2 x2 x3 y3 x2 y3 x3 x4 y4 x3 y4 x4 done tells XPPAUT the file is complete Appendix E Complete command list Here we lay out the tree structure of all the commands They are not described however All keyboard shortcuts are capitalized and indicated in boldface Thus to get the command to fit data you could click U H I or choose nUmerics stocHastic fIt data E 1 Main menu Initialconds Range 2 par ranges Last Old Go Mouse Shift New sHoot File formUla mIce DAE guess Backward Continue Nullcline New Restore Auto Manual Freeze Freeze Delete all Range Animate Dir field flow Direct Field Flow No dir fld Colorize Scaled Dir Fld Window zoom Window Zoom In Zoom Out Fit phaAsespace All None Choose Kinescope Capture Reset Playback Autoplay Save Make Anigif Graphic stuff Add curve Delete last Remove all Edit curve
87. overcomes the seed density and the seed floats up The difference between the seed density and the density of the soda water provides the driving force We assume that the soda is sufficiently viscous so that we can ignore inertial effects Once the seed hits surface the bubbles pop more so if the velocity high Let Vo denote the volume of the seed with no bubbles and let d gt 1 be the density with the density of the soda water equal to 1 Let V t be the volume of bubbles accumulated and we will let their density be 0 The density of the seed is Vo ds d Vo V t The velocity of the seed is x k ds 1 Q e 2 The function Q prevents the seed from rising above the soda water and thus it is kept below e We will use the step function for Q The accumulation of bubbles is V c where cis just a constant Let f x denote the fraction of bubbles that remains once the seed hits the surface Then when x 0 we replace V t by f V t A linear f seems to be a good approximation f amp max fo fit 0 Try writing an ODE file and simulating it I made a nice seed oscillator with e 0 1 fo 0 05 k Vo 1 d 2 and c 0 1 Project idea The drinking duck is a famous toy which consists of a glass tube filled with freon and decorated to look like a duck When the duck s head is moistened evaporative cooling causes the freon to rise up to the duck s head and this makes the duck swing down into a wa
88. plots 2 ase didi hays re 221 9 2 2 Plotting results from Range Integration 223 9 3 Fitting a simulation to external data 224 9 4 The data browser as a spreadsheet 226 9 5 Oscillators phase models and averaging 227 9 5 1 Computing a limit cycle and the adjoint 228 9 5 2 AVETAgIE er e aa ts e e de 230 9 5 3 Phase response curves o e 230 9 5 4 Phase models 0 ae da Se ee a do E 233 9 6 ALCA Lt as A it A ee 236 9 6 1 Iterating with fixed variables 236 9 6 2 Timers ii a a eee A rn 238 9 6 3 Initial data depending on parameters 239 9 6 4 Poincare maps revisited 245 9 7 Don t forgetias iiia se dete ae jor ce iglesia pea Ab de Ge ee io 246 9 8 Dynamic linking with external C routines 249 9 8 1 An array example 252 Colors and linestyles 255 The options 257 Numerical methods 261 C 1 Fixed points and stability o a a 261 C 2 TMnt srators ss a ea EE 262 C 2 1 Delay Equations 265 C 2 2 The Volterra Integrator 266 C 3 How AUTO works o e 0020002 eee 266 D Structure of ODE files 269 Complete command list 275 E 1 Main ment soa Once onan ee ee amp OM a a ee ee et a 275 E 2 AUTOR AA Ss te ae ct gree alas ae te hy et ie His Be DBR a A 276 E 3 Browser commands coi ae ah aha e a da Se 277 F Error messages 279 F 1 C
89. problems in which the natural setting is not a differential equation but rather an integral equation XPPAUT can solve a variety of Volterra integral equations The most general forms that can be solved by XPPAUT are u t f t f K t s u s ds T7 f t f K t s u s ds 5 1 Functional Equations 95 where f t is an arbitrary time dependent function including dependence on u or other variables In general if there is a way by differentiating to convert the integral equation to a differential equation then XPPAUT will be much more efficient than it is solving the integral equation The reason is basically that to solve an integral equation you have to integrate over an increasingly longer interval There are a number of ways to speed XPPAUT up For example if the integral equation involves a convolution then you can tell XPPAUT this fact and the integration goes much more quickly Let s solve an example that is interesting due to the bifurcations that it undergoes as a parameter is varied The equation is u t f r f et 5 7 u s 1 u s ds The function f t ae is transient and serves to force u away from 0 which is clearly a solution in absence of f t This is a useful trick to know if you are only interested in long time behavior This transient pushes u away from zero Here is the ODE file chao_int ode a chaotic integral equation I take advantage of the convolution to make a lookup table x t cxex
90. routines which we will explore in detail later Clear all the graphs from the main XPPAUT window by clicking Graphics Remove all G R Clear the transpose by clicking File Transpose and click on the Cancel button Create a file called cable100 ani that has the following five lines animation for the array cable100 ani vtext 8 95 t t fcircle 1 100 100 u j1 1 2 2 02 j 100 end and save it in the same directory as the ODE file This is an animation file and is used to tell the XPPAUT animator what to do at each time step For now don t worry about the syntax Click on Viewaxes Toon V T and a new window will appear on the screen This window is the animation window and will have the name of a random cartoon on the title bar In the animation window click on the File button Assuming you named the animation file cable100 ani type this in for the file name or select it from the list Click on the Ok button in the animation window 144 Chapter 6 Spatial problems PDEs and boundary value problems Assuming there are no errors you should be able click on the Go button and see an animation of the spatial profile as it goes from a cosine wave to a damped exponential 6 2 2 The issue of stiffness Discretization of PDE s using the method of lines often leads to sets of differential equations which are numerically stiff See the appendix for definitions That is if you use a fixed step integrator you will have to ch
91. set of coupled ODEs The method of lines is quite suitable so that is what we will use uj ga 4 Quy uj 1 Uj FSN Here h L N is the grid size and N is the number of grid points The boundary conditions are coded in the equations for the end points uy and uy 1 We can write the boundary condition at j 0 as Co aouo bo ur uo h obtaining Uo co bour h ao bo h 140 V x Chapter 6 Spatial problems PDEs and boundary value problems 100 p 90 80 70 60 50 40 30 20 20 30 40 50 60 70 80 90 Figure 6 5 Steady state behavior of a nonlinear dendrite gbar 0 8 gbar 0 55 Mo gbar 0 gbar 0 5 0 1 0 2 0 3 0 4 0 5 0 6 0 7 gbar representative solutions Bottom Flux at x 0 as a function of 9 Similarly we can solve for uy41 obtaining Given these two end conditions all variables in the discretization above are defined UN 1 cL brun h ar bi h at j 1 and j N Here is the ODE file for a grid of 100 points 0 8 Top Some 6 2 Partial differential equations and arrays 141 cable100 ode cable equation with different BC s cO a0 u bO u_x cl al u bl u_x h 0 1 L 100 h u0 c0 b0 u1 h a0 b0 h u 1 100 d h h u j 1 2 u j u j 1 u j u101 cl bl u100 h al b1 h par d 1 h 1 c0 1 a0 1 b0 0 c1 0 al 0 b1 1 total 5 dt 005 done Arrays are s
92. size and ii adaptive step size These can each be divided further into two classes i explicit and ii implicit This latter distinction is best illustrated by considering the two simplest fixed step integrators Euler and backward Euler Euler s method solves dx dt s f z t by the approximation a t h a t hf x 6 t where h is the step size This is the easiest method to code and to understand However it suffers from the two major trouble spots of numerical integration ac curacy and stability Any numerical analysis book will tell you that the accuracy is only O h That is let T be a fixed interval of time Let n be defined by nh T Then 2 T on O h where x is the nt iterate of Euler s method Thus to halve the error you must halve the step size This problem can be somewhat rectified by using higher order methods The best known of these are the Runge Kutta RK methods which make several calls to the right hand side For example XPPAUT uses Heun s or modified improved Euler method a second order RK method defined by Y Tn hf an t Ya n hf y1 t h 1 Inti Un ya Note that this requires two evaluations of the right hand sides but gives O h ac curacy The advantage of this is that halving the step size quadruples the accuracy Figure C 2 illustrates this point Try to reproduce this figure with the following ODE file a first order equation whose exact answer is kn
93. solved using the results from the Newton method For example consider this trivial example d o y O ytx 1 z 0 0 y 0 1 The solution to this is x y 1 exp t exp t In XPPAUT we would write the following dae1 ode simple DAE x y O y x 1 solve y 1 init x 0 aux yyy done NOTE The line solve y 1 tells XPPAUT which variable to solve for and also provides an initial guess for Newton s method I have added the line aux yy y so that I can also plot the solution y In XPPAUT these are kept as internal variables so that you must make them plottable Run XPPAUT with this file Integrate it and check to see if it is close to the actual solution This next example is more complicated dae2 ode shows dramatic failure until the tolerances are tightened x y O 0 001 5 y 3 y x 1 solve y 1 init x 0 aux A deaf done Now to do this analytically we would have to solve the cubic for y This would result in a horrible formula Let XPPAUT do the work Run this and you should see 120 Chapter 5 More advanced differential equations that XPPAUT fails dramatically This is because Newton s method is not finding the roots with as much accuracy as you might want The superfluous small number 0 001 multiplying the relationship between y and x is making the Newton solver think it has found a root This can be easily remedied by tightening the tolerance of the Newton solver Click on nUmerics sIn
94. standard features are extant There is a binary X version for Windows which is identical to the full UNIX version and I recommend that you use that instead as this book describes the X version See the next section 1 3 X windows version on Windows This is the recommended way to run the program in the Windows environment It is only slightly more difficult to install It does not use the Windows API but works identically to the UNIX version NOTE If you have only used an X windows emulator to log into another machine this may be a bit of a surprise You can run local programs which are properly compiled X windows programs right on your PC with the X emulator running You do not have to be on a network to run this program on your Windows PC Before you download XPPAUT on a Windows machine you should have X windows emulator There are a number of them available at a cost or as demos There are at least three that are very inexpensive X WINPRO The demo version runs for 30 minutes at a time and the full version is 90 URL http www labf com index html X WIN32 The demo version runs for 120 minutes at a time I use this product at home Prices range from 50 for students to 200 for corporations URL http www starnet com productinfo MI X This is the smallest and has the fewest features The demo lasts for 15 days The cost is 25 URL http www microimages com They are all pretty simple to install and take up very little di
95. sum is 5 the returned value is f 5 1 Fire up XPPAUT with this file Click on Initialconds Formula I U and choose U 0 100 as the first and heav ran 1 5 as the formula This turns on half the cells Hit Enter at the next prompt and the simulation will be run 6 2 Partial differential equations and arrays 159 al dale 0 lt t lt 100 y i Figure 6 11 Cellular automata simulation Use the array plot to look at the space time evolution of the automata Here is the easy way in the Initial Data Window click on the square next to u0 Scroll down to U100 and click in its square Then click on the arry button at the bottom of the Initial Data Window Using a text editor create a new table file called ca5 _III tab that has the form Oorrroocdo o This rule produces a so called class III cellular automata and is quite pretty In XPPAUT click on nUmerics looKup U K and type in f for the lookup table name For the filename type ca5_III tab and click Enter twice and then Esc to get to the main menu You have just replaced the original lookup table with the new one you edited Click I G to rerun the simulation Click on Redraw in the array window to redraw the result You should see a series of triangles Play around with your own tables and change the initial conditions See if you can classify all the possible types of patterns Note that there are only 64 possible rules Maybe you can get a MacArthur awa
96. than a parameter so that we can range through it and keep a record it is like a bifurcation parameter We will define a parameter To which is the unperturbed period We will use the ability of XPPAUT to find maxima of solutions to ODEs and have the computation stop when the maximum of x is reached The time at which this occurs is the time T 7 and so we will also track an auxiliary variable 1 t To which is the PRC Here is the ODE file vdpprc ode PRC of the van der Pol oscillator init x 2 x y y x y 1 x 2 a pulse t tau tau 0 pulse t heav t heav sigma t par sigma 2 a 0 par t0 6 65 aux prc 1 t t0 dt 01 done Note that the function pulse t produces a small square wave pulse To compute the PRC you want to integrate the equations for a bit to get a good limit cycle with no perturbation a 0 Then start at the maximum value of x and integrate until the next maximum This will tell you the base period Then you want to apply the perturbation at different times and compute the time of the next maximum and then from this get the PRC Fire up XPPAUT with this file Follow these simple steps 1 Integrate it and then integrate again using the Initialconds Last I L command to be sure you have integrated out transients 232 Chapter 9 Tricks and advanced methods 2 Now find the the variable of interest in this case x To do this in the Data Viewer click on Home to go to the top of the data
97. that a stable and unstable limit cycle coalesce here Thus for a small range of currents there is bistability between the upper fixed point and the stable limit cycle The stable limit cycle solid circles terminates on the fold giving rise to a SNIC saddle node infinite period cycle In fact the frequency of the limit cycle goes to zero as the saddle node is approached To see this click on Axes Frequency fill in the dialog as follows and close it ais VY 2nd Parm phi Xmin 05 l The lower branch looks just like a square root which is what the theory of this type of bifurcation predicts Figure 7 4 shows the frequency of both the stable and unstable limit cycles We will now look at the two parameter diagram for this model Click on Axes Two par fill in the dialog box as follows and close it ais Y Wain Pamm T 2nd Parm phi ECT Ymin 0 Ymax 3 This creates a graph with 1 along the horizontal and phi along the vertical axis Now click on Grab and hit the Tab key until you come to the first limit point LP which should be labeled 2 Look at the bottom of the window as you move through the diagram until the desired point is found Press Enter to grab it Click on Run and a vertical line will appear The line is vertical because the limit point is independent of the parameter phi which has no effect on the value of the fixed point but can affect the stability Again click on G
98. the Parameter Window In the Main Window erase the screen and redraw the nullclines Erase Nullclines New E N N The fixed point has moved up Check its stability using the mouse Sing pts Mouse The fixed point should be 0 1 0 4 Use the mouse to choose a bunch of initial conditions in the plane All solutions go to a nice limit cycle That is they converge to a closed curve in the plane representing a stable periodic solution Let s make a nice picture that has the nullclines the direction fields and a few representative trajectories Since XPPAUT only keeps the last trajectory computed we will freeze the solutions we compute You can freeze trajectories automatically or one at a time We will do the former Click on Graphic stuff F reeze 0 n freeze G F O to permanently save computed curves Up to 24 Chapter 2 A very brief tour of XPPAUT 26 can be saved in any window Now use the mouse to compute a bunch of trajecto ries Draw the direction fields by clicking Dir field flow D irect Field D D Finally lets label the axes Click on Viewaxes 2D V 2 and the 2D view dialog will come up Change nothing but the labels the last two entries and put V as the Xlabel and w as the Ylabel Click on Ok to close the dialog Finally since the axes are confusing in the already busy picture click on Graphic stuff aXes opts G X and in the dialog box change the 1 s in the entries X org 1 on and Y org 1 on to 0 s to turn o
99. the phase space the parameter y0 should be less than 0 and y1 should be greater than zero so that y 0 the x axis will be visible The function tr x is used by the animator If you run this file the usual z t versus t will be plotted The fun begins when we look at the animation In the animation file I plot the function f x and then animate a little ball along the x axis here is the animation file animation file for a phase line phase line ani permanent this code draws a function ff x line 0 99 01 f j j 1 01 j 1 RED 2 66 Chapter 4 XPPAUT in the classroom a straight line along y 0 line 0 y0 y1 y0 1 y0 y1 y0 BLACK transient following the dancing ball fcircle x x0 x1 x0 y0 y1 y0 02 BLUE end Since we have not looked at animation files too much I will explain this one in detail However you may want to look at the chapter on animation Chapter 8 first Lines that begin with are comments The line permanent tells the animator that whatever follows is static and is not animated For example titles and other things in the animation are usually not dependent on time The code starting with line O 99 is somewhat mysterious so we will pick it apart The general form for the line command is line x1 y1 x2 y2 color thick This will draw a line from x1 y1 to x2 y2 in color color with thickness in pixels thick The lower left and upper right coordinates o
100. the secondary parameter is b Since there 7 1 Standard examples 169 are only two parameters there is no need to invoke the Parameter item in the AUTO window We eventually want to compute a two parameter bifurcation diagram to do this we must first compute a one parameter diagram From here on in all commands that I mention concern the items in the AUTO window Click on Axes and choose HiLo this option plots both the maximum and minimum of orbits which for the present problem is irrelevant since there are no periodic orbits In the resulting dialog box you tell AUTO the variable to plot the main parameter for 1 parameter diagrams and the second parameter for 2 parameter diagrams as well as the dimensions of the plot Fill it in as follows sx Nein Para a 2na Paros b ain 25 on 3 imax 25 3 _ and then click 0k We have told AUTO that the main bifurcation parameter is a and that X is the variable to plot The graph will show a on the horizontal axis and x on the vertical with a range of 2 5 2 5 x 3 3 Next click on the Numerics item Change only one item in this dialog box Par Min the minimum value of the parameter which you should change from 0 to 2 Par Min and Par Max tell AUTO how far to continue solutions Then click OK Now you are ready to run Click on Run Steady state This tells AUTO you are continuing along a steady state or fixed point of the system A thick black line will go
101. to destroy the diagram e If AUTO seems to not be able to continue change dsmin to make it smaller for periodic orbits and boundary value problems change ntst and make it larger e If AUTO seems to miss a bifurcation point that is clearly there clear the dia gram make dsmax smaller so that AUTO doesn t miss it Reset the diagram and recompute 7 1 3 Exercises 1 Set phi 1 2 and compute the one parameter bifurcation for ml ode diagram using J as the parameter Make sure that Par Min is 0 5 Note how the periodic orbit intersects the middle branch of fixed points at a homoclinic orbit Set J to be near this intersection and look at the phase plane of the Morris Lecar model See if you can understand the global dynamics for this value of I Next observe that for J 0 072 the system appears to be tri 178 Chapter 7 Using AUTO bifurcation and continuation stable There are two stable fixed points and one stable periodic orbit Draw the complete phase portrait with 0 072 Include unstable periodics which you can compute by integrating backwards Compute the bifurcation diagram using the current as a parameter for the di mensionless Morris Lecar equations using the parameter values in the above example but setting v3 0166 v4 25 phi 333 and looking in the win dow 0 5 lt v lt 0 5 and 0 lt I lt 0 5 Make sure that you start at a fixed point when J 0 Compute the branch es of periodic solutions emanating fr
102. unique stable fixed point Vp wr One is interested in the existence of traveling pulses that is solutions that are functions of ct and homoclinic to the rest state A classic way to analyze nerve equations is to assume that the recovery variable w is slow that is is a small parameter Then one studies the reduced system holding w at its rest value and looks for a front that joins the rest state to the active state in this case V Vea We do this first Here is the ODE file for the traveling front morris lecar front 138 Chapter 6 Spatial problems PDEs and boundary value problems mlfront ode param c 0 params vi 01 v2 0 15 gca 1 33 params vl 5 iapp 04 g1 5 minf v 5 1 tanh v v1 v2 f g1 vl v gca minf v 1 v iapp v vp vp c vp f init v 1 xlo 1 xhi 1 ylo 1 yhi 1 xp v yp vp done Here is your first task Find two values of c such that the unstable manifold from the fixed point V V 1 0 either goes off to infinity c too low or is pulled into a stable fixed point c too high Hint start with c 0 and c 2 Next use these two values to successively refine your guess of the velocity until you have it to within two decimal places We now turn to the full equations Shooting in this case is more subtle and in fact there is no mathematical proof of the existence of traveling wave solutions to this particular model Traveling waves satisfy cV V f
103. up 0577 c 0 initial fixed points init uleft 1 upleft 0 uright 0 upright 0 total 1 01 dt 01 xp u yp up xlo 25 xhi 1 25 ylo 75 yhi 25 some AUTO parameters epss 1e 7 epsu le 7 epsl 1e 7 parmax 60 dsmax 5 dsmin 1e 4 ntst 35 done I have added a few AUTO numerical settings so that I don t have to set them later Run XPPAUT with this ODE file and integrate the equations We will now continue this approximate heteroclinic in the parameter per Fire up AUTO File Auto and click on Axes Hi Put con the y axis and make Xmin 0 Xmax 60 Ymin 2 Ymax 2 Then click OK As above we will also keep solutions at particular values of per by clicking on Usr period 3 choosing per 20 per 40 per 60 and then OK Now we are ready to run Click on Run Homoclinic When the dialog box comes up set the following Left Eq uleft Right Eq uright NUnstable 1 NStable 1 and then click OK You should see a nice straight line go across the screen Grab the point labeled per 40 and then click on File Import Orbit Look at it in the Main Window by clicking Restore and freeze it Now in the AUTO window click on Axes Hi and change the Main Parm to a and Xmax 1 Click OK and then click on Numerics to get the Auto Numerics dialog Change Dsmax 0 1 Par Max 1 and click OK Now click on Usr Period 4 and make the four user functions a 75 a 9 a 25 a 1 and click OK Now click on Run and watch a line drawn across the screen This
104. variable The solution to this is r ro t so that when t 0 r ro We choose ro small and positive to avoid division by zero The boundary condition for r at 0 forces r rg The boundary condition bndry a r0 ap forces a ro roda dr ro which is only approximate but the error is small as rg 0 Finally note that the interval of integration here is ro 1 ro which is a bit longer than 1 but doesn t matter since the diffusion coefficient d can always be rescaled Run XPPAUT on this file and solve the boundary value problem Click Bndryval No show and it should draw a nice solution Change the plot variable to k by clicking Viewaxes 2D and change the Y axis to k Now let s look at the solution over a range of values of g Click on Bndryval Range which will solve the boundary value problem as a parameter ranges over some values Fill it in as follows Range over a Steps 10 stare 0 mas o Cycle color Y N Y Side 0 1 0 Movie Y N N Then click on Ok You will see a rainbow of parabolas that are the solutions to the boundary value problem More examples are discussed in Chapter 7 130 Chapter 6 Spatial problems PDEs and boundary value problems 6 1 2 Infinite domains Boundary value problems on infinite domains are not generally possible to do in XPPAUT However some types of problems can be numerically solved by putting them on a large finite domain or by shooting from a fixed point Consider the
105. variables but not vice versa In the oscillator example above we could make the fixed variable R available for plotting by adding a statement aux myr R so that myr is the user accessible version of R 3 4 2 Exercises 1 Write an ODE file for the Fibonacci recurrence fear fnt fn 1 fo f 1 2 Write an ODE file for the Lotka Volterra equations dz dy q er bay q TY t day with positive parameters a b c d Track the quantity Q aln y cln x by dx with initial data x y 2 3 Write a differential equation file for the Morris Lecar equations defined as follows dV C gL V EL gcaMo V V Eca gkw V Ex 1 dw wo V w de b Tw V Moo V 1 1 exp V V1 Va exp V V3 V4 cosh V V3 2V4 a 5 II e e Di Tw V where 0 8 gk 8 Ica 4 4 gL 2 C 1 Ex 84 Eca 120 Ez 60 Vi 1 2 Vo 9 V3 2 V4 15 and J 90 Integrate them 3 5 Discontinuous differential equations 35 and analyze them along the lines that you did in chapter 2 for the Fitzhugh Nagumo equations Look in the V w phase plane and show that there is a coexistent periodic solution and a stable fixed point If the ODE file is too hard for you to figure out then as a last resort download the ODE file mlex ode 4 Write the third order system 2 ax ba ex q as a system of first order equations Hint le
106. your browser or using software like animate and xanim on Linux and QuickTime Player and RealPlayer on Windows Note that the execrable Media Player doesn t support animated GIFs Now you can fill your web page with fancy dynamical systems animations and further clog the bandwidth 8 2 2 Important notes The animator is aware of all of the state variables in your system but is not aware of any of the auxiliary quantities However it is aware of fixed variables Thus if you want to pass on a complicated quantity to the animator it is best to define it in the ODE file rather than in the animation file Furthermore you should define it as a fixed variable rather than an auxiliary variable For example in the pendulum ODE file we could add the lines xb 5 3 sin theta yb 5 3 cos theta to define the endpoints of the bob Then the animation file would be much simpler line 5 5 xb yb BLUE 3 fcircle xb yb 04 RED 204 Chapter 8 Animation 8 2 3 A spatial problem revisited Recall in chapter 6 that we looked at a spatially distributed neural network with a delay uj uj fe y W Du j 1 ciuj t 7 l m The ODE file was called nnetdelay ode Run this again Here we will create an animation file which plots a series of small filled circles at each spatial point along the x axis and whose height is a scaled version of the state variable u Here is the file nnetdelay ani use with nnetdelay ode a one liner
107. 0 Now integrate the equations Graphic stuff Edit crv and change the color to color 8 and click on Ok Graphic stuff Add Crv and make Y axis d and Color 9 in the dialog box Window the figure so that all curves are visible You will see 4 curves two are the data curves and two are the components of the simulation This is the before picture Now we will run the curve fit to see if we can do better Click on nUmerics stochastic fIt data and you will get a large dialog box Fill it in as follows File modda ross gt Tiwa qd Te Gol 23 Parans a DUKIDESE Params Tolerance 0 001 Epsilon Te ps OOOO Mas iter 20 This tells XPPAUT the name of the file how many columns the file has the names of the variables you want to fit to their respective columns Thus c is fit to column 2 and d to column 3 You also tell XPPAUT the names of the parameters that will be varied The tolerance is used to determine when further changes are no longer necessary Let x1 x2 be two successive least square values If either x1 xal or lx1 xal xa are less than the tolerance then the program assumes success Don t make this too small as otherwise you are wasting time The parameter Epsilon is used in numerical derivatives The parameter Max iter sets the maximum number of iterates to try to achieve the desired tolerance Finally Npts is the number of data points rows that you want to use XPPAUT uses the first column of data in the data
108. 1 u 2 pi u 0 done I have initialized u to its resting value assuming that there are no inputs or noise We want to record the times t at which a spike occurs that is when u t crosses 7 so we will create a Poincare section Click on nUmerics Poincare map Section and choose u as the variable and 3 14159 as the section Esc to the main menu and integrate Look in the Data Viewer and you will see a few times recorded We will now integrate this 100 times and store all of the spike times Click on Initialconds Range and choose sig as the parameter with 25 and both the starting and ending values Choose 100 steps and most importantly choose N for the Reset Storage option so that the browser is not reset after every run Click OK and it will start It may take a few minutes to finish Once you are done you will have a list of about 700 times an average of seven spikes per stimulus Now will will make a histogram Click on nUmerics sTochastic Histogram Choose 100 for Number of bins 0 for Low 50 for High T for the variable to histogram and Enter for the condition In a second the histogram is computed Esc to the main menu and plot U vs t since the first column contains the bins and the second the number per bin You should see a histogram with several peaks trailing off to zero See Fig 5 9 left Repeat this with more or less noise to see how the histogram changes This technique can be applied to any model neuronal system in which the spi
109. 10 10 exp c abs t a exp b abs t par cp 4 cm 4 tau 0 r 3 ut 25 par a 5 b 1 c 25 init u 45 55 5 special k conv periodic 100 10 w u0 f u 1 1 exp r u ut ulO 100 ulj cp k j cm delay u j tau dt 05 total 100 delay 10 done 6 2 Partial differential equations and arrays 151 0 lt t lt 100 Figure 6 9 Solutions to the delayed neural net equation with a delay of 1 As in all of our examples there are 101 units We will integrate this system for 100 time steps We have initialized the middle 10 units to be 0 5 Because the network is periodic we need to introduce a spatial inhomogeneity if we are to obtain any spatial patterns since homogeneous solutions are preserved The last line tells XPPAUT that the maximum delay will be 10 By default u t 0 for t lt 0 We have initialized the delay parameter 7 to be zero Run this problem Use the array plot to see the space time behavior Notice that the solution is a series of stripes the medium has broken into a spatially periodic pattern Now we will increase 7 and see what happens Set 7 5 and reintegrate the equations The edges of the stripes have small damped oscillations Plot u25 as a function of time to see this Question How do you know that these oscillations are real Make the timestep DeltaT smaller nUmerics Delta T and note that they seem to damp out faster You will find that eventually they remain for small timesteps so
110. 1000 tells XPPAUT to only look at the final point The last line sets up the range integration so that you won t even have to fill in a dialog box Run this with XPPAUT Click on Initialconds Range I R The dialog box should look like this Range over bY Stes 200 smo ma i Reset storage N Click on Ok The Devils staircase will be drawn for you If you want a really detailed picture make the number of steps 2000 Then zoom into different regions and see that there are many flat regions corresponding to constant rotation numbers If the amplitude parameter is larger than 1 27 0 16 the map is no longer invertible and the assumptions for Denjoy s theorem no longer hold Change a to 0 4 and redraw the staircase 4 2 5 Complex maps in one dimension The Mandelbrot set has often been called the most complex object in mathematics While this is likely to be false the computation of this set has led to a huge number of little applications which will draw it for you to any desired resolution Many of these applications offer little explanation of the set nor of its mathematical significance Here I will offer a brief description of what the set means First let s consider the logistic equation that we looked at above Ln 1 4Lp 1 Lp Set a 4 and make the following transformation u 2x 1 Then Un 1 1 2u 4 2 Discrete dynamics in one dimension 57 1 m P J Z o 0 8 F
111. 2 We initialize V 1 so that it is not identical to Va to break the symmetry 3 Each time Vi crosses Vr from below V is reset to 0 and s s1p is incre mented by b A similar condition is set for Va 4 I have set a whole lot of options so that the user doesn t have to worry about them and can just run the integration I have set the integration time step to 0 01 the total integration to 100 time units and I only will start storing data after a transient of 80 time units I set the low and high limits of the x axis to 80 and 100 respectively I set the lower limit of the y axis to 0 I tell XPPAUT to plot 2 curves in the window and tell it the second curve to plot is Va Note by default the x component of a plot is time t and the y component is the first variable defined in the file Run this equation in XPPAUT and note that both V and V2 are synchronized Change the initial conditions however you want but make sure that V 0 lt 1 and the solution will always go to synchrony Change b to 1 and integrate again The oscillations alternate synchrony is not stable Now change b back to 5 and change g to 2 Integrate again Notice that synchrony is unstable Now change b to 1 again Set all variables to 0 except Vi Set Vi 1 and integrate You should see synchronous oscillations Set V 4 and integrate the oscillations alternate This result was proven by van vreeswijk et al 1994 See Figure 3 2 3 5 2 Clo
112. 2 p x1 r x2 236 Chapter 9 Tricks and advanced methods par wi 1 w2 3 total 100 fold x1 fo1d x2 xp x1 yp x2 x10 0 y10 0 xhi 6 3 yhi 6 3 done As in the previous example I have set this up so that the projection is onto the torus phase plane Here are some things to do e Integrate the equations and note how solutions tend to the diagonal There is a stable phase locked synchronous solution e Change the integration time step to 05 and integrate the equations again Note that there is another phase locked solution that is out of phase Change the integration time step back to 0 05 e Change the coupling strength a from 3 to 3 and repeat the first two exercises Note how synchrony is unstable when a lt 0 e Change a to 3 again Draw the nullclines They don t intersect Change w1 the frequency of the first oscillator to 0 1 Redraw the nullclines Determine the stability of the fixed points and draw all the invariant manifolds for the saddle point Where do the stable manifolds go They tend to vertical lines in the middle of the phase plane That is there is a repelling invariant circle in which oscillator two fires repetitively and oscillator 1 just sort of wobbles around near m All stable solutions end up at a fixed point This is called phase death e Now change w1 back to 1 and change w2 to 3 Integrate the equations Note that for every three cycles of oscillator two there is one cycle of oscillator one
113. 3 m par m 1 aux e m y 2 2 x 4 4 x7 2 2 xp x yp y xlo 1 5 ylo 1 5 xhi 1 5 yhi 1 5 done Do the following steps 78 Chapter 4 XPPAUT in the classroom Figure 4 11 Double well potential 1 Click on Graphic stuff Freeze On freeze 2 Click on Initial conds mIce and choose 10 or so initial conditions to get some nice trajectories 3 Click on nUmerics Colorize Another quantity and choose E as the quan tity and select Choose set the minimum to 25 and the maximum to 75 Escape to exit to the main menu 4 Click on Dir field flow Colorize and choose a grid of 100 5 Redraw by clicking Restore You should see a nice figure 8 such as in figure 4 11 4 5 Nonlinear systems 79 4 5 2 Homework Here are some equations to study Some of these are integrable so you should find the integral For each of these draw the nullclines the direction fields the stable and unstable manifolds of the saddle points 1 x y 1 22 y 3 2 y in the window 3 3 X 3 3 This is an interesting vector field since on one side it looks like it is integrable while on the other side the two fixed points are nodes 1 x y 1 27 y x 3 x y in the same window x 4r 2 y y x in the window 3 lt x lt 3 and 6 lt y lt 6 How many limit cycles are there and how many fixed points Hint to find the unstable limit cycle s you should integrate backwards in time clic
114. 31 A 0 Figure 6 2 The connection of the unstable dashed and stable solid manifolds of a fixed point as the parameter varies this restriction to one dimension is not as bad as it seems In the next chapter we show how to find more general homoclinic orbits Let s turn to several increasingly difficult examples We will start with the standard example the Fisher equation Ut Ure u 1 u which is a model for the spread of an advantageous gene There are two constant steady states u 0 1 We are interested in a constant speed traveling wave front that joins the unstable state 0 to the stable state 1 We look for solutions of the form u x t U x ct leading to the differential equation cU U U 1 U This can be written as a system U V V cV U 1 U There are two fixed points 0 0 and 1 0 Linearizing about these fixed points we see that 0 0 has eigenvalues c 2 Vc 4 2 so that this is either a stable node c gt 2 or a spiral 0 lt c lt 2 Thus the stable manifold is 2 dimensional The fixed point at 1 0 is a saddle point there is a one dimensional stable manifold and a one dimensional unstable manifold Since the traveling coordinate is x ct and c gt 0 we seek a wave going from U 1 to U 0 so that we want to follow the unstable manifold of the point 1 0 into the point 0 0 Since the stable manifold of 0 0 is two dimensional we expect that this will happen for a w
115. 5 In any case we see that the Lorenz equations can be well approximated by a one dimensional map How do we find other periodic points If we change the X axis shift to 2 for example we will be plotting z _2 against z so that intersections 86 Chapter 4 XPPAUT in the classroom Figure 4 13 Poincare map for the Lorenz attractor looking at successive values of the maximum of z Period 1 2 and 3 points are illustrated with the diagonal are period 2 and period 1 points Try this and you will find two period 2 points one period 2 orbit Now for the final blow lets find the period 3 points Change the X axis shift to 3 in the Ruelle plot and redraw the graph You should see 6 intersections besides the trivial one corresponding to two period 3 orbits Figure 4 13 shows the numerically computed map and the second and third iterates In chapter 9 section 9 6 4 we present a different way to make Poincare maps which can be significantly faster Through three lines of evidence i sensitivity to initial conditions ii com plex spectrum iii and period 3 orbits you can be reasonably sure that the Lorenz equations are chaotic In fact there are only a few rigorous results on this system 4 6 3 Homework 1 Explore the simple map fiorenz equation 4 5 and verify that the maximal Liapunov exponent is about 0 64 so that it is not as chaotic as the real Lorenz system Find values for the perio
116. A and you will get a dialog box to edit Fill it in as follows Pos 120 mos 201 Rousp 5 gt Autoplot 0 1 0 Colskip 1 and click on 0k A new window appears with a number of buttons Click on Redraw and you should see a space time plot of the variable u x t Space runs horizontally and time runs vertically Since we have integrated the equations with a time step of 0 005 for 5 0 time units there are 1000 rows of data We plot every 142 Chapter 6 Spatial problems PDEs and boundary value problems u 00 Redrau Edit Print tyle 0X lt t lt 5 Leal 1 Figure 6 6 The array plot window showing the space time behavior of the cable equation 5th row About halfway through the simulation it appears that the solution has reached a steady state You should see something like figure 6 6 Cool shortcut in the Initial Data Window click on the little box to the left of the variable U1 and scroll down until you see the variable U100 and click on its little box Then click on arry in the Initial Data Window This will automatically set up the array for you Let s look at a spatial plot at different times XPPAUT has a transpose command which transposes the data in the data browser to allow you to plot the spatial distribution of the variable u Let s plot u x 0 u x 5 u x 2 5 and u x 5 As above these represent respectively rows 0 100 500 and 1000 Click on File Transpose
117. All variables that you want to mod out can be set by the fold name directive This automatically tells XPPAUT to look for modded variables The default period is 27 so we don t have to change it If you want to change the period to say 3 set it with the command tor_period 3 Run XPPAUT on this using the mouse to set some initial conditions Note how all initial data synchronizes along the diagonal implying 6 02 Change the intrinsic frequency w2 and see how big you can make it before phase locking is lost Pulsatile coupling Winfree 1967 introduced a version for coupling oscillators using the phase response curve of the oscillator assuming that the interaction between oscillators was only through the phase and took the form of a product Ermentrout and Kopell 1991 proved that this was a reasonable model when certain assumptions were made con cerning the attractivity of the limit cycle Here is a pair of pulse coupled phase models de Fp P 02 R 0s de Er wz P 01 R 09 Think of R 0 as the phase response curve of the oscillator and P 0 as the pulse coupling For example if P 0 is a Dirac delta function then this model reduces to a one dimensional map This instance of coupling is simulated in 9 6 2 The following ODE is a case in which the coupling is through a smooth function phasepul ode pulsatile phase model r x a sin x p x exp betax 1 cos x par a 3 beta 5 x1 w1 p x2 r x1 x2 w
118. Boundary value problems 135 and continue in this way until all four invariant sets are computed The first two trajectories computed represent the unstable manifolds of the fixed point How do I know this XPPAUT draws the stable unstable manifolds in blue yellow and always computes the unstable branches first Use the Initialconds sHoot command to recreate these trajectories if you want integrate forward in time for unstable and backward for stable manifolds To integrate backward click on nUmerics Dt and change the integration stepsize from positive to negative When you choose Initialconds sHoot you are given 4 choices pick 2 as the second trajectory is the branch of the unstable manifold desired Freeze this curve in memory Graphics Freeze Freeze and accept the defaults Now change c to 0 5 Once again click on S G to compute the fixed point answer No to the Print eigenvalues prompt and Yes to the Compute invariant sets prompt As before the first trajectory becomes unbounded and requires a response But the second trajectory does not go off to infinity Rather is is attracted to the fixed point at a 0 XPPAUT always integrates until a solution goes out of bounds when invariant sets are computed Since this solution will never go out of bounds you must interrupt the integration by tapping Esc the escape key Finish the computation of the stable manifolds As before you can compute this new trajectory using the Initial
119. Contents Preface 1 Installation 1 1 Installation on UNIX 1 1 1 Installation from the source code 1 1 2 Installation from binaries 1 1 3 Additional UNIX setup 1 2 Native MS Windows NT 95 98 2000 1 3 X windows version on Windows 000 1 4 Installation on MacOSX 0 0 0 000000 1 5 Environment variables 1 5 1 Resour erfiles ot ta te ke Rie hh we 2 A very brief tour of XPPAUT 2 1 Creating the ODE file 2 2 Running the progra e 2 2 1 The main window o 2 2 2 Quitting the program 0 2 3 Solving the equation graphing and plotting 2 4 Changing Parameters and Initial Data 2 5 Looking at the numbers the Data Viewer 2 6 Saving and restoring the state of XPPAUT 2 6 1 Command summary 2 7 A nonlinear equation 2 2 2 7 1 Direction fields o o 2 7 2 Nullclines and fixed points 2 7 3 Command Summary 2 8 The most important numerical parameters Exercises ay cat das Sos wh wie ia a e de A a le Writing ODE files for differential equations 3 1 Introduction s i doi A A do E A AA ala ot GE 3 2 Ordinary differential equations and maps 3 2 1 Nonautonomous systems ry NOTWW NP NRH eS
120. Enter to grab this point Now click on Run Periodic to track the branch of periodic solutions emerging from the Hopf bifurcation This branch terminates on the middle branch at a homoclinic orbit The picture will be similar but not identical to Figure 7 4 Now for the cool part We will save this diagram and then import it into the bursting file so that the J v dynamics can be plotted superimposed on the bifurcation diagram In the AUTO window click on File Write pts and pick a file name such as mlhom dat to save the points of the diagram Exit XPPAUT Run XPPAUT with the file mlsqr ode which is the previously viewed bursting equation Integrate the equations to get something like the top right figure 7 9 Now click on Graphics Freeze Bif Diag and choose the previously saved diagram e g mlhom dat Voila You will see the nice burst superimposed on the bifurcation diagram This explains the mechanism for bursting When the potential v is below vo the current J increases causing the voltage to follow the lower stable fixed point This continues until the knee is reached and the potential jumps up to the branch of periodic solutions The average voltage is greater than vo so that I begins to decrease until it hits the homoclinic point The potential then drops to the stable rest state where it repeats the cycle 7 5 1 Exercise Repeat the above trick with the canonical elliptic burster u u A R R vt e v v A R R ute
121. Here is the file gs ode grey scott equations par d 25 al 1 a 2 h2 100 u1 d u2 u1 a u1 2 z1 u1 al z1 h2 z2 z1 ul ul zit 1 z1 2 199 u j d u j 1 2 xu j u j 1 axu j u j z j u j al z j hb2 z j 1 2 z j z j 1 z j 1 u j u j 1 u200 dx u199 u200 a u20072 z200 u200 al 2200 h2 2199 2200 z200 1 u200 u200 1 init z 1 200 1 init ui 1 2 meth cvode tol 1e 5 atol 1e 4 bandup 2 bandlo 2 dt 25 total 200 xhi 200 yhi 25 yp u100 done Run this and time how long it takes Then click on nUmerics Method and choose Cvode but turn off the banded option Rerun the simulation make sure you have a copy of War and Peace to read in the meantime It is about 30 times slower Explore the behavior of the continuous approximation to the class III cellular automata k 5 1 k 5 with periodic boundary conditions and 1 FC a a exp alu 2 with 0 lt z1 lt zg lt 1 and a gt 0 Note that you will have to define the delayed variables as fixed variables Try T 4 a 20 z 2 z2 6 and initialize a few variables to 1 Integrate it for a while to speed things up use Euler Here is an ODE file ctsca ode continuous approximation to class 3 CA du O 100 delay u j tau table w 11 5 5 1 11 f u 1 1 exp ax u z1 1 exp a u z2 par tau 4 a 20 z1 2 z2 6 init u 45 55 1 special k conv period 101 5 w du0 164
122. I have taken the liberty of setting some of the AUTO parameters so that you can just run this with no hassle I have also set nout 7 so that AUTO looks at the 7th iterate of the map Start up XPPAUT and iterate it for a few times to make sure you have no transients Then click on File Auto and in the AUTO window click on Run to compute the diagram It looks complicated but in reality it simply shows that for 2 1764 lt a lt 2 20 there are two period 7 points one is stable and the other is unstable The reason there appears to be many branches is that for a period 7 point there are 7 possible starting values Figure 7 5 shows the diagram We will now do a two parameter bifurcation analysis of the above problem You can either edit all the parameters in the above file or simply load in the stripped down version del_log2 ode delayed logistic map simple 2 parameter stuff y x x axx 1 y eps par a 1 5 eps 0 init x 333 y 333 meth discrete total 100 done Run this model Then start up AUTO Use the Axes Hi Lo and change them so the Xmin 0 Ymin 1 Xmax 4 Ymax 1 In AUTO Numerics change Par max to 4 Then run the continuation You should see a so called Hopf point labeled 2 Note that it says the period is 6 This may mean that there is a period 6 point lurking nearby Let s draw the two parameter diagram using this heretofore worthless param eter e Grab the Hopf point by clicking on Grab moving to it and then t
123. K when you are done Note that you could have filled in the labels if you had wanted but for now there is no reason to You should see a nice elliptical orbit in the window This is the solution in the phaseplane cf Fig 2 3 Shortcuts Click on the button ICs in the Main Window to bring up the Initial Data Window This window shows the current initial data There is a very simple way to view the phaseplane or view variables versus time Look at the Initial Data Window Fig 2 4 You will see that there are little boxes next to the variable names Check the two boxes next to X and Y Then at the bottom of the Initial Data Window click the XvsY button This will plot a phaseplane and automatically fit the window to contain the entire trajectory This is a shortcut and does not give you the control that the menu command does For example the window is always fit to the trajectory and no labels are added or changed Nor can you plot auxiliary quantities with this shortcut To view one or more variables against time just check the variables you want to plot up to 10 and click on the XvsT button in the Initial Data Window You should have a phaseplane picture in the window If not get one following the above instructions or using the shortcut Click on Init Conds Mouse I M Use the mouse to click somewhere in the window You should see a new trajectory drawn This too is an ellipse Repeat this again to draw another trajectory If you
124. MIN AUTOYMAX AUTOVAR The last is the variable to plot on the y axis The x axis variable is always the first parameter in the ODE file unless you change it within AUTO COLOR MEANING 0 Black White 1 Red 2 Red Orange 3 Orange 4 Yellow Orange 5 Yellow 6 Yellow Green 7 Green 8 Blue Green 9 Blue 10 Purple KEYWORDS You should be aware of the following keywords that should not be used in your ODE files for anything other than their meaning here sin cos tan atan atan2 sinh cosh tanh exp delay ln log log10 t pi if then else asin acos heav sign ceil flr ran abs del _shft max min normal besselj bessely erf erfc argl arg9 shift gt lt gt lt not int sum of i These are mainly self explanatory The nonobvious ones are e heav arg1 the step function zero if arg1 lt 0 and 1 otherwise e sign arg which is the sign of the argument zero has sign 0 ran arg produces a uniformly distributed random number between 0 and arg 286 Appendix F Error messages e besselj bessely take two arguments n x and return respectively J x and Y 1 the Bessel functions e erf x erfc x are the error function and the complementary function e normal arg1 arg2 produces a normally distributed random number with mean argi and variance arg2 e max arg1 arg2 produces the maximum of the two arguments and min is the minimum of them e if lt exp1 gt then lt exp2 gt else lt exp3 gt eval
125. Now try the following interesting experiment Use the Initialconds formula command to initialize ul u100 to exp ct where c 1 0 1 0 05 By plotting u80 and u90 in the same window show that the velocity of this front depends on the initial conditions Measure the time difference between crossing of u 0 5 Thus show that the front is slower with the sharper initial data 6 2 Partial differential equations and arrays 161 Repeat the previous exercise with u 1 u replaced by u u a 1 u where a 02 and see that the velocity is independent of the initial condition it is unique Obtain a traveling wave solution to the Morris Lecar equations by solving the initial value problem in the excitable regime You should try using the equa tions and parameters used in the shooting exercise Use Neumann boundary conditions and work on a domain of size 10 with 100 grid points Assume that the diffusion coefficient is 0 1 Initialize the voltages to rest V 0 4 and initialize the first four points in the grid to 0 to initialize the wave Run the simulation and observe the traveling wave Try computing the velocity by finding the amount of time it takes to travel m grid points The velocity is mh y d T where h 10 100 is the grid size and T is the time to travel the m grid points Compare this to the velocity you obtained by shooting As in the previous exercises try to make the ODE file yourself before looking at the answer
126. S etc dX dT A X Bx Y dY dT Cx X Dx Y Once you quit XPPAUT you can start it up again and then use the File Read set to load up the parameters etc that you saved Now you should quit the program We will look at a nonlinear equation next find fixed points and draw some nullclines and direction fields To quit click on File Quit Yes F Q Y 2 6 1 Command summary Initialconds Go computes a trajectory with the initial conditions specified in the Initial Data Window I G 20 Chapter 2 A very brief tour of XPPAUT Initialconds Mouse computes a trajectory with the initial conditions spec ified by the mouse Initialconds m I ce lets you specify many initial conditions I M orI I Erase erases the screen E Restore redraws the screen R Viewaxes 2D lets you define a new 2D view V 2 Graphic stuff Postscript allows you to create a postscript file of the current graphics G P Kinescope Capture allows you to capture the current view into memory and Kinescope Save writes this to disk Window zoom F it fits the window to include the entire solution W F File Quit exits the program F Q File Write set saves the state of XPPAUT F R File Read set restores the state of XPPAUT from a saved set file F R 2 7 A nonlinear equation Here we want to solve a nonlinear equation We will choose a planar system since there are many nice tools available for analyzing two dimensional systems A clas sic m
127. The parameter eps is for computing the numerical derivative a g x g x CC acme eae Let s use this to compute the period three bifurcation curves Run XPPAUT Click on Graphics Edit G E and choose curve 0 Change the linetype to 0 so that dots will be drawn and click Ok Next click on Numerics Poincare map Section Fill in the dialog box as Variable ar Direction 1 Stop on section y 4 2 Discrete dynamics in one dimension 53 This tells XPPAUT to only plot the points once the quantity err crosses zero This means that the value of the root is within the specified tolerance It also guarantees that Newton iterates which do not converge to a root will not be plotted Now click on Esc to exit the numerics menu We now will set up the two parameter range integration Click on Initialconds 2 Par range I 2 to get a rather unwieldy dialog box Fill this in as Varyi R Reset storage N Start 0 Use old ic s Y Enar I Cycle color N Vary2 a Movies N Crv 1 Array 2 2 Steps 200 Seep OL ______ This dialog tells XPPAUT which pair of parameters you want to vary The entry Crv 1 Array 2 allows you to vary the two parameters together or indepen dently Click on 0k and the period 3 points will be plotted If you want to compute at a finer resolution increase the number of steps for the bifurcation pa rameter Steps2 If you compute for period 2 points then you should al
128. UT quit fq corresponding to File Quit The remaining options can be set from within the program They are Plotting options e LT int sets the linetype It should be less than 2 and greater than 6 e XP name sets the name of the variable to plot on the x axis The default is T the time variable e YP name sets the name of the variable on the y axis e ZP name sets the name of the variable on the z axis if the plot is 3D e NPLOT int tells XPP how many plots will be in the opening screen e XP2 name YP2 name ZP2 name tells XPP the variables on the axes of the second curve XP8 etc are for the 8th plot Up to 8 total plots can be specified on opening They will be given different colors e AXES 2 3 determine whether a 2D or 3D plot will be displayed e PHI value THETA value set the angles for the three dimensional plots e XLO value YLO value XHI value YHI value set the limits for two dimensional plots defaults are 0 2 20 2 respectively Note that for three dimensional plots the plot is scaled to a cube with vertices that are 1 and this cube is rotated and projected onto the plane so setting these to 2 works well for 3D plots e XMAX value XMIN value YMA X value YMIN value ZMA X value ZMIN value set the scaling for three d plots Numerical options e SEED int sets the random number generator seed e TOTAL value sets the total amount of time to integrate the equations default is 20 e DT value sets
129. V w cw g V w with V 00 w 00 V 00 V wr 0 0 4 0 0 It is easy to show that this equilibrium point has a one dimensional stable manifold and a two dimensional unstable manifold Thus as with the front we don t generically expect that there will be an intersection if the one dimensional manifold with the two dimensional manifold in the three dimensional phase space However we have the free parameter c to vary We must establish the correct shooting set Here is the ODE file mlwave ode morris lecar traveling wave param c 5 params vi 01 v2 0 15 v3 0 1 v4 0 145 gca 1 33 phi 333 params vk 7 vl 5 iapp 04 gk 2 0 g1 5 om 1 minf v 5x 1 tanh v v1 v2 ninf v 5 1 tanh v v3 v4 lamn v phi cosh v v3 2 v4 f v w g1 vl v gk w vk v gca minf v 1 v iapp g v w lamn vw ninf vw w v vp vp c vp f v w w g v w c init v 4 w 0 vp 0 O xp v yp w xlo 5 xhi 3 ylo 1 yhi 6 done 6 2 Partial differential equations and arrays 139 Note that the one dimensional manifold is a stable manifold in this case Since is reasonably small we expect that if there is a traveling wave then it will have a velocity close to that of the front which is the velocity of the pulse in the limit as 0 Set the velocity c to 0 5 and shoot using the Sing Pts Go command Observe what the right branch of the unstable manifold does Change
130. You can read these in and plot them in your favorite graphing program You can export the entire simulation data set by clicking the Write button in the Data Viewer In the AUTO window you can write data from the current view by clicking on File Write pts XPPAUT can read this data and interpret it so that you can place a color bifurcation diagram into the XPPAUT main window for labeling etc The file consists of five columns of data x y1 y2 n b The first three coordinates are the x coordinate of the current view and the maximum and minimum y values respectively n 1 2 3 4 for stable fixed points unstable fixed points stable periodic orbits or unstable periodic orbits b is the branch number More complete information about a bifurcation diagram is obtained by using the File All info 221 222 Chapter 9 Tricks and advanced methods command The result is a series of rows of numbers with the following information n b p1 pa T ytt YR go y 11 M1 rN My where n is the type of point b is the branch number pj p2 are the active parameters T is the period y y are all the max and min values of the coordinates of the system and r mjy are the real and imaginary parts of the eigenvalues at that point So for example if you had a 3 dimensional system and you wanted to plot the real part of the eigenvalues as a function of the main parameter you would want to plot the third column p and the 12th 14th and 16th
131. a built in tutorial about the nature of the fixed points 9 8 Dynamic linking with external C routines 249 x y the linear planar systems planar ode ax xx bx y c xtd y par a 1 b 0 c 0 d 2 init x 2 y 0 xp x yp y xlo 5 ylo 5 xhi 5 yhi 5 here is a tutorial Linear planar systems There are several different behaviors for the phase plane of a linear 2D system To see these click on the and then DirField Flow 5 a 1 b 0 c 0 d 2 1 Stable node two real negative eigenvalues a 0 b 1 c 3 d 0 2 Saddle point a positive and negative eigenvalue a 1 b 3 c 2 d 0 3 Stable vortex complex eigenvalues with negative real parts a 5 b 2 c 2 d 25 4 Unstable vortex complex with positive real parts a 5 b 0 c 5 d 5 5 Unstable node two positive eigenvalues a 5 b 2 c 2 d 5 6 Center imaginary eigenvalues a 1 b 1 c 2 d 2 7 Zero eigenvalue Try your own values and classify them done Note the lines that have the within them will be seen by the user as so that what is being done is invisible Clicking on the 7 Zero eigenvalue will result in everything within the curly brackets being done 9 8 Dynamic linking with external C routines XPPAUT has a way to allow you to define the right hand sides of your ODE di rectly in C code Why would you want to do this Occasionally you may have a tremendously complex right hand side tha
132. a file that has a special structure To do this you declare table w junk tab where junk tab is a file containing the information for the lookup function w The file has the following format h ct cot Z PNR 148 Chapter 6 Spatial problems PDEs and boundary value problems f_N where N is the number of values you will define t_1 is the lowest value of domain of the function t_2 is the highest value and f_1 etc are the values at the N discrete points File based lookup tables allow you to determine how intermediate values of t are defined In function based tables only linear interpolation is allowed For file based tables there are two other ways to interpolate piecewise constant and cubic splines XPPAUT determines which is used from the first character in the file If the first character is s then it uses splines if it is i it uses piecewise constant Otherwise it uses linear interpolation Thus for a table with 100 values the first line of the table will be one of the following three 100 s100 i100 and the intermediate values of the function will be defined by respectively linear interpolation splines and piecewise constant File base tables are an excellent way to drive a simulation with experimental data XPPAUT can produce table files from columns of numbers that are in the data browser Use the Read command in the Data Viewer to load the browser with data Then use the Table command in the Data Viewer t
133. aluate this convolution and also provides several different ways to treat the endpoints Suppose that we are given an array of variables u 0 N and we want to convolve this with the 2m 1 weights W m W m First we must use the table command in the ODE file to create the weight table W Then we use the special command to create a function whose return value is the desired convolution A brief aside on tables In XPPAUT you can define functions as lookup tables by using the table command You can either have XPPAUT read the values from a file or you can have XPPAUT create the table using a function Suppose you want to create a table of length N for a function f t defined for t lt t lt t2 That is the N values of the table are f t1 f ti dt f t2 where dt t2 t1 N 1 Let s call the table w and suppose N 51 t 25 and t 25 Then you write the following in your ODE file table w 51 25 25 f t The usefulness of tables is that the function could be very complicated so that there can be a big speed increase by using a lookup table for the function w is treated like a function in XPPAUT so that w 17 returns 17 Note the dummy argument for the function is always called t For tables defined by functions linear interpolation is used to obtain values of the function that are not on the grid points of the table Thus w 17 5 will return f 17 f 18 2 You can also define a table using a dat
134. alysis of two dimensional linear systems The general linear system has the form x ax by y cx dy 70 Chapter 4 XPPAUT in the classroom where a b c d are parameters The behavior of this system is completely determined by the eigenvalues of the matrix a b ree If the eigenvalues are both real and the same sign the origin is a node if they are real and opposite signs it is a saddle and if the eigenvalues are complex with nonzero real parts the origin is a vortex or spiral In the cases of nodes and spirals if the real part of the eigenvalues is negative positive the node or spiral is attracting repelling There are several degenerate cases If there is a zero eigenvalue then the eigenvector corresponding to the zero eigenvalue is a line of fixed points for the system Finally if there are pure imaginary eigenvalues then the origin is called a center Before starting with the computer we recall a few facts about linear two dimensional systems Define the trace to be T a d the determinant to be D ad be and the discriminant to be Q T 4D Then the following holds f D lt 0 then the origin is a saddle point fT lt 0 D gt 0 and Q gt 0 the origin is a stable node fT gt 0 D gt 0 and Q gt 0 the origin is an unstable node fT lt 0 D gt 0 and Q lt 0 the origin is a stable spiral fT gt 0 D gt 0 and Q lt 0 the origin is an unstable spiral f T 0 and D gt 0 the origin is a ce
135. ames were played Both games are losers in that repeatedly playing them leads to a net loss However when the games are played sequentially or chosen randomly with 50 probablility then they produce a winning strategy Game one consists of flipping a coin that has a slightly less than 50 chance of winning Thus over the long haul you lose Game two works like this If the amount of money you have is a multiple of three than you flip a very badly biased coin If the amount of money you have is not a multiple of three then you flip a coin with a good probability of winning Ultimately the very badly biased coin costs you and this game is a long term loser Now consider the third game At each turn you flip a fair coin If heads then play game 1 and if tails play game 2 This turns out to be analogous to the ratchet we described above but it is a discrete analogue Let s write an ODE file for each of these games Here is game 1 gamel ode this plays game 1 game p e if ran 1 lt p e then 1 else 1 par eps 005 pa 5 ca ca game pa eps total 100 meth discrete dt 1 done The function gamel p e returns 1 according to whether or not you win The variable ca tracks the amount of cash you have The method is discrete You play 100 games in each simulation Since the probability of winning is 0 495 over the long haul this is a losing game Simulate this 1000 times and look at the mean value of the amount of cash as a function
136. amiltonian systems They should not be used for other types of systems In addition to the fixed step integrators XPPAUT has several adaptive in tegrators Four of them Stiff Gear Rosenbrock and Cvode are implicit and work well for stiff differential equations Stiff is based on the stiff algorithm in Numerical Recipes in C Gear is a direct translation of FORTRAN code found in Gear s book CVODE is based on LSODE and was obtained at the web site http netlib bell labs com netlib ode index html and Rosenbrock is based on an algorithm called ODE23S found in The other three adaptive integrators Dormand Prince 8 3 Dormand Prince 5 and QualRk are all explicit DorPri8 3 is based on a Runge Kutta method of order 8 with automatic step size control It was developed by Prince and Dormand and is described in Hairer 1993 It provides a piecewise polynomial approximation of seventh order to the solution DorPri 5 is similar and of lower order Both methods are complicated and the reference to Hairer should be consulted for details Both require a relative and an absolute tol erance Relative tolerance is scaled by the magnitude of the output while absolute tolerance is not If problems occur during the integration XPPAUT puts out error messages that suggest possible remedies QualRk is an adaptive step fourth order Runge Kutta method It is based on code in Numerical Recipes The interested user should consult that reference However I poin
137. an be rectified by changing the numerical parameters You can exit from Grab mode by tapping Esc which does not grab the point or by tapping Enter which 170 Chapter 7 Using AUTO bifurcation and continuation loads the point into memory AUTO can generally be restarted only from special points So lets make this diagram extend to the left Grab the first point in the diagram Grab Enter Click on Numerics and change Ds from 0 02 to 0 02 and click on OK This tells AUTO you want to change the direction since the sign of the numerical parameter Ds determines the starting direction for AUTO Now click on Run You should now see a sideways cubic curve In particular there are two points labeled 3 and 4 occurring at a 3 84 If you click on Grab you will discover that the two special points are labeled LP which means that they are limit points also known as fold points Note that as you move past them the eigenvalue goes through the unit circle The curve of fixed points along the middle branch is unstable Grab the leftmost limit point 3 We will now track this fold point as a function of the second parameter b producing a two parameter bifurcation curve Click on Axis and select Two param Change Ymin from 3 to 0 5 Since the plot type is Two Par the vertical axis contains the secondary parameter b and the horizontal the main parameter a Click on OK Click on Run You should see a thin curve that rises and goes to the
138. an der Pol with epsilon 0 Ww v_ O v_ 1 v_ v_ w solv v_ 1 aux v v_ init w 0 NEWT_ITER 1000 NEWT_TOL 1e 3 JAC_EPS 1e 5 METH qualrk total 10 dt 01 xlo 1 5 xhi 1 5 ylo 5 yhi 5 xp v yp w done Look at the data in the browser and verify that the jump in V is discontinuous 122 Chapter 5 More advanced differential equations Chapter 6 Spatial problems PDEs and boundary value problems In space no one can hear you scream Ad for the movie Alien Spatially distributed models are common in physics biology and chemistry The numerical solution of one dimensional problems is possible to a limited extent using XPPAUT You first have to discretize the system to convert it to a system of ODEs Then you can take advantage of XPPAUT s numerical and graphics capabilities to solve the resulting systems Often you are only interested in finding steady state solutions to a partial differential equation PDE In this case the resulting equations are sometimes an ordinary differential equation with special boundary conditions This leads to a boundary value problem XPPAUT and AUTO are both able to solve complex boundary value problems 6 1 Boundary value problems Boundary value problems hereafter BVPs present a more difficult problem both numerically and theoretically than do the sorts of initial value problems that we have already described In BVPs conditions are given at both ends of the time interval Often this
139. anchester F W 1908 Aerodonetics London TY Li and JA Yorke 1975 Period three implies chaos Amer Math Monthly 82 985 992 P Linz Analytical and Numerical Methods for Volterra Equations SIAM 1985 R Macey and G Oster MADONNA http www berkeleymadonna com JD Murray Mathematical Biology 1988 Springer Verlag Paullet Joseph Ermentrout Bard Troy William The existence of spiral waves in an oscillatory reaction diffusion system SIAM J Appl Math 54 1994 no 5 1386 1401 W H Press S A Teukolsky B P Flannnery and W T Vetterling Numerical Recipes in C Second Edition Cambridge 1992 Werner Rheinboldt MANPAK www netlib org contin manpak Shaw Robert The Dripping Faucet as a Model Chaotic System Aerial Press Santa Cruz USA 1984 L F Shampine and M W Reichelt The MATLAB ODE Suite SIAM Journal on Scientific Computing 18 1 1997 S Strogatz Nonlinear Dynamics and Chaos with Applications in Physics Biology Chemistry and Engineering Addison Wesley 1994 C van Vreeswijk L F Abbott and B Ermentrout J Computational Neuroscience 1 313 321 1994 Bibliography 289 34 West B Strogatz S McDill JM Cantwell J 1997 Interactive Differential Equations Chapt 19 Addison Wesley Interactive US 35 T L Williams and G Bowtell J Computational Neuroscience 4 47 55 1997 36 Winfree A T 1967 Biological rhythms and the behavior of populations of coupled oscillators J Theo
140. anted the derivative of the voltage Then replace the voltage by first choosing v as the column to replace and then typing in v for the formula Then you can plot the derivative of the voltage Note that differentiation and integration are post integration com mands only That is you cannot use these in a new column formula or right hand side Occasionally you need a sequential list of numbers You can replace a column with such a list The formula a b creates a list of numbers the same length as the number of rows in the Data Viewer starting with a and ending with b The formula a b creates a list starting with a and incrementing by b for each row For example replace t with 0 6 283 and then replace V with sin t Plot V versus t and you will see a nice sine wave Pretty lame huh One trick is to read in data from a file and manipulate it with the data browser and use XPPAUT as a fancy graphing program 9 5 Oscillators phase models and averaging One of the main areas of my own research concerns the behavior of coupled nonlinear oscillators In particular I have been interested in the behavior of weakly coupled oscillators Before describing the features of XPPAUT that make such analyses easy I will discuss a bit of the theory Consider an autonomous differential equation X F X which admits as a solution an asymptotically stable periodic solution Xo t Xo t P where P is the period Now suppose that we couple two such osci
141. apping Enter Then Axes Two par and click on Ok in the dialog box the window is fine Click on Run and you will see a curve of Hopf points in the eps a plane and another lower curve which is tangent to this curve at a point approximately a 2 8 182 Chapter 7 Using AUTO bifurcation and continuation and e 0 3 Along the lower curve there is an eigenvalue of 1 and along the upper curve are eigenvalues on the unit circle but not real 7 2 2 Exercises 1 The Ricker model is another discrete population model that is often used Tn In41 Zne Starting at a 1 2 and x 0 18232 compute the sequence of period doublings up to period 8 using the 8th iterate The parameter a should be allowed to range between 0 and 20 and the population x lines between 0 and 8 2 The modified Nicholson Bailey system models a predator prey interaction En41 Tn explr 1 2n k ayn Yn 1 En 1 exp ayn where r k a are positive parameters Start at r 5 k 5 0 lx 1 y 0 and create the bifurcation diagram for 0 lt k lt 5 Note that k is the carrying capacity and represents the amount of prey that exists when there are no predators Solve the system when k 3 2 Does the resulting periodic solution have a period roughly like predicted from the Hopf point Set k 5 r 1 6 and verify that there is a period 7 orbit starting at roughly x 376 y 198 Continue this period 7 orbit holding r fixed and letting 4 5 lt k lt
142. arameter for the numerical computation of the ma trix J called epsilon in XPPAUT If the difference between successive iterates falls within the tolerance or G Xy falls within the tolerance then convergence is assumed and the root is found The matrix J is found by perturbing each of the variables by an amount proportional to epsilon and then using this perturbation to approximate the derivative Once a fixed point is found the matrix J is evaluated one last time and the eigenvalues of J are computed using standard linear algebra routines See Numer ical Recipes for example XPPAUT uses this information to compute the stability of solutions One dimensional stable and unstable invariant manifolds are com puted by finding the eigenvector corresponding to the real eigenvalue A small perturbation from the fixed point along this vector is made and the equations are integrated The eigenvector is computed by inverse iteration That is suppose is an eigenvalue Let M J vI where v is a number close to A XPPAUT chooses a random unit vector and multiplies it by M7 and normalizes the result to get 261 262 Appendix C Numerical methods a new vector This iterative scheme continues until convergence is achieved The resulting vector is an approximate eigenvector C 2 Integrators The numerical solutions of ordinary differential equations is an old and well studied field XPPAUT has essentially two classes of integrators i fixed step
143. at they come out in this format Instead of using the u 1 100 syntax we instead use the 1 100 command which tells XPPAUT that everything that follows is an array Here is the above PDE discretized to 100 points with Neumann boundary conditions rd2 ode rd2 ode generic 2 variable reaction diffusion with Neumann conds f u v u 1 u u a v g u v u d v b c u1 f u1 v1 dux u2 u1 h 2 vi1 g ul vi1 dv v2 v1 h 2 4 2 99 ulj f u j v j du u j 1 u j 1 2 u j h 2 v j g u j v j dv v j 1 v j 1 2 v j h 2 u100 f u100 v100 du u99 u100 h 2 6 2 Partial differential equations and arrays 145 v100 g u100 v100 dv v99 v100 h 2 par a 25 d 0 b 4 c 1 du 2 dv 1 h 1 meth cvode bandup 2 bandlo 2 done Everything between the signs is considered an array these equations are expanded as though you had written u2 f u2 v2 du u3 u1 2xu2 h72 v2 8 u2 v2 dv v3 v1 2 v2 h 2 u3 f u3 v3 dux u4 u2 2 u3 h 2 v3 g u3 v3 dv v4t v3 2 v3 h 2 Note how they are interlaced so that the resulting system is nicely banded I have told XPPAUT to use the banded form of CVODE by informing the program about the upper and lower bandwidths The problem with this is that when you use the array plot or the transpose both u and v will be interlaced The cure for this is to skip every other column in the array plot or transpose That is change Colskip fr
144. be produced on your disk You can view this with xanim or other viewing software or just open it with your web browser The Mandelbrot set We have used XPPAUT to animate Julia sets For open sets of parameters the Julia set is connected The set of parameters c for which 60 Chapter 4 XPPAUT in the classroom the Julia set is connected is the Mandelbrot set Another way to look at it is to define it as the set of parameters c such that iterates of f z z c starting at the origin remain bounded Since the Julia set is connected there is no escape to infinity This set is typically computed by choosing a parameter c and iterating a number of times until zn exceeds some bound If after this fixed number of iterates Zn 18 still bounded then c is considered to be in the Mandelbrot set The number of iterates needed to determine whether a point is in the Mandelbrot set is becomes large near the boundary of the set Often points that are not in the set are color coded according to the number of iterates it takes to become unbounded To compute this set in XPPAUT we will use a number of features which control the output and color Here is the ODE file for computing the Mandelbrot set mandel ode mandel ode drawing the mandelbrot set x x 2 y 2 cx y 2 x y cy so that the parameter axes are plottable cx cx cy cy init x 0 y 0 if this crosses 0 then we are out of the set aux amp x72 y 2 4 xp cx
145. c bifurcations The equations are a A ar by cz dw x eyzw y A ar bz cw da y exzw z A ar bw ca dy z exyw w A ar ba cy dz w exyz where r x y 2 w and a b c d e are parameters The parameter A is the bifurcation parameter The equations are notable because for some 4 6 Three and higher dimensions 89 values of a e as A crosses 0 there is bifurcation to instant chaos Try a 1 b 0 1 c 05 d 015 e 55 and vary A Use initial conditions x y w 0 1 and z 0 2 You should integrate these for a long time Here is a possible XPPAUT file field worfolk equations l a xr bxy 2 c z 2 d w 2 xt e y z w L a xr b z 2 c w 2 d x 2 y e x Z W l axr bxw 2 c x 2 d y72 z ex x yx w 2 l a r b x 2 c y 2 d z 2 w exx y z r x 2 y 2 z 2 w 2 par 1 1 a 1 b 1 c 05 d 1 e 1 set fw1 1 1 a 1 b 1 c 05 d 1 e 1 set fw2 1 1 a 1 b 1 c 05 d 015 e 55 set fw3 l 1 a 1 b 1 c 085 d 005 e 572837 init x 1 y 1 z 2 w 1 maxstor 20000 dt 5 meth qualrk tol 1e 5 total 4000 done 2 2 3 N Try the second set of parameters fw2 As a final example consider the simplest equation that I know of that has chaotic behavior and approaches it through a series of period doubling bifurcations y II aq y z ce by az q Choose a 1 b 2 and let c vary fr
146. c3 00005 c4 5 init y1 1000 y2 2000 compute the cumulative reactions p1 c1x1 p2 p1 c2x2 y1 p3 p2 c3 y1 y2 y1 1 2 p4 p3 c4 y1 choose random s2 ran 1 p4 z1 s2 lt p1 Z2 s2 lt p2 amp s2 gt p1 Z3 s2 lt p3 amp s2 gt p2 Z4 s2 gt p3 time for next reaction tr tr log ran 1 p4 yi max 1 y1 z1 z2 z3 z4 y2 max 1 y2 z2 z3 bound 100000000 meth discrete total 1000000 njmp 1000 O xp y1 yp y2 xlo 0 ylo 0 xhi 10000 yhi 10000 done x1 gt y1 c1 5 2 Stochastic equations 109 y Y 10000 8000 10 6000 4000 2000 pl O al Me owt Kot NL wat Malin tro d 80 EA et Vua ath vandal A o 0 2000 4000 6000 8000 10000 o so 100 150 200 250 yl time Figure 5 6 The Gillespie method applied to the Brusselator left and to a membrane model Since X 1 X2 are held constant I have combined the products c X 1 and c2X into single parameters Next I compute the cumulative sum P Note that P4 ay since it is the sum of all the rates I choose a random number s2 between 0 and ag Now comes the tricky part I define 4 numbers z These are 1 if reaction j takes place and 0 otherwise Thus if s2 lt P then reaction 1 takes place If Py lt s2 lt Pa then reaction 2 takes place and so on Next we determine the time of the next reaction Then Y Y are incremented or decremented according to the reaction For example reactions 1 and 3 produce a
147. ced methods Flagged output Newer versions of XPPAUT incorporate a feature for that allows you to make Poincare maps and many other calculations very rapidly The dis advantage of the following method is that you need to add a special line in your ODE file and this cannot be done within the program Recall from section 3 5 that XPPAUT allows you to catch crossings of variables and based on this take some action One such action is to output a point to the data browser Just include the action out_put 1 within the list of actions and when the event occurs XPPAUT will send output to the data browser Since you generally want to suppress all other output you should for example set Transient to some large number from the numerics menu This guarantees that XPPAUT won t print out any other data The advantage of this over a standard Poincare map is that there can be many pos sible events that are flagged It is generally faster than the Poincare map feature since graphic output is suppressed 9 7 Don t forget 1 Initial conditions and parameters can be typed in at the command line as formulae Use the Parameter command and then type the name of the parameter When asked for a value type in the sign first and then the formula Use the Initialconds New command to type in the initial value in the same manner 2 There is a stupid little calculator built in to XPPAUT Click on File Calculator to get it You can type in the usual math type ex
148. chaotic behavior Change the parameter a to 1 and solve it Note the equations go to a fixed point Increase a to see the periodic orbit Note the period doubling between a 4 and a 5 The default value of a seems to be chaotic Note that in the above ODE file we cannot write fr t since the right hand side contains nothing to tell XPPAUT that this is a functional equation rather than the definition of a simple function of time XP PAUT looks for symbols like int to distinguish function definitions from functional equations 5 1 6 Exercises 1 Explore the Mackey Glass equations a t ya t Bf a t 7 where f u u 1 u This is a model for circulating white blood cells For humans y 0 1 8 0 2 n 10 and the delay is 7 6 Integrate to 600 keeping output every 10 Total 600 Nout 10 in the nUmerics menu Be sure to start x 0 away from 0 as x 0 is a fixed point Observe the small oscillations Change T to 20 and look at the new behavior Try this set T 1 and integrate to the fixed point Click on Initialconds Last to be sure you are on it Determine the stability and notice that the fixed point 5 1 Functional Equations 101 Figure 5 4 Solution to the Fabry Perot equations in the chaotic regime is stable Click on sIng pts Range and choose 7 as the range parameter going from 1 to 6 in 10 steps Click on Ok You will see in the Equilibrium Window that the rest state has lost stabili
149. cks regular and not There are many other examples of models that involve discontinuous right hand sides The simple clock is an excellent example We model a pendulum clock as 38 Chapter 3 Writing ODE files for differential equations follows It consists of a decaying sinusoid that receives a kick each time it reaches its maximum on the left hand side Let x be the angle of the pendulum and y be the velocity We model the decaying sinusoid as d ax by ay be dt with the kick occurring whenever y 0 and x lt 0 The kick is toward the left with magnitude k Thus when y t 0 and z t lt 0 then z t x t k The ODE file is the clock model a linearly decaying spiral that is kicked out by a fixed amount x a x b y y a y b x heres the kicker global 1 y x x k par a 1 b 1 k 5 init x 5 y 0 change some XPP parameters to make a nice 2D phase plane total 150 xlo 1 5 ylo 1 xhi 1 5 yhi 1 xp x yp y done As usual I have set some internal parameters so that you are looking at a phase plane projection Let s take a look at the global declaration The linear equation without the kick this is a linear equation but since the kick is dependent on the values of the variable the ODE is nonlinear is a spiral source and since b gt 0 the flow is counter clockwise Thus when y 0 and x lt 0 the variable y crosses 0 from positive to negative Thus the choice of 1 in the global stateme
150. close to a Gaussian A rather jagged one Freeze this if you want to compare it to the next case Now change s0 1 to turn on the refractory period Repeat the range integration You needn t change any of the dialog entries Find the Fano factor and make a histogram 1 found a mean of 35 5 and a deviation of 4 25 with a Fano factor of 0 5 which is much more regular than Poisson Make a histogram of the counts with 30 bins in the range 20 to 50 on the variable count and you will see a much narrower histogram cf Fig 5 9 The refractory period makes the spike count more regular 5 2 5 Exercises 1 Implement the Gillespie algorithm for the Lotka Volterra equations Xi in Y 2Y1 YI Yo 2Y2 Y Z where c1 X 10 c2 0 01 c3 10 and initially Y 1000 Y 1000 Note that only Y Y2 are variables and X is held fixed 2 Implement the fully deterministic version of the neural model by adding the following differential equation for x the fraction of sodium channels dx 1 Z te0 V s V where zo and 7 are defined in the ODE file as xinf and tauxi respectively Figure out the minimum current for which oscillations occur For low channel numbers the stochastic model will fire spikes occasionally even though the current is quite far below the critical value for repetitive oscillations Finally set J 10 in the deterministic model and compute the period of the spikes Do the same for the stochastic model usi
151. cluding occasional warnings that you can safely ignore Tf you get no errors then you probably have succeeded in the compilation If the compilation stops very quickly then you probably you will have to edit the Makefile according to the architecture of your computer Look at the README file and the Makefile which has suggestions for many platforms Step 4 If you successfully have compiled the program then you should have a file xppaut in your directory To see type ls xppaut If you see something like xppaut listed then you have succeeded If you don t see this then the compilation was unsuccessful Consult the README file for a variety of possible fixes Also there are many comments in the Makefiles that are included with the package I have not yet found a computer on which I cannot compile the program Common problems are the wrong path to the X Windows libraries nonexistence of ranlib among others Step 5 Once you have compiled it just move the executable to someplace in your path The usual is usr local bin but you must have root privileges to do this XPPAUT needs no environment information 1 1 2 Installation from binaries Some binaries are available at one or both of the above URLs You should download these as well as the source code above The source code has many examples and the XPPAUT reference manual Download a binary e g xppaut4 6 hpux gz and uncompress it with the command gunzip xppaut4 6 hpux gz and copy
152. cond order equation as a pair of first order equations ii we have added an additional differential equation for the free param eter w iii the boundary conditions attempt to make v 0 0 and dv dx 0 1 and v 1 0 iv I set the known initial conditions to the proper values and guess the unknown omega 1 and v I have set the total amount of time to 1 00 6 1 1 Solving the BVP Start XPPAUT with this file Try integrating it Notice that the solution does not come close to v 0 at t 1 Click on Bndryval Show You will see a few trajectories flash by and then the last one is shown It is the computed solution Look in the initial condition window and see that w has been computed to be 3 141592648 which is very close to the true solution of 7 Try a guess of w 5 and see what happens You will get an eigenvalue close to 27 and the second harmonic Try guessing w 35 and you will get an eigenvalue close to 117 There are other options in the Bndryval command The option you have chosen draws intermediate guesses while the option No show only draws the last one As we noted above not every boundary value problem has a solution Consider the equation u ku u 0 1 u 1 0 There is no solution to this However run the ODE file u k u k 0 b u 1 b u init u 1 k 1 dt 01 total 1 000 xhi 1 done and you will find that XPPAUT finds a solution in which k is 12 or so You will notice that u 1 6 107
153. condition for x to be 1 and integrate so that all you see is the horizontal line x 1 Click on Sing pts Range and fill in the resulting dialog box as follows Range over tan Steps 20 Start 0 gt ma Shoot OAN Stability col 2 Movie 4 0 N and then click on Ok A bunch of numbers should scroll by in the console window If you look in the Data Viewer you will see the numbers 0 to 2 in increments of 0 1 in the Time column and some other numbers in the X column The first col umn contains the values of tau through which you ranged and the second column contains the real part of one of the infinitely many eigenvalues for the linearized equation There are infinitely many such eigenvalues since the characteristic equa tion is transcendental and involves exponentials Click on X vs t and choose X to plot You will see that the real part of the eigenvalue crosses the axis at about 0 8 Thus for 7 gt 0 8 the fixed point is unstable and there is probably a Hopf bifurca tion For 7 lt 0 8 the rest state is stable Try setting tau 75 and integrating with x 0 8 and then doing the same for tau 85 In the former case you should see a damped not damped much oscillation while in the latter the solution goes to a periodic orbit If you click on Sing pt Go you should see the little equilibrium window indicate the fixed point is unstable If you look at the console window you will find that the unstable root is 0 066 1 88
154. conds sHoot command and choosing solution 2 Freeze this one too See Figure 6 4 We now have defined our shooting sets For c close to zero the unstable manifold hits the u axis before it hits the u axis For c large enough the unstable manifold hits the u axis before the u axis By continuity with respect to parameters there must be a value of c between 0 and 0 5 for which the stable manifold crosses both axes at the same time that is it hits 0 0 our shooting target One must avoid the cases for which the trajectory is tangent to the axes Therein lies the hard analysis required to make this argument into a rigorous proof So the procedure is clear Refine your choice of c according to which of these two conditions hold If the solution hits the u axis increase c and if it hits the u axis decrease it Try c 0 25 Notice that it hits the u axis so it is too small Choose a point halfway between 0 25 and 0 5 c 0 375 and notice that this time c is too big Thus we know that the velocity of the traveling wave is between 0 25 and 0 375 Again choose c halfway between these values c 0 3125 and run again It is too small Make it bigger say 0 34375 This turns out to be too small also Several more guesses narrow the values of c to between 0 34375 and 0 359375 Figure 6 4 shows three trajectories one of which is very close to the correct value of c 6 1 3 Exercises 1 Reconsider the eigenvalue problem Ung
155. cribe the other windows as the tutorial progresses Clicking on the Close button closes this window 2 2 2 Quitting the program To exit XPPAUT click File Quit Yes F Q Y 2 3 Solving the equation graphing and plotting Here we will solve the ODE use the mouse to select different initial conditions save plots of various types and create files for printing Computing the solution In the Main Window you should see a box with axes numbers The title in the window should say X vs T which tells you that the variable X is along the vertical axis and T along the horizontal The plotting range is from 0 to 20 along the horizontal and 1 to 1 along the vertical axis When a solution is computed this view will be shown Click on Init Conds Go I G in the Main Window A solution will be drawn followed by a beep As one would expect given the differential equations the solution looks like a few cycles of a cosine wave 2 3 Solving the equation graphing and plotting 13 08 06 04 02 0 02 04 0 6 0 8 Figure 2 3 Phase plane for the linear 2d problem Changing the view To plot Y versus T instead of X just click on the command Xi vs t X and choose Y by backspacing over X typing in Y and typing Enter Many times you may want to plot a phase plane instead that is X vs Y To do this click on Viewaxes 2D V 2 and a dialog box will appear Fill it in as follows leads Xu Xlabel Ylabe1 Click on O
156. ct to no more than 4 other cylinders since branching always consists of splitting into two The connectivity is of the form Y lv V Yi a Vj 6 2 Partial differential equations and arrays 157 Figure 6 10 A dendritic tree rendered in the animation window Thus for N compartments two tables of length 5N are created that describe the weights and the connectivity We pad the weights for compartments that connect to fewer than 4 others with zeros The input file for this nice little utility consists of the a y coordinates for the endpoints of each cylinder and the diameter The program uses the endpoint information to decide which cylinder is connected to which and it uses the dimensions of the cylinder to determine the relative weights of the coupling The lengths are scaled and then an animation file is created that draws the cylinders The animation files draws each cylinder color coded according to some user specified function The C code and some examples are available on the XPPAUT web site An example is shown in figure 6 10 One dimensional cellular automata Cellular automata enjoyed some popular ity a number of years ago and are still quite amusing to play with A simple class of model CA consists of a line of cells that take on values of O or 1 At each time step the sum of m neighbors is taken which is thus between 0 and 2m 1 and a rule is provided that maps each possible sum to either 0 or 1 Typically the bou
157. d 1 2 and 3 orbits Integrate this for a long time and try to show that there are 6 different period 5 orbits 2 Analyze the Rossler equation gv y 2z y x ay z bz cz rz 4 6 Three and higher dimensions 87 where a 0 36 b 0 4 c 4 5 with initial data x y z 0 4 3 0 054 First integrate it for a total of 200 with a timestep of 0 1 Look at the time series and various phase space projections Look at it in three dimensions Note that the Window zoom Fit is invaluable since this will compute the scales for you Compute the Liapunov exponent Compute the power spec trum You should see a big peak and some side peaks but there are many other frequencies However it is not nearly as chaotic as the Lorenz system as evidenced by the dominance of certain frequencies and the Liapunov exponent that is close to 0 Compute a Poincare map for this attractor as follows Instead of selecting Max min choose Section Choose X as the variable 0 as the section and 1 as the direction a point will be stored each time that x crosses 0 with a positive derivative Integrate for a total of 2000 Plot y _1 against y using the Ruelle plot The resulting map is essentially one dimensional and looks like an upside down logistic map Find period 2 and period 3 points For fun simulate the map Yn 1 6 0 65 yn 3 5 with yo 5 which is a reasonable fit to the true map Find the period 1 2 and 3 points for this
158. d another plot in which I keep the z value constant This is a useful trick and lets one plot a variety of interesting projections against the coordinate planes Only zmin and the parameter zlo need to be changed in order to change the position of the planar projection By plotting z y and holding x at some fixed value you can similarly plot the y 2 projection See figure 4 1 for the result 4 2 Discrete dynamics in one dimension Many discussions of dynamical systems begin with the study of one dimensional maps These have the form Inti f Ln One of the standard ways to illustrate the dynamics of these maps is to draw a cobwebbing diagram This is a plot of 7 41 versus n along with the line y x and 4 2 Discrete dynamics in one dimension 49 the function y f x XPPAUT does not have a specific cobwebbing function built into it but it is very easy to fool the program into making a cobweb plot Every other point shares either the same x value or the same y value thus the cobweb plot consists of a series of horizontal and vertical lines Once this is done you also want to plot the function and the line y x Here is an example cobweb map for the logistic function cobweb ode way to fool XPP into cobwebbing first I define a function that every other step evaluates the map in the alternate steps it just keeps the same value so that it alternates between horizontal and vertical jumps g x y if mod t 2 lt
159. d you will see a series of labeled points along the bottom of the figure Click on Numerics and change Dsmax to 10 and then click on Ok Grab the first branch point BP labeled 2 Then click on Run again AUTO automatically switches when asked to continue from branch points You will see a branch rising up from this point When AUTO is done grab the next branch point labeled 3 and continue this Do this for the branch points at r 3 4 as well Now click on Numerics and change Ds to 0 2 to change directions click on Ok Grab each of the branch points and run again For some of them it will appear like nothing has happened this is because the maximum value of u is the same on each branch so that they are not distinguishable Click on Axes Hi and change the Y axis to v instead of u click on Ok and then reDraw and you will see a slightly different view TRICK Distinguishing branches in AUTO Here is a trick that you can use to avoid the symmetry problem in graphing the curves of solutions As we saw in this example the things that AUTO plots maximum of a solution and the norm of a solution cannot always lead to distinctions So here is a way to fool AUTO into letting you plot something else in the bifurcation diagram For the above problem the value of v at the origin is zero for the trivial branch of solutions but for the solutions that emanate from each branch point it is non zero and it is necessarily different The reason for this
160. dent variable occupies the left most column and the dependent variables filling in the remaining windows Click on the top of the Data Viewerto make it the active window The arrow keys and the PageUp PageDown Home and End keys as well as their corresponding buttons do all the obvious things Left and right keys scroll horizontally a useful feature if you have many variables I mention three buttons of use Find brings up a dialog box prompting you for the name of a column and a value If you click on Ok XPPAUT will find the entry that is closest and bring that row to the top You can find the maximum and minimum for example of a variable by choosing a really big or small number in the Find dialog Get loads the top line of the Data Viewer as initial data Write writes the entire contents of the browser to a text file that you specify 2 6 Saving and restoring the state of XPPAUT Often you will have a view a set of parameters and initial data that you want to keep You can save the current state of XPPAUT by clicking on File Write set F W in the Main Window This bring up a file selection box Type in a filename the default extension is set The resulting file is an ASCII file that is human and computer readable The first and last few lines look like HH Set file for linear2d ode on Fri Aug 4 13 53 31 2000 2 Number of equations and auxiliaries 4 Number of parameters Numerical stuff 1 nout 40 nullcline mesh RH
161. discretize the piecewise constant wave model oO OY Sota ies H u a y 0 dy Ot 20 Jo where H u is the Heaviside step function This nonlocal equation has a traveling front solution The discrete analogue is 1 2m 1 u ui y H uj i 0 We will discretize this into 101 points and choose m 5 We first define a table for the weights table w 11 5 5 1 11 This creates a lookup table of length 11 and each entry is just 1 11 a constant weight Next we create an array in which the step function is applied to u h O 100 heav u j theta Thus h0 h1 is a set of fixed variables whose values are just the step function applied to the corresponding u variables Now we define the discrete convolution special k conv zero 101 5 w h0 This says that we will set the values of of h to be zero outside the domain Finally we define the right hand sides using the special operator k j u o 100 u j x 31 Putting this together yields the ODE file convwave ode define a table for the convolution table w 11 5 5 1 11 define h u theta h O 100 heav u j theta define the special convolution operator special k conv zero 101 5 w h0 u o 100 u j x 3 init u0 1 u1 1 u2 1 par theta 1 done 150 Chapter 6 Spatial problems PDEs and boundary value problems NOTE that a common error is to write k j instead of k j In XPPAUT the brackets are ignored so that k 10 would be interpr
162. dition must be provided to fix a phase mE on 0 268 Appendix C Numerical methods where 9 o are the previously found solutions Thus we now have a 3n 2 dimen sional system to solve Continuation for maps is done in a similar manner For differential equations on the interval 0 1 AUTO solves an algebraic system that is obtained by intelligently discretizing the ODE and then solving the resulting al gebraic equations The parameter Ntst in the AUTO Numerics menu defines the number of points in the discretization The discretization is not uniform rather more points are placed where the solutions is changing rapidly Thus there is never really any integration of solutions of the equations in AUTO instead continuation of systems of algebraic equations is done Here is a detailed example of how AUTO continues equations Consider flu A A u2 0 Suppose we start at A 1 and u 1 The normalized nullvector for fu A 0 is given by 1 1 A 5 1 2u i 4 1 20 where we use the fact that fu 2u We must solve 1 As 1 4u3 4 uj ug 1 1 As Aj 1 205 1 and the Jacobian of this pair of equations at the guessed value u is 1 2u 1 Ma 1 4u7 4 1 2451 The determinant of this matrix is 1 4uuj 1 1 4u _ which will be nonzero if we move a small enough step so that u and u _1 are close More details on AUTO and the methods used can be found in the references
163. ds the maximum Then click on Get which loads this point as an initial condition Integrate again and look for the time of the next maximum This is the period Alternatively hold the mouse in the window with the button pressed down and move it around you can read off the coordinates at the bottom of the window 3 Change the total amount of integration time to the approximate period nUmerics Total and integrate the equations one more time 4 Fireup AUTO and set up the bounds for the parameter as well as the graphical viewpoint 5 In the AUTO window click on Run Periodic 172 Chapter 7 Using AUTO bifurcation and continuation 6 Keep your fingers crossed Let s apply these techniques to the isola problem Here is the ODE file an isola of limit cycles isola ode f r 25 r 1 72 a72 r x 2 y72 x f r x y y f r y x par a 0 init x 1 224 y 0 ylo 1 5 yhi 1 5 done Run XPPAUT and integrate the equations Use the mouse to see that the period is roughly 6 3 it is actually 27 Change the integration time to 6 3 and run again by clicking Initialconds Last starting at the end of the last integration this assures that the transients are integrated out Start up AUTO by clicking File Auto F A Since there is only one parameter you don t have to worry about setting the active parameters From here on in all commands will be in the AUTO window Click on Axes Hilo and fill the dialog as follows
164. dual Also the centroid the average voltage and recovery variable are plotted as a bigger black circle Run the animation and watch the cells spread out to almost uniformly fill in the limit cycle the centroid stays near the inside and doesn t change much Now change phi from 1 to 0 5 and re integrate the equations Initialconds Go Now look at the animation Use the techniques of weakly coupled oscillators see Chapter 9 to explain the difference between phi 1 and phi 0 5 8 3 My favorites 209 You can make this file even better by superimposing the nullclines onto the animation It is best to do all the calculations needed in the ODE file rather than in the animation file so that we add the following lines of code to the ODE file nullcline stuff for animation define the nullclines wnull v winf v vnull v I gl el v gcax minf v eca v gkx v ek now crudely draw them for the animation vv 0 20 6 1 1 j 20 vn 0 20 vnull vv j 25 wn 0 20 wnull vv j 25 done I have drawn 20 line segments for each nullcline and also scaled them identically to the phaseplane drawn in the ODE file The points vv j vn j are the coordinates for the v nullcline while vv j wn j are those for the w nullcline With these added lines in the ODE file the animation file now has the form m110x ani use with m110x ode PERM line 0 19 20 vn j j 1 20 vn j 1 BLUE 3 line 0 19 20 wn
165. e whose solutions are closed curves of various amplitudes and aspect ratios The parameter a determines the aspect ratio the parameter f the frequency around the 214 Chapter 8 Animation Figure 8 3 The Zipper carnival ride Left is the manufacturer s schematic right shows the Zipper as rendered by XPP system and the initial data determine the overall size of the track The positions x y of this system determine the position on the track of the car To get the zipper to rotate we will solve the harmonic oscillator Tp WYhb Yp WT where w is the rotation frequency Once we have the rotation position and the position on the track we know the position of the pivot of the car The position of the bob of the pendulum corresponding to the center of mass of the riders is then just as with the regular pendulum We can then derive equations for the angle of the pendulum relative to the y axis ml 6 mglsin0 ml cos 0 mge sin 0 where te Y is the second derivative of the position of the car on the track The ODE file is a bit complicated by the fact that we need to draw the track Here is the file zipper ode this is a model for the zipper carnival ride zipper ode table of 16 points on the zipper table zx zipx tab table zy zipy tab here is the track RHSs 8 3 My favorites 215 xdot ax f y abs y p ydot f x abs x p second derivatives xddot a f pt 1 ydot abs y
166. e instead you will see that the plots are all modulo 27 Switch the view to the phaseplane for the two variables 01 02 In the Initial Data Window click on the box to the left of each variable and then on the xvsy button You will see that the trajectories seem to converge on a diagonal line which is slightly shifted upward Try integrating using the Initialconds mIce I I command to choose a variety of initial conditions They should converge to the same line This is an example of a phase locked solution an invariant circle on the torus Change a from 0 15 to 0 08 and integrate the equations again Notice how the trajectories do not converge to an attractor Instead the whole torus will gradually fill up Phase locking no longer occurs A derived example Now let s turn to the example that we derived in the previous section and see if a pair of oscillators will phaselock when we use the interaction function defined in 9 2 phase_app ode 9 5 Oscillators phase models and averaging 235 phase_app ode phase model for two coupled oscillators using numerically computed H h x 3 34 3 05 cos x 29 cos 2 x 3 61 sin x 33 sin 2 x thi wi a h th2 th1 th2 w2 a h thi th2 par wi 1 w2 1 a 1 total 100 fold thi fold th2 xlo 0 ylo 0 xhi 6 3 yhi 6 3 xp thi yp th2 done I have added a few new commands The fold th1 directive tells XPPAUT to make a circle out of the variable thi that is to mod it out
167. e ie V crosses threshold from below then V is reset to 0 The line dt 01 ylo 0 sets the timestep to be 0 01 and the minimum value of the y axis to be zero Run this file and integrate the equations an you will see a nice oscillation Move the mouse inside the graphics window and hold the left button down while moving the mouse At the bottom of the main window the coordinates of the mouse are shown You can use this to compute the period of the oscillation about 1 8 Let s write an ODE file for the coupled system The question is how to im plement the alpha function Since b exp bt is a solution to s 2bs b s 0 s 0 0 s 0 b this suggests letting s evolve according to this equation with the condition that each time the voltage crosses threshold s t is incremented by b Here is the ODE file iandf2 ode two integrate and fire models coupled with alpha functions b 2 t exp bx t 3 5 Discontinuous differential equations we solve these by solving a 2d ode vi v1 il g s2 v2 v2 i2 g s1 si sip sip 2 b sip b b s1 s2 s2p s2p 2 b s2p b b s2 init vi 1 par 11 1 1 12 1 1 vt 1 g 2 par b 5 global 1 vi vt v1l 0 s1p slp b b global 1 v2 vt v2 0 s2p s2ptb b dt 01 total 100 transient 80 xlo 80 xhi 100 ylo 0 nplot 2 yp2 v2 done Here are some comments on the ODE file 1 We rewrite the second order equations for s as a pair of first order ODEs 37
168. e rest of the commands will be in the AUTO window Click on Axes Hilo to set up the graphics axes Fill in the dialog box as follows and then close it as Y Wain Para T ond Paras phi EEC am Poa 5 Yoox 5 gt Next open the Numerics dialog box change Par Min to 5 and Dsmax to 0 05 close it Now you should be OK to run Click on Run Steady state You should see a cubic like curve with a few special points on it see figure 7 4 Click on Grab and move through the diagram You will see limit points LP at 7 0 08326 and I 0 02072 as well as a Hopf bifurcation point HB at 7 0 20415 One expects to pick up a branch of periodic solutions at a Hopf bifurcation When you get to this point click Enter to grab it Now you can try to compute the periodic orbit emanating from this point Click on Run and you will get a list of choices Choose Periodic and when you see a sort of smudge at the end of the branch of limit cycles click on ABORT to stop AUTO You should see something like Figure 7 4 The smudge is due to AUTO s difficulty at computing the long 7 1 Standard examples 175 period oscillation which terminates on the lower fold point This shows a number of interesting features that are quite important in bifurcation theory First note that the branch of periodic orbits that comes out of the Hopf point is unstable open circles and then it turns around at a limit point of oscillations at I 0 242 so
169. e a series of such maps An bn along with probabilities pn for choosing each one then this is called an iterated function system IFS Starting with some point x is the plane choose a map apply it to get a new point and plot the point Do this thousands of times and the result will often be an interesting image Barnsley and others have developed image compression algorithms based on these ideas However we will use them to create fractal pictures As an illustrative example that requires no computer lets look at a one dimensional example At each step multiply x by 1 3 Then flip a coin If it is heads add 2 3 otherwise add 0 From this it is clear that the middle third of the interval can never be plotted Similarly the middle third of the left and right intervals can never be plotted Thus the points that are plotted consist of the interval with all the middle thirds removed the Cantor set This is a simple example and you can look at it via the following ODE file cantor ode 4 2 Discrete dynamics in one dimension 63 plots the Cantor set x x 3 2 heav ran 1 5 3 y 5 xlo 0 xhi 1 xp x yp y ylo 4 yhi 6 meth discrete total 40000 1t 0 maxstor 40001 done Notes The first equation is exactly the iteration The term heav ran 1 5 is zero if the random number between 0 and 1 is less than 5 and 1 otherwise The equation for y is a dummy since we want to plot a 1 dimensional set we keep the y coordinate c
170. e can later be altered and varied in XPPAUT Never put spaces between the name value and equal sign For example par a 1 b 2 c 2 big_g 20 number namel valuel name2 value2 defines named parameters with default values These are fixed and cannot be changed in XPPAUT For example 271 number faraday 96485 rgas 8 3147 tabs0 273 15 name x y f x y defines a function of up to 9 variables This function can be used in any right hand side or other functions For example f x if x gt 1 then 1 1n x x 1 e1lse 0 g v w 2v72 1 w w 10 table name filename defines a lookup table called name defined the values found in the file filename This behaves as a one variable function which uses linear interpolation on the values of the tabulated file The file has the structure npts xlo xhi y xlo y xhi The domain of the function is 10 Epa table name npts xlo xhi f t defines a precomputed table called name using npts and evaluating f t at t Ljo hi For example table h 101 O 6 283 sin t 6 4 cos t 2 sin 2 t Tables are automatically re evaluated whenever you change parameters unless you turn off the AUTOEVALUATE flag wiener namei name2 defines a series of normally distributed random variables with mean 0 and standard deviation dt where dt is the internal time step The advantage of using wiener is that when you change the time step the magnitude of these variables is automatically
171. e direction Grab the starting point labeled 1 Run again and when the curve gets fairly close to the vertical axis ABORT the calculation You will see a parabola that intersects the horizontal axis at about d 0 295 This tells us that if the diffusion is too great it is impossible to support rotating waves Similarly as d gets smaller the magnitude of the rotating waves increases they seem to exist in arbitrarily large domains 7 3 1 Homoclinics and heteroclinics A recent version of AUTO includes a library of routines called HOMCONT which allow the user to track homoclinic and heteroclinic orbits XPPAUT incorporates some aspects of this package The hardest part of computing a branch of homoclinics is finding a starting point Consider a differential equation x f x a where a is a free parameter Homoclinics are codimension one trajectories that is they are expected to occur only at a particular value of a parameter say a 0 We suppose that we have computed an approximate homoclinic to the fixed point z which has an n dimensional stable manifold and an n dimensional unstable manifold We assume ns Ny n where n is the dimension of the system The 186 Chapter 7 Using AUTO bifurcation and continuation remaining discussion is based on Sandstede et al The way that a homoclinic is computed is to approximate it on a finite interval say 0 P We rescale time by t Ps We double the dimension of the system so t
172. e nature of the origin Click on Sing pts Go S G and answer Yes to the dialog box The singular point or fixed point or equilibrium is computed If you look in the terminal window from which you started XPPAUT you will see that the eigenvalues are printed and have values of 1 2 11 93 A new window appears which gives a summary description of the stability of the fixed points the nature of the eigenvalues and value of the fixed point From this window you see that the origin is stable and that it has two complex eigenvalues with negative real parts as shown by the text c 2 Change the parameter c from 3 to 3 Draw the direction fields and a flow diagram This is a saddle point since D 8 is the determinant Determine the stability of the origin Click on Sing pts Go and this time answer No to the Print eigenvalues dialog Another message box comes up This asks if you want to draw invariant sets Answer yes to this Then press Enter four times You will see 4 trajectories coming out of the origin They go northeast northwest etc Two are yellow these are the unstable manifolds of the saddle point and since this is a linear system are identical to the eigenvector associated with the positive eigenvalue The other pair of trajectories are in blue they are the branches of the stable manifold and are identical to the eigenvectors associated with the negative eigenvalue When XPPAUT drew these the first two were the unstable manifold
173. e nose makes with the horizontal These equations 8 2 Some first examples 205 are due to Lanchester 1908 and the model is taken from the book by West et al 1997 The equations are Po e KE Rey dt dt The Dv is the viscous drag and works against the speed The term v is the lift The component of gravity along the direction of the plane is mgsin and since the equations are dimensionless we set the coefficients to 1 The term vd6 dt is centripetal force and the cos term is the component of gravity in the angular direction These equations only give the speed and the attack angle We also need to track the position of the plane which is found by integrating the velocity y vcos vsin 0 dy g Ueo q Using Simple glider We can go ahead and solve these equations numerically using XPPAUT and then create an animation file For our first version we will plot a thick line to represent the glider The position on the screen is given by a scaled version of x y and the angle of the line is 0 Here is the ODE file glider ode glider v sin th d v 2 th v 2 cos th v x v cos th scale y v sin th scale par d 25 scale 1 lg 04 these initial data correspond to a nice loop init th 0 25 v 3 init x 1 y 2 some stuff for the animation glider is just a thick line 2 lg length glider xl mod x 1 lg cos th yl mod y 1 lg sin th xr mod x 1 1g cos th yr mod y 1 1g sin th
174. e of the velocity c such that this equation has a traveling wave solution The associated ODE is cU U f U We will shoot from U 1 U 0 forward in time and attempt to hit U 0 U 0 A linearization reveals that both points are saddle points The unstable manifold of 1 0 and the stable manifold of 0 0 are both one dimensional so we expect that there will not be a solution unless c is chosen appropriately Here is the ODE file for this problem bistable ode classic example of a wave joining O and 1 u_t u_xx u 1 u u a cu u u 1 u u a f u u 1 u u a par c 0 a 25 u up up c up f u init u 1 xp u yp up xlo 5 xhi 1 5 ylo 5 yhi 5 done We have written the ODE as a pair of first order equations The initial data is set at 1 0 so that we want the unstable manifold of this saddle point to hit the point 0 0 The window is set up as the phase plane with an appropriate view Now we let XPPAUT compute the one dimensional invariant manifolds Click on Sing Pts Go S G and XPPAUT will quickly ask if you want the eigenvalues printed Click on No Next you will be prompted as to whether you want to compute the invariant sets Since that is the whole goal of this exercise we will answer yes You should see a trajectory drawn off to the top right and you will get an out of bounds message Click Ok and the next invariant set will be computed Again click Ok 6 1
175. e pendulum PERM the frame settext 3 roman BLACK text 2 9 The pendulum line 5 1 9 1 GREEN 2 line 5 5 7 5 GREEN 2 line 7 5 7 1 GREEN 2 TRANS the pendulum line 5 5 5 3 sin theta 5 3 cos theta BLUE 3 fcircle 5 3 sin theta 5 3 cos theta 04 RED done Note that the settext command takes three arguments the text size 0 5 the font roman symbol and a color The default is black roman with size 0 which is really tiny NOTE I have set the animator back to transient mode before the commands for actually drawing the pendulum If you don t do this only the first frame will be drawn Load this animation into the animator and make it go We will introduce one more bell and whistle In small text we will type the current time of the simulation Here is the final version pendulum ani goes with pendulum ode animation for the pendulum PERM the frame settext 3 roman BLACK text 2 9 The pendulum line 5 1 9 1 GREEN 2 line 5 5 7 5 GREEN 2 line 7 5 7 1 GREEN 2 settext 1 roman BLACK TRANS the pendulum 8 2 Some first examples 203 vtext 05 05 t t line 5 5 5 3 sin theta 5 3x cos theta BLUE 3 fcircle 5 3 sin theta 5 3 cos theta 04 RED done I have first set the text back to black and set the size to 1 The command vtext takes four arguments the position on the screen a text string and a formula which will be eva
176. e periodic orbits of period 3 and period 5 on the right 51 Maximal Liapunov exponent as a function of the parameter a for thelogistic map sad raa a aa a EO da 55 Rotation number for the standard map and a expanded view over a small range of the parameter b o a 57 Mandelbrot set oa 2 og e a a 62 Direction fields and sample trajectories for the equation xz x 1 x 69 Phase plane showing trajectories and nullclines for the two dimensional nonlinear example Below the stable and unstable manifolds of the fixed points 4 22 en a Qe Nk ee eee ee be oo 73 Direction fields and nullclines of a predator prey example 75 4 10 4 11 4 12 4 13 6 1 6 2 6 3 6 4 6 5 6 6 6 7 6 8 6 9 6 10 6 11 7 1 7 2 7 3 List of Figures Colorized conservative system Greyscale indicates log of the integral 77 Double well potential oaaae 78 The Lorenz attractor Spay ee eb dy ar ae Es 82 Poincare map for the Lorenz attractor looking at successive values of the maximum of z Period 1 2 and 3 points are illustrated 86 The initial condition window for delay equations 93 Solutions to Ken Cooke s delay equation 94 Three dimensional reconstruction of the chaotic integral equation 96 Solution to the Fabry Perot equations in the chaotic regime 101 The probability of existence of the mutant species as a function of GME e a E RS da E ag ak Ae peas amp oR 106 The Gi
177. e returning the value of the map Thus it is possible to use XPPAUT to continue branches of periodic points of maps rather easily We will thus look at the 8th iterate of the map Fire up AUTO File Auto Click on Axes can choose Hi Lo Fill in the dialog box as follows and then click Ok esis x Nain Parar a 2na Para a om 25 mino oaa ma _ Click on Numerics in the AUTO window and change Dsmax from 0 5 to 0 05 and change Par max from 2 to 4 Now click on Run and you will see a rather messy looking diagram There are three distinct branch points at a 3 3 449 3 544 corresponding to the period 2 4 and 8 bifurcations You should see a figure like Fig 7 5 We can use a similar trick to find a set of period 7 orbits in a two dimensional map the delayed logistic map Tn 1 Yn Un 1 avy 1 i Yn e 7 2 Maps boundary value problems and forced systems 181 Here there is an additional parameter e which will play no role in our analysis At a 2 177 there is a period 7 orbit that starts at x y 1115 12111 We will let AUTO find the entire branch of solutions for this model Here is the ODE file delayed logistic map illustrates a period 7 continuation for the delayed logistic map y x xX axx 1 y eps par a 2 177 eps 0 init x 1115 y 12111 meth discrete total 100 nout 7 autoxmin 2 17 autoxmax 2 21 autoymin 0 autoymax 1 dsmax 025 dsmin 00001 parmin 2 parmax 2 5 done
178. e slider The equations will be integrated Move the slider some more and click on the go button to get another solution The sliders can also be attached to initial data Just choose a variable instead of a parameter 2 5 Looking at the numbers the Data Viewer In addition to the graphs that XPPAUT produces it also gives you access to the actual numerical values from the simulation You can bring up the Data Viewer by clicking on the turquoise button labeled Data at the top of the Main Window The Data Viewer shown in Figure 2 8 has many buttons some of which we will use later in the book The main use of this is to look at the actual numbers 18 0 050000001 0 1 0 15000001 0 2 0 25 0 30000001 0 34999999 0 40000001 0 44999999 0 5 0 55000001 0 560000002 0 64399998 0 69999999 0 75 0 80000001 0 85000002 0 89999998 n9tggagaa Chapter 2 A very brief tour of XPPAUT e A EL 0 9987503 0 9950042 0 9587711 0 9800666 0 9689124 0 9553365 0 9393727 0 921061 0 9004471 0 8775826 0 8525245 0 8253356 0 7960838 0 7648422 0 7316889 0 6967067 0 6599832 0 62161 A E01 cOz1 0 04997917 0 09983341 0 1494381 0 1986693 0 247404 0 2955202 0 3428978 0 3894183 0 4349655 0 4794255 0 5226873 0 5646425 0 6051564 0 6442177 0 6816388 0 7173561 0 7512804 0 7833269 A DIZAT Figure 2 8 The Data Viewer 2 6 Saving and restoring the state of XPPAUT 19 from a simulation The indepen
179. e that you create is separate from the ODE file and for a given ODE you can load in many different animation files experiment with them and change them without having to exit XPPAUT The idea behind the animation files which have an extension ani is to draw simple geometric objects whose coordinates and colors depend on the dynamic variables that you have integrated The cartoon is made by redrawing the picture at each time step after changing the coordinates appropriately Animations can be very simple one liners or complex pictures although perhaps not complex enough to give Disney any competition In this chapter I will take you through the basics of animation show how it can be used for classroom demonstrations research problems and then present a bunch of my own which are based on toys and amusement park rides 8 2 Some first examples 8 2 1 The pendulum A good first example is the nonlinear pendulum which we looked at earlier ml 6 mglsin 8 The ODE file is called pendulum ode Run this with an initial value of 0 3 and 9 0 v 0 0 You should see a nice big oscillation Let s make some physical sense out this Recall that 0 is the angle between the end of the pendulum and the y axis Thus the coordinates of the bob of the pendulum are x y lsin 0 1 1 cos 0 To depict the pendulum we should draw the rod for the pendulum as well as the bob Perhaps we should also draw a stand for the pendulum as w
180. eal root a y 1 0 and the complex conjugate roots x y 1 2 v3 2 Thus we can ask the following question Given an initial guess o yo which of these roots will the guess converge to If we color code the point 29 yo according to the root that it is attracted to the resulting picture looks like that in figure 9 3 This picture can be made with XPPAUT by taking advantage of the ability to vary initial data parametrically Here is the ODE file used to produce the figure cube ode newtons method in the complex plane to find cube roots of 1 z 3 1 x 3 3 xy 2 1 3x 2y y 3 0 draws fractal basin boundaries H HH HH OH tolerances par eps 001 discretization of the phase space par dx 0 dy 0 the three roots par ri 1 s1 0 par r2 5 s2 8660254 par r3 5 s3 8660254 the functions f x73 3 x y7 2 1 g 3x xx 2 y y73 the derivatives of the functions fx 3 x72 y72 fy 6 x y gx 6 x y gy 3 x 2 y 2 the Jacobian used in the inverse det fx gy fy gx a euclidean distance function dd x y x x y y if close to the root then plot otherwise out of bounds for each root 242 Chapter 9 Tricks and advanced methods Figure 9 3 The basin boundaries for the iteration which arises from the application of Newton s method to the equation 23 1 in the complex plane Each point in the plane is colored according to the root that it converges to aux xp 1 3 1 if dd x r jl y s j1 lt ep
181. eb page and save it as fhn ode 2 7 1 Direction fields Run this by typing xpp fhn ode The usual windows will pop up One of the standard ways to analyze differential equations in the plane is to sketch the direction fields Suppose that the differential equation is x f x y y g x y The phaseplane is divided into a grid and at each point in the grid x y a vector is drawn with xz y as the base and 1 sf zx y y sg x y as the terminal point where s is a scaling factor This so called direction field gives you a hint about how trajectories move around in the plane XPPAUT lets you quickly draw the direction field of a system Click on Dir field flow D irect Field D D and then accept the default of 10 for the grid size by clicking Enter A bunch of vectors will be drawn on the screen mainly horizontal They are horizontal because e is small so that there is little change in the w variable The length of the vectors is proportional to the magnitude of the flow at each point At the head of each vector is a little bead If you want to have scaled direction fields which don t take into account the magnitude of the vector field just click on Dir field S caled Dir Fld D S and use the default grid size I prefer pure direction fields but this is a matter of taste Click on Initialconds m I ce and to experiment with a bunch of different trajectories Note how the vectors from the direction field are tangent to the trajec
182. ee the curve go up and to the left It crosses the right line of fold points and terminates on the left line The intersection with the right line of fold points is irrelevant since the Hopf occurs on the upper branch of solutions and the right fold line represents the loss of the lower branch of fixed points The termination of the curve of Hopf points on the left line of folds signifies a new higher order bifurcation point the Takens Bogdanov bifurcation There is a double zero eigenvalue the Hopf bifurcation and the saddle node have coalesced This bifurcation often signals the presence of homoclinic orbits For phi below this curve and between the two vertical lines the upper branch of fixed points is unstable Outside the vertical lines there is only one fixed point On the right it is stable above the Hopf curve and unstable below On the left the fixed point is always stable For example choosing I 15 phi 5 is in the regime where there is one fixed point and it is unstable Since you can readily prove that all solutions stay 7 1 Standard examples 177 bounded for this model this implies that there must be at least one stable periodic solution Look at the phaseplane in this case and verify that you are right Here are some hints for computing the complete diagram for a system e ALWAYS make sure you are starting at a fixed point or clearly defined limit cycle For fixed points click on Initialconds Go and then Initialconds Last a
183. ell say something 199 200 Chapter 8 Animation like Figure 8 1 Let s do this in steps First we will draw the rod and then gradually build on this basic picture Here is an animation file for the rod pendulum ani goes with pendulum ode animation for the pendulum line 5 5 5 3 sin theta 5 3 cos theta BLUE 3 done The first two lines are comments and the last line tells the animator that the file is done Thus there is only one real line of code Before discussing this line I want to set out the coordinate frame The animation window is the unit square by default so that 0 0 is at the lower left and 1 1 the upper right Thus the coordinate 0 5 0 5 is in the middle This will be our pivot point for the pendulum We will scale the length of the pendulum so that it is 3 10 of the length of the frame This is more a matter of aesthetics however the animation looks pretty ugly if the drawings go out of the square since there is no clipping done All coordinates and arguments for the drawing commands are separated by semicolons The line x1 y1 x2 y2 command draws a line from x1 y1 to x2 y2 Thus the command tells the animator to draw a line from 0 5 0 5 to 0 5 0 3 sin 0 0 5 0 3 cos 6 at each time step using the value of the variable 0 to compute the coordinates The last two arguments of the line command tell the animator to make the rod blue and draw it with a line that is three pixels wide In the Mai
184. ems called normal forms We briefly mention the main bifurcations of interest 7 1 Standard examples 167 e Bifurcations at a zero eigenvalue for differential equations When Ao has a zero eigenvalue then new fixed points often arise These can be transcritical pitchfork or saddle node bifurcations Only the latter is generic the other two occur when there is symmetry The transcritical and pitchfork are branch point bifurcations and AUTO will automatically detect and follow them The saddle node or fold or limit point bifurcation occurs when a branch of solutions bends around e The Hopf bifurcation This occurs when Ap has a pair of imaginary eigenval ues Generically a small amplitude limit cycle will emerge from the branch of fixed points e Period doubling bifurcation This occurs for maps when an eigenvalue of Ag is 1 For a limit cycle the result is that the period of the limit cycle doubles e Torus bifurcation When the eigenvalues of Ao for a map cross the unit circle at values other than 1 7 or the cube roots of 1 then a torus bifurcation arises Chaos and other complex behavior often appear AUTO detects all of these bifurcations and in some cases can create a two parameter curve along which the bifurcation occurs In the next few sections we show how to use XPPAUT and AUTO together on a variety of differential equations maps and boundary value problems 7 1 Standard examples As a first example we c
185. ends on other parameters It is only evaluated when the parameters are changed The second notable point is that I have added auxiliary variables for the square magnitude of z If you make h small you will probably have to switch to a stiff integrator Run this as is and array plot rr0 You will see the interesting pattern shown in Figure 6 8 6 2 3 Special operators XPPAUT has a number of operators that make it easy to solve discrete convolution equations that arise in physical systems Here we will numerically solve a neural network model with delayed inhibition The model has the following form duj Sl Y WO Dult em uj t 7 6 4 i 0N This equation arises as a discrete approximation of a continuous convolution equa tion e g W x y u y dy 6 2 Partial differential equations and arrays 147 We could use the XPPAUT sum operator to evaluate this discrete convolution with the code sum 0 N of w j i shift u0 i but this will be quite slow and would be a real waste if for example the function W j vanished for j large enough Recall that shift u0 n returns the value of the nt variable defined after the variable u0 Suppose that W j is nonzero for j lt m Then we could rewrite the code fragment as sum m m of W i shift u0 j i provided we knew how to treat the shift u0 i when i is negative or exceeds the length of the array of variables XPPAUT provides a fast way to ev
186. ential equations The stability of the particular branch of solutions is readily obtained by analyzing the linearization This too is automatically accomplished by AUTO If a fixed point or limit cycle exhibits a change in its stability this is often a sign that new qualitatively different behavior could happen For example new fixed points might emerge from the branch of solutions or a limit cycle may emerge These qualitative changes in the local and global behavior are called bifurcations and their detection 165 166 Chapter 7 Using AUTO bifurcation and continuation during continuation is the subject of much mathematical research AUTO provides a number of tools for the automatic detection of bifurcations of fixed points and limit cycles A thorough description of local and global bifurcations can be found in the excellent book by Kuznetsov Consider a differential equation of the form de F x u TER 7 1 dt where u is a parameter Suppose that 7 1 has a fixed point xo o Let A uo Ao be the linearized matrix for F with respect to x about the fixed point xo po There are two important insights gained by looking at Ao First if Ao is invertible then we can follow the fixed point xp as a function of the parameter u in some neighborhood of uy and this is the only fixed point near x9 This fact is a consequence of the implicit function theorem The second point is that if none of the eigenvalues of Ap lie on the imagina
187. er XPPAUT will not use the Volterra solver if it cannot find any integrals so we will define one that fools the program With this in mind here is the ODE 100 Chapter 5 More advanced differential equations fp ode fabry perot cavity init tau 1 x 1 08 needed to fool XPP into using the Volterra solver junk int 0 x ode for the delay tau v ctdelay v tau x0 xsta sin th 2 1 cos th 2 2 cos th cos xs x v v v q x x0 ax fr fr fi fi sin th 2 volt fr 1t cos th cos delay x tau delay fr tau sin delay x tau delay fi tau volt fi 1t cos th sin delay x tau delay fr tau cos delay x tau delay fi tau par q 11 xs 1 07 a 11 459 th 1 r 1 2221 c 120 eps 1 dt 05 total 200 trans 50 delay 4 vmaxpts 1 done Most of this file is pretty straightforward I fool XPPAUT into using the Volterra solver with the dummy integral junk int 0 x The volt declaration tells XP PAUT to treat these two equations as though they were Volterra integral equations I have added the statement vmaxpts 1 which tells XPPAUT to change the max imum number of points stored by the Volterra integral solver to 1 which speeds everything up since no time is wasted evaluating the dummy integral You can also change this number within XPPAUT with the nUmerics Method Volterra command type in 1 when prompted for MaxPoints Integrate the equations Plot fr fi v in a nice three d plot to see the
188. es condition A again and so on Hitting Escape many times will sometimes work but occasionally you need to kill XPPAUT All curves used When you freeze a plot there are only 26 per graph window Delete some 279 280 Appendix F Error messages Bad condition Ignoring occurs in the computation of histograms If the formula for the condition is bad XPPAUT just ignores it Out of memory Time to upgrade Out of film In the Kinescope command only so many screen shots can be saved No prior solution You tried the Initialconds Last command without first computing a solution with new initial data No shooting data available You have to find a fixed point to your system with a one dimensional invariant manifold before you can use this Incompatible parameters You tried to load a saved set file that is not associated with the ODE file you are running F 1 Cheat sheet F 1 Cheat sheet FILE FORMAT comments d lt name gt dt lt formula gt lt name gt lt formula gt lt name gt t lt formula gt volt lt name gt lt formula gt lt name gt t 1 lt formula gt arrays x n1 n2 j j 1 markov lt name gt lt nstates gt t01 t02 tOk 1 t10 tk 1 0 tk 1 k 1 aux lt name gt lt formula gt parameters defined as formulae lt name gt lt formula gt lt name gt lt formula gt parameter lt namel gt lt valuei gt lt name2 gt lt value2 gt wiener lt namei gt
189. esolve this better Click on the Window zoom Zoom W Z command and zoom by clicking the mouse into an area which contains all the the equilibria A good window is 5 4 X 2 5 Click on Erase to erase the screen Redraw the nullclines and the direction fields N N D Enter Enter You can now see clearly that 74 Chapter 4 XPPAUT in the classroom there are 4 fixed points Now lets assess the nature of the fixed points indicated by the intersections of the nullclines Click on Sing Pts Mouse and click with your mouse on the upper right fixed point Answer No to the question about eigenvalues and Yes to the question about invariant sets A yellow trajectory one branch of the unstable manifold is drawn and a window appears warning you that the trajectory is out of bounds Press any key and the other branch of the unstable manifold is drawn Press a key and a branch of the stable manifold blue is drawn Press the Esc key to get the other branch You should see four trajectories for this saddle point two in yellow the unstable manifolds and two in blue the stable manifolds Repeat this at the other 3 fixed points two more of them are saddles and the fourth one is an unstable vortex You should see something like Figure 4 8 In the figure for clarity I have split the picture into two parts showing the flow and nullclines to the left and the stable and unstable manifolds to the right Arrows go out of all the yellow orange trajector
190. ess obscure After the iteration s contains the number of 1 s We then just get M by modding by 2 Here is a fragment of XPPAUT code that implements this iteration s0 0 x0 n 11 16 s j s j 1 mod x j 2 x j flr x j 1 2 These are all fixed variables so they are evaluated at every time step The block delimited by the does the iteration 16 times so that this will work as long as n lt 216 65536 The quantity s16 has the desired sum We have used the fact that the fixed variables are evaluated in the order in which they are defined Here we have defined 34 fixed variables With this bit of code the rest of the ODE file is easy Here is mt ode mt ode the morse thule sequence of 0 s and 1 s the sum mod 2 of the 1 s in the binary expansion of the integers look at the power spectrum for z init n 0 s0 0 x0 n 1 16 s j s j 1 mod x j 2 x j flr x j 1 2 n n 1 z mod s16 2 aux ss s16 meth discrete total 10000 maxstor 68000 bound 100000 238 Chapter 9 Tricks and advanced methods xlo 0 xhi 10000 ylo 0 yhi 16 yp ss done Run this ODE file and look at the nice self similar nature of the sums which are plotted The sequence itself is just the auxiliary variable ss Now the really cool part of this is to look at the power spectrum of the sequence ss Click on nUmerics stocHast Power and choose ss as the variable to transform Click on Esc to get
191. et up by using the x n m construction which defines m n 1 variables xn xm Thus in this example u50 would be the variable in the middle of the domain The name of the array variable is j and thus u j 1 defines the variable u51 if j 50 In the present example we have set the boundary conditions to be u 0 t 1 and uz L t 0 The initial data are all zero in this case Once one deals with PDEs there are a number of practical issues that have to be considered First you certainly do not want to type in all the initial condi tions Second you may want to look at spatial profiles rather than temporal ones Third you may want to look at the entire space time picture XPPAUT has some commands that make this pretty easy to do Run XPPAUT with this file Let s put initial data in that has the form u x 0 cos rx L Since there are 100 el ements then the initial data is u_j cos pi j 100 To input this you use the Initialconds formUla I U command At the prompt for the variable type in u 1 100 to indicate you want to set initial data for u1 u100 Then when asked for the formula type in cos pi j 100 Don t forget to put brackets around j and hit Enter Since there are no other variables whose initial data you need to solve click on Enter again The equations will be integrated Let s get a big picture of the entire space time behavior To do this we use the Array plot options for XPPAUT Click on Viewaxes Array V
192. eted as k10 which makes no sense since k is actually a function However k j is interpreted correctly since j is evaluated for j 0 100 as the integer j The array that results from this convolution is zero offset so that the first element is k 0 Run this file and integrate it Plot the array of values from u0 to u100 using the array plot to see the wavefront Either use the short cut by checking the boxes for u0 and u100 in the Initial Data Window and then clicking on the Arry button or use the Graphics Array command and fill in the dialog box In the Main Window plot a few spatial profiles such as u10 u30 u60 and measure the times of crossing u 0 5 for u30 u60 in order to measure the velocity of the wave We now turn to the convolution problem 6 4 We will define the weights as follows W exp ell aexp bl where c a b are positive parameters Once again here are the steps 1 Define the table of weights using the variable t as the dummy table w 21 10 10 exp c abs t a exp b abs t 2 Apply whatever function you want to the variables you want to convolve Optional 3 Define the convolution with the special operator special k conv periodic 101 10 w u0 Here I use periodic conditions on the array 4 Define the right hand sides using k j u O 1007 u j ce k j ci delay ul j tau Here is the complete ODE file nnetdelay ode a neural net with delays table w 21
193. ets Let s make a whole bunch of Julia sets and animate the result as the parameter varies We set cx 0 and vary cy from 1 to 1 First change the total number of iterates back to 2000 Click on Initialconds Range I R and fill in the dialog as follows Range over ey Notice that you should type in Y in the Movie entry since this tells XPPAUT you want to save the contents of the window Also make sure that no windows other than perhaps the dialog window obscure the main graphics window Click on Ok and 20 Julia sets will be computed Each screen image is saved in memory so that you can play them back Now click on Kinescope Playback K P and you can step through each of the plots one at a time by clicking on a mouse button Click Esc to exit To get a smooth animation of the Julia sets click on Kinescope Autoplay K A The tell XPPAUT how many times you want to repeat the animation e g 2 and how many milliseconds between frames e g 100 After this the animation will be repeated as a smooth movie You can save the animations in several ways One of these requires that you have a separate software program that can encode a series of still images into a movie An easier way but one which is not quite as efficient is to create an animated GIF Animated GIF s can be played on most web browsers such as Netscape or Explorer Click on Kinescope Make Anigif and your movie will be replayed At the same time a file called anim gif will
194. ework o e a a a o o a More advanced differential equations Functional Equations e 5 1 1 Delay equations 5 1 2 Integral equations 5 1 3 Making 3D movies 00 5 1 4 Singular integral equations 5 1 5 Some amusing tricks 5 1 6 Exercises cua catar Poke a ls A A E e Stochastic equations o o e e eee eee 5 2 1 Markov Processes o o 5 2 2 Gillespie s method 00 4 5 2 3 Ratchets and games o o 5 2 4 Spike time statistics o 5 2 5 Exercises u lt a de A Ya eee 5 3 Differential algebraic equations o o Contents vii 5 3 1 Exercises af acs at ee te MO BA Oe Be 121 6 Spatial problems PDEs and boundary value problems 123 6 1 Boundary value problems o 123 6 1 1 Solving the BVP 126 6 1 2 Infinite domains 130 6 1 3 Exercises ans Dada ee A a RRS 135 6 2 Partial differential equations and arrays 139 6 2 1 An animation fle 143 6 2 2 The issue of stiffness o 144 6 2 3 Special operators o o e 146 6 2 4 Exercises uan a a ee Be we EG ede 160 7 Using AUTO bifurcation and continuation 165 7 1 Standard examples e 167 7 1 1 Aylumitraycler eat o a a 170 7 1 2 A real
195. ex1 gt lt ex1 gt are evaluated and their integer parts are used as the lower and upper limits of the sum The index of the sum is i so that you cannot have double sums since there is only one index lt ex3 gt is the expression to be summed and will generally involve i For example sum 1 10 of i will be evaluated to 55 Another example combines the sum with the shift operator sum 0 4 of shift u0 i will sum up u0 and the next four variables that were defined after it 11 12 13 Bibliography http www cs ruu nl wais html na dir sci nonlinear faq html con tains a lengthy list of software packages as well as some comments on the content J M Aguirregabiria Dynamics Solver http tp 1c ehu es jma ds ds html P Blanchard R L Devaney and G R Hall Differential Equations Brooks Cole 1998 R L Borrelli and C S Coleman Differential Equations A Modeling Perspec tive John Wiley 1995 J Bower and D Beeman The Book of GENESIS Telos 1997 K E Brenan S L Campbell and L R Petzold The Numerical Solution of Initial Value Problems in Differential Algebraic Equations Elsevier Science Publishing Co 1989 S D Cohen and A C Hindmarsh CVODE 1 0 ftp sunsite doc ic ac uk Mirrors netlib att com netlib ode E Doedel AUTO ftp ftp cs concordia ca pub doedel auto Ermentrout G B and Kopell N 1991 Multiple pulse interactions and aver aging in systems of coupled neural
196. example o 173 7 1 3 EXCrGiS S mamasita da a a eee 177 7 2 Maps boundary value problems and forced systems 179 7 2 1 Maps ll la a a Bk 4 4 A 179 7 2 2 EJXETCISOS uk was e a o ae da 182 7 3 Boundary value problems o o e e 182 7 3 1 Homoclinics and heteroclinics 185 7 3 2 Exercises i b 4 3 4 Ole edo 4 BAA A eee 192 7 4 Periodically forced equations o o o 193 7 4 1 EX6rcisesi e a a eee ee ek A 195 7 5 Importing the diagram into XPPAUT 195 7 5 1 EXCTCISC 3 4 eee hoe e ee ad Y 197 8 Animation 199 8 1 Introduction 2 s ss Sac eae bh des SNA Bah as 199 8 2 Some first examples o 199 8 2 1 The pendulum o 199 8 2 2 Important notes o 2 000 203 8 2 3 A spatial problem revisited 204 8 2 4 A glider and a fancy glider 204 8 2 5 Coupled oscillators 207 8 3 My favoritesy ee 4 4 dad cda Pe e a Soe Ee Be ae Gd 209 8 3 1 More pendula Aaa ee BA ee ob Eke 209 8 3 2 A roller coaster o o e e 212 8 3 3 Th Zipper s td Ne he he ecard rad de 213 8 3 4 The Lorenz equations 216 8 4 The animator scripting language o 218 9 Tricks and advanced methods 221 9 1 Introduction seana aaa a de 221 viii Contents 9 2 Graphics trieka 4 s 2 feo o e ee eed AAA ee 221 9 2 1 Better
197. example consider the logistic map In41 atrp 1 n f z a with parameter a If we were to run this with AUTO it would say that at a 3 there was a Hopf bifurcation with period 2 Since this is really a period doubling bifurcation we could try to follow it However AUTO cannot continue from Hopf points of maps Thus how could we follow this apparent period 2 point Since this period two point arises from the primary branch of solutions if we look at f x a f f x a a then the period two orbit is just a branch of the second iterate and AUTO will follow it Thus if we wanted to find all the periodic orbits dividing 8 we could just look at the 8th iterate of the map with a as the parameter Here is our version of the logistic map logisticmap ode the classic logistic map x t 1 a x 1 x 180 Chapter 7 Using AUTO bifurcation and continuation A 08 A e os E 06 AAA 06 E o4 F o2 F o2 F Figure 7 5 Bifurcation diagram for period 8 orbits of the logistic map left and period 7 orbits for the delayed logistic map right par a 2 8 init x 64285 total 200 meth disc done Run XPPAUT with this We have started pretty close to the fixed point Click on nUmerics in the main XPPAUT window and change nOutput from 1 to 8 and exit the numerics menu When AUTO is run within XPPAUT it looks at the numeric parameter nOut and uses this to iterate the map nOut times befor
198. f Volterra equations 5 1 1 Delay equations We will start with some delay equations Consider the delayed logistic equation d AT rz t 1 z t 7 where 7 gt 0 is a delay The way to write this in XPPAUT is to use the delay function delay x r returns z t r where r is a non negative number Here is the ODE file 91 92 Chapter 5 More advanced differential equations delx ode delayed logistic equation x r x l delay x tau par r 2 tau 0 init x 5 aux dlx delay x tau total 50 delay 10 yhi 6 xhi 50 done NOTES I have added an extra plottable quantity d1x which represents the delayed value of x I have also added the line total 50 delay 10 yhi 6 xhi 50 This tells XPPAUT to integrate for a total of 50 time units the maximum that the delay will ever be is 10 set the plot maximum for the y axis to 6 and set the maximum for the x axis to 50 XPPAUT by default sets the maximum delay to 0 You can also set this maximum from within the program using the nUmerics dElay command U E to set the delay Start this up and integrate it Then change the parameter 7 to 1 Integrate it again note the oscillation Try a smaller value of 7 Find the critical value of T above which the constant steady state is unstable You can do this automatically in XPPAUT XPPAUT can track the stability of fixed points of delay equations and will attempt to find a positive eigenvalue if it exists Try this set the initial
199. f the front plot U against t Tap X and choose U instead of V and press Enter Figure 6 3 shows a complete picture of the unstable and stable manifolds for the fixed point 1 0 including the trajectory that represents the traveling front Arrows indicate the directions these are added using the Text Arrow command see Chapter 9 134 Chapter 6 Spatial problems PDEs and boundary value problems The bistable reaction diffusion equation The Fisher equation is in a sense rather trivial as the wave exists over a whole range of parameters In most systems you must choose the velocity exactly right This requires that you repeat the steps above many times as you attempt to home in on the correct parameter The key is that you want to find a parameter that is below the correct one and one that is above the correct one How can you tell In most one parameter shooting problems choosing the parameter low will lead to the trajectory going in one way and choosing it high will lead to it going in another way Thus you can usually decide if you are too high or too low I will illustrate this with the bistable reaction diffusion equation Ut Ure f u 6 1 where f u has three roots 0 lt a lt 1 with f 0 lt 0 f 1 lt 0 and f a gt 0 Fife and McLeod proved the existence uniqueness and stability for traveling front solutions to this equation where u x t U x ct U oo 1 and U oo 0 That is there is a unique valu
200. f the animation win dow are 0 0 and 1 1 respectively The 0 99 notation is new this just tells XPPAUT to repeat this line 100 times with a dummy index going from 0 to 99 See chapter 6 for more details on this notation Thus 100 lines will be drawn The first coordinate of the first line is 0 the first coordinate of the second line is 1 01 and so on The vertical coordinate of the first line is ff 0 since j refers to the index of the current line which is 0 The last line in this 100 line sequence is expanded by XPPAUT to be line 99 01 ff 99 100 01 f 100 RED 2 The function ff u is defined in the ODE file Thus this code simply draws the function f x The next line of code tells the animator to draw the horizontal axis scaled to be at y 0 The code transient tells XPPAUT all that follows should be animated A filled circle is drawn at the scaled x coordinate of the dynamical system with radius 02 units that is 2 of the whole animation window and it is colored blue Click on View axes Toon V T to pop up the animation window Click on File in the animation window and load the animation file phase line ani Unless there is an error you should be ready to run the animation If there is an error then you did not type in the animation file correctly Click on the Go button in the animation window and the graph of f will show up along with a little ball showing the progress of the dynamical system Notes
201. f the invariant manifolds emerging from the saddle point The dynamics under the flow of the equation is such that points get all mixed up like cake batter being stirred We can see this happen as follows Choose a region in the phase plane Color each point in the phase plane black white if the solution with that initial data is in the x gt 0 x lt 0 half plane at some fixed time to Do this for a variety of different times to see how the flow mixes everything up This is like the basin boundaries we computed above but only for finite times The XPPAUT file is like the one in the previous section but simpler duffbas ode a trick to compute the basin boundaries of the duffing equation better be patient it takes awhile here is the ODE x y y 15xy bx x 1 x72 f cos 8333 t dx dy are the increment sizes f is the forcing amplitude par dx dy f 1 here are the initial data evaluated once per cycle initial data lie in the square 2 4 2 4 x 1 2 1 2 x0 2 4 dx 4 8 yO 1 2 dy 2 4 save the initial data at different time slices aux xp 1 10 if x gt 0 amp abs t 2 j lt 1 then x0 else 100 aux yp 1 10 if x gt 0 amp abs t 2 j lt 1 then y0 else 100 set initial data with parameter glob 0 t x x0 y y0 set some options total 22 xp xp5 yp yp5 xlo 2 4 ylo 1 2 xhi 2 4 yhi 1 2 1t 1 maxstor 500000 trans 1 dt 2 meth 8 done The lines that start with are treated l
202. f the stochastic membrane model is persistent sodium current ala Gillespie napgill ode we use the exponential integration method c gt o rate a o gt c rate b definition of a b a xinf tau 1 b tau 1 a xinf 5 1 tanh v vxt vxs tauxi phix cosh v vxt 2 vxs a xinf tauxi b tauxi a the cumulative probabilities pi a xn o p2 pitb o which reaction s2 ran 1 p2 z1 s2 lt p1 5 2 Stochastic equations 111 Figure 5 7 An asymmetric periodic potential when is the next reaction dt log ran 1 p2 the fraction of open channels x 0 xn the total conductance gv gl gkx w gnax x the reversal potential vrev gl vl gk w vktgna x vnati gv that is v gv v vrev init v 78 w 3 here is the exponential method v vrev v vrev exp gv dt w winf v w winf v exp dt tauw v update reaction time tr tr dt and the channels o max 1 0 2 z1 1 parameters par xn 100 par i 0 gk 10 gl 1 gna 2 5 vk 85 v1l 70 vna 120 par vxt 52 vxs 15 vwt 65 vws 20 phix 10 phiw 1 functions winf v 5 1 tanh v vwt vws tauw v 1 phiw cosh v vwt 2 vws some XPP settings meth discrete total 20000 bound 10000000 njmp 10 xlo 0 xhi 200 xp tr yp v yhi 20 ylo 85 done Run this Change the total number of channels xn as well as the total number of reaction steps Since more channels means that the rates of reacti
203. fcircle 05 9x 0 99 100 05 8 u j 01 done Recall that the 0 99 statement is expanded by XPPAUT into 100 lines and each j is substituted by the corresponding integer Thus this animation code represents a long file fcircle 05 9 0 100 05 8 u0 01 fcircle 05 9 1 100 05 8x u1 01 fcircle 05 9 99 100 05 8x u99 01 I have scaled everything so that it lies in the window Load this animation file and look at the result In fact this is not an optimal way to view the time evolution It would be much better to draw lines between the points This is easy as well with the one line file nnetdelay2 ani use with nnetdelay ode a one liner line 05 9 0 98 100 05 8 u j 05 9 j 1 100 05 8 u j 1 BLACK 2 done Note that I only go from 0 to 98 rather than 0 to 99 This draws a black line with thickness 2 between successive x y values with x the spatial grid point and y the magnitude of u Load this animation file and try it 8 2 4 A glider and a fancy glider The little balsa wood glider that is popular with everyone is readily modeled as a pair of differential equations I will not go through a complete derivation of the equations Consider a glider that is moving in the a y plane so that it can only tilt up and down and cannot veer out of the plane The dimensionless equations are for the speed v a scalar quantity which is the magnitude of the velocity and the attack angle 0 the angle th
204. ff the plotting of the X and Y axes Click Ok when you are done Now create a postscript file Graphic stuff P ostscript G P and accept all the defaults Name the file whatever you want and click on Ok in the file selection box Figure 2 10 shows the version that I have made Yours will be slightly different If you want to play around some more turn off the automatic freeze option Graphic stuff Freeze Off freeze G F 0 and delete all the frozen curves Graphic stuff Freeze Remove all G F R 2 1 3 Command Summary Nullcline New draws nullclines for a planar system N N Dir field flow D irect Field draws direction fields for a planar system D D Sing pts Mouse computes fixed points for a system with initial guess specified by the mouse S M Sing pts Go computes fixed points for a system with initial guess specified by the current initial conditions S G Graphic stuff Freeze On Freeze will permanently keep computed trajecto ries in the current window G F O Graphic stuff Freeze Off Freeze will toggle off the above option G F O Graphic stuff Freeze Remove all delete all the permanently stored curves G FR Graphic stuff aXes opts lets you change the axes G X Viewaxes 2D allows you to change the 2D view of the current graphics window and to label the axes V 2D 2 8 The most important numerical parameters XPPAUT has many numerical routines built into it and thus there are many nu merical paramete
205. file to determine the time points of the simulation These should be in increasing order but are not required to be equally spaced Once you have put the values into the dialog box click on Ok and a bunch of stuff will flash by in the terminal window After a few seconds a message box should pop up that says Success This means convergence was achieved and a minimum was found However notice that in the Parameter Window two of the kinetic constants ka k4 are negative This is non physical So what can one do about this The easiest strategy is to set them both to zero or some very small positive number Then redo the fit only this time do not include them as parameters in the fitting algorithm they will be left alone Click on stochastic fIt data 226 Chapter 9 Tricks and advanced methods 0 L Le Figure 9 1 Curve fit to experimental kinetics model Top show the initial guess and bottom shows the parameters found by curve fitting again and edit the Params entry by deleting k2 k4 from the list and clicking Ok Again the fit will run successfully This time no physical constraints are violated Escape from the numerics menu and click on Initialconds Go and look at the new solutions you have computed They are much better Figure 9 1 shows the results of the initial and final choices of parameters 9 4 The data browser as a spreadsheet The Data Viewer has a number of interesting features that allow
206. floating point num ber it can be a formula that depends on the variables In all the commands the color is optional except settext The other way of describing color is to use names which all start with the symbol The names are WHITE RED REDORANGE 0RANGE SYELLOWORANGE YELLOW YEL LOWGREEN GREEN BLUEGREEN BLUE PURPLE BLACK The number following the speed declaration must be a nonnegative integer It tells the animator how many milliseconds to wait between pictures Thus it is really anti speed The dimension command requires 4 numbers following it They are the coordinates of the lower left corner and the upper right The defaults are 0 0 and 1 1 The settext command tells the animator what size and color to make the next text output The size must be an integer 0 1 2 3 4 Y with 0 the smallest and 4 the biggest The font is either roman or symbol The color must be a named color and not one that is evaluated The remaining ten commands all put something on the screen e line x1 y1 x2 y2 color thickness draws a line from x1 y1 to x2 y2 in user coordinates These four numbers can be any expression that involves variables and fixed variables from your simulation They are evaluated at each time step unless the line is permanent and this is scaled to be drawn in the window The color is optional and can either be a named color or an expression that is to be evaluated The thickness is also optional but if y
207. function A A in a big square in the right half plane From the argument principle we can thus obtain the number of roots in the contour and thus the sta bility of the fixed point XPPAUT then uses Newton s method to find the root of A A closest to 0 C 2 2 The Volterra Integrator This is based on a method in Peter Linz book on solving integral equations It is an implicit first order method I will show how it works for the scalar integral equation u t f t f F t s u s ds Let t hn where h is the time step Then we can write the discretization of this as n 1 Un fnth 5 F nh nj uj hF nh nh un j 0 Note that un appears on the right hand side Thus we have to once again use New ton s method to find un This method is stable for the same reason that backward Euler is stable Consider the singular integral nh P nh s u s A f nh s g where 0 lt r lt 1 Then h P nh s u s 5S era j 0 jh 1 D F nh nj u nj nh jh 5 nh G 1 h 5 gt j 0 This trick changes the singular integral into a regular sum C 3 How AUTO works AUTO is capable of finding fixed points of systems and tracking them as a parameter varies as well as tracking solutions to differential equations In the latter case the C 3 How AUTO works 267 problem is always regarded as a boundary value problem and as such is always solved on the unit interval Consider first the solutions to algebraic p
208. function using the original variable names for the unshifted terms and primed names for the shifted terms 9 5 3 Phase response curves One of the most useful techniques available for the study of nonlinear oscillators is the phase response curve or PRC for short The PRC is readily computed ex perimentally in real biological and physical systems The PRC is defined as follows 9 5 Oscillators phase models and averaging 231 Suppose that there is a stable oscillation with period T Suppose that at t 0 one of the variables reaches its peak This can always be done by translating time At a time 7 after the peak one of the state variables is given a brief perturbation that takes it off the limit cycle This will generally cause the next peak to occur at a time T r that is different from the time it would have peaked in absence of the perturbation The PRC A r is defined as A T 1 T r T If A T gt 0 for some 7 we say that the perturbation advances the phase since the time of the next maximum is shortened by the stimulus Delaying the phase occurs when A r lt 0 Experimental biologists have computed the PRCs for a variety of systems such as cardiac cells neurons and even the flashing of fireflies We will use XPPAUT to compute the PRC for the van der Pol oscillator 2 4 1 27 subjected to a square wave pulse of width o and amplitude a Here is how we will set up the problem We will let 7 be a variable rather
209. g pt ctl U I and change the Newton tolerance from 0 001 to le 8 Exit the numerics menu and integrate the equations again This is much more reasonable The main point of this little exercise is that you cannot always believe the numerics and if there is a surprising result then you must take every precaution to be sure that it is not an artifact Sometimes you have a fully nonlinear equation of the form dx F 0 Rewrite this as d L F q TY 0 F y z Consider a simple model for an artificial liver device The model is a one dimensional PDE The steady state is thus two point boundary value problem w 0 Use Vus w u 0 ki y Uai with boundary conditions u 0 uo uz L 0 We could solve this for w and the result is a rather cumbersome expression Instead we let XPPAUT do the work We choose L 5 in this example We first write this as a system of first order equations Here is the file for the BVP dae_ex3 ode a boundary value DAE model u up up v up lamux u w DAE stuff O k w 1 w lamv u w solve w 235 some extra stuff init u 1 up 34 aux WW w aux z k w 1 w lamv u w par k 1 lamu 25 lamv 25 v 2 u0 1 bdry u u0 bdry up total 5 001 jac_eps 1e 5 newt_tol 1e 5 dt 005 done To solve this click on Bndryval Show and watch it converge to a solution 5 3 Differential algebraic equations 121 5 3 1 Exercises 1 In this example you can watch
210. gs for many versions of the program that have appeared throughout the years Finally I want to thank my wife Ellen for her patience and the boys Kyle and Jordan without whom this book would have a 2000 copyright Chapter 1 Installation Like all Holme s reasoning the thing seemed simplicity itself when it was once explained Arthur Conan Doyle The Stock broker s Clerk Installation of XPPAUT is done either by downloading the source code and compiling it or downloading one of the binary versions I will give sample installa tions for UNIX Windows and MacOS X If you are totally clueless at compiling source code it is best to either have your system administrator install it for you or download a precompiled binary for your computer There are compiled versions available for Linux SUN HP Windows and Mac OSX 1 1 Installation on UNIX 1 1 1 Installation from the source code Create a directory called xppaut and change to this directory by typing mkdir xppaut cd xppaut Step 1 Download the compressed tarred source code xppaut latest tar gz into this directory from one of the two URLs e http www math pitt edu asympbard xpp xpp html e http www cnbc cmu edu xbard files html Step 2 Uncompress and untar the archive gunzip xppaut_latest tar gz tar xf xppaut_latest tar This will create a series of files and subdirectories 2 Chapter 1 Installation Step 3 Type make and lots of things will scroll by in
211. hat we can simultaneously solve for the equilibrium point as the parameters vary We want to start along the unstable manifold and end on the stable manifold Let Lu be the projection onto the unstable subspace of the linearization of f about the fixed point and let Ls be the projection onto the stable space Then we want to solve the following system dx ds dLe ds f xe 0 Ls x 0 ze 0 Lu x 1 ze 1 Pf x a II 7 10 Io l O O OLS Note that there are 2n differential equations and 2n boundary conditions n for the equilibrium ns at s 0 and n at s 1 There is one more condition required Clearly one solution to this boundary value problem is x s xe s 7 which is pretty useless However any translation in time of the homoclinic is also a homoclinic so we have to somehow define a phase of the homoclinic Suppose that we have computed a homoclinic s Then we want to minimize the least squares difference between the new solution and the old solution to set the phase This leads to the following integral condition E E E EN 7 11 This is one more condition which accounts for the need for an additional free pa rameter XPPAUT allows you to specify the projection boundary conditions and by setting a particular flag on in AUTO you can implement the integral condition Since the XPPAUT version of AUTO does not allow you to have more conditions than there are differential equations you should pic
212. he cross is on the first point in the upper branch Change the sign of Ds in the Numerics dialog of the AUTO window Click on Run Extend to get the rest of the branch You should see something like Figure 7 8 7 5 Importing the diagram into XPPAUT 195 7 4 1 Exercises 1 Periodically forced oscillators lead to remarkably complex behavior Here we will look at the periodically driven Fitzhugh Nagumo equations see chapter 3 Here is the appropriate ODE file fhnfor ode Fitzhugh Nagumo equations with sinusoidal forcing v I vx 1 v v a w cxu w eps v gamma w u ux 1 u 2 z7 2 f z z zx 1 u 2 z2 2 fxu init u 1 v 0505 w 1911 par c 05 f 8 par 1 0 2 a 1 eps 1 gamma 25 O xp V yp w xlo 25 xhi 1 25 ylo 5 yhi 1 total 7 86 dsmax 05 parmin 5 parmax 1 autoxmax 1 autoxmin 0 autoymin 1 5 autoymax 1 5 done Run this a few times to make sure you are locked on a periodic Then Run AUTO using the above parameters You should see that the periodic solution loses stability at a point which AUTO marks as PD this means that there is a period doubling instability Grab this point and Click on Run choosing Period doubling from the menu choices Be ready to click on the ABORT key since it will circle around the period 2 orbit 2 Find a problem in a research paper that involves periodic forcing and analyze the bifurcation behavior Or see what kind of phenomena you can find by adding a term like ccos wt
213. he function to get some other polar curves Other parametric plots are just as easy to plot For example this plots a t y t in the plane parametric ode f t cos a t g t sin b t par a 4 b 1 gis aux x f s aux y g s xp x yp y xlo 2 ylo 2 xhi 2 yhi 2 total 50 done Similarly three dimensional parametric plots are also possible Here is one where I have set up all the axes Change parameters and plot away param3d ode 3d parametric plots f t cos a t g t sin b t h t sin c t d par a 4 b 1 c 2 d 75 gis aux x f s aux y g s aux z h s O xp x yp y zp z xlo 2 ylo 2 xhi 2 yhi 2 axes 3d xmax 1 25 ymax 1 25 zmax 1 25 xmin 1 25 ymin 1 25 zmin 1 25 total 50 done As a last example of plotting in three dimensions here is a nice trick in which I plot the three dimensional parametric plot along with a two dimensional projection below it param3d2 ode 3d parametric plots with 2d projection f t cos a t g t sin b t h t sin cxt d par a 1 34 b 1 12 c 2 28 d 75 48 Chapter 4 XPPAUT in the classroom Figure 4 1 Three dimensional plot illustrating a projection on the coor dinate plane par zlo 4 s 1 aux x f s aux y g s aux z h s aux zplane zlo O Xp X yp y zp z x10 2 y1l0 2 xhi 2 yhi 2 axes 3d xmax 1 25 ymax 1 25 zmax 1 25 xmin 1 25 ymin 1 25 zmin 4 nplot 2 xp2 x yp2 y zp2 zplane total 100 done I have adde
214. he spatially homoge neous solution is unstable to nonhomogeneous perturbations New solutions arise which are spatially inhomogeneous Here is an ODE file for the Brusse lator the brusselator reaction diffusion model brusspde ode reaction terms f u v a b 1 x u v u 2 g u v b u v u 2 par a 1 b 1 5 equations u0 u1 u 1 100 f u j v j du u j 1 2 u j u j 1 h h u101 u100 vO v1 v 1 1007 89 u 3 v 3 1 dv v j 1 2 v j v j 1 h h v101 v100 spatial parameters par h 2 du 1 dv 4 initial data init u 4 100 1 init u 1 3 1 2 init v 1 100 1 5 numerical stuff meth cvode atol 1e 5 tol 1e 6 total 400 dt 25 done It is very similar to the simple cable model discussed above Because diffusion equations tend to be numerically stiff due to the division by h I have added a line at the end of the file to use the stiff integrator CVODE to specified tolerances with a timestep of 0 25 for a total of 400 time units Run this and look at the steady states Change D making it smaller and verify that the spatially homogeneous state becomes stable again Note that you can make this much faster by using the banded method and rewriting the discretized system like the example 6 3 7 Asalast PDE example consider the Grey Scott equations This is an example 6 2 Partial differential equations and arrays 163 which makes it very clear why you want to use the banded option
215. he spike time statistics of single neurons and populations of neurons XPPAUT has a number of features that let one simulate and analyze equations with noisy inputs We will look at histograms for Poisson processes with and without refractoriness post stimulus histograms and spike time autocorrelation functions Post stimulus time histograms When an experimentalist records from a neuron that is receiving some input the spike time histogram PSTH is often used as a measure of the response One looks at the distribution of spike times as a function the onset of the input Many trials are run and a histogram of the times is made I present a simple example here A simplified model neuron the theta model which like the integrate and fire lies on a circle Here are the equations we will simulate u 1 cosu 1 cosu Io ow t f t where w t is normally distributed noise Ip is the bias and f t is the stimulus We will use J 0 2 0 0 25 and f t 0 4t exp t 8 By definition the theta model spikes when u crosses 7 It is defined modulo 27 as well so when it hits 27 it is reset to 0 Here is the ODE file psth ode 116 Chapter 5 More advanced differential equations psth ode using an alpha function input and the theta model with noise wiener w par 0 2 sig 25 amp 4 tau 8 init u 84 f t amp t exp t tau meth euler total 50 dt 1 u 1 cos u 1 cos u idt tsig wtf t global
216. he time until the next reaction is computed by drawing a random number X is decremented by 1 We don t let X fall below 1 since then ay 0 and the reaction stops We initialize for 1000 reaction steps We plot X against the variable t which represents real time I have added an auxiliary variable which represents the solution to the mean field equation dX de Run this and graph the continuous approximation along with it Note that they are quite close Try this again but let X 10000 initially and set xin 10000 Only plot every tenth reaction point Click on nUmerics nOut and change this to 10 Also change Total to 10000 and then click on Esc exit and integrate again They are even closer cX X 0 1000 108 Chapter 5 More advanced differential equations Consider the classical Brusselator chemical reaction XA Y X2 Y Ya 2Y Yo 2Y Y Z with rates c c2 C3 C4 respectively Only Y1 Y2 vary with X1 X2 fixed The rates used to compute the probabilities are ay Cc Xy a2 c2 X2 Y1 ag c3 Y2Y Y 1 2 a4 c4 Y Note the third reaction once one Y is chosen there are Y 1 remaining and we must divide by 2 since we are counting pairs of Y s and don t want to count the same pair twice Here is the corresponding XPPAUT file gillesp_bruss ode gillespie algorithm for brusselator x2 y1 gt y2 Z c2 2 y1 y2 gt 3 y1 c3 y1 gt Z2 c4 par c1x1 5000 c2x2 50
217. heatsheet a kaa ee ee ee a ee ee al a 281 Bibliography 287 2 1 2 2 2 3 2 4 2 5 2 6 Dl 2 8 2 9 3 1 3 2 3 3 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 List of Figures The main XPPAUT window 00 005 0 ae 11 The Equation Window 0 02000000 12 Phase plane for the linear 2d problem 13 The initial conditions window 04 14 Fil select0r 2 23 23 A EEE OSE 15 The parameter window 000002 eee 16 Top Unused parameter slider Bottom Used parameter slider 18 The D ta Vie web o e mna whe el a o a 18 Direction fields and some trajectories for the Fitzhugh Nagumo equatlON Sari O o ee a es A eet ee 22 Nullclines direction fields trajectories for J 0 4 in the Fitzhugh Nagumo equations ooo yb Goad ee ee RA ed ed 23 Phase plane of the forced Duffing equation 30 A pair of integrate and fire neurons coupled with alpha function synapses With slow excitatory coupling synchrony is unstable with fast inhibition synchrony is also unstable with slow inhibi tion synchrony is stable o o 42 A kicked clock and its chaotic brother 43 Three dimensional plot illustrating a projection on the coordinate plane nso Dag eR Bee ee Se oS os a go Te The ae ae 48 A cobweb plot of the logistic Map 4 50 The logistic map showing an orbit diagram on the left and a plot of th
218. here are many realistic models of this but they all work on the same principle illustrated by the above example The equations for this are dx zV a dt od where z is a two state Markov process that has switching rates a 8 for 0 1 and 1 0 respectively Here is the XPPAUT file ratchet ode a simple thermal ratchet wiener w par a 8 sig 05 alpha 1 beta 1 piecewise linear potential with slope tt 1 from 0 to a and slope a 1 a from a to 1 f V f x if x lt a then 1 else a 1 a x z f mod x 1 sig w z is two states markov z 2 0 alpha beta 0 meth euler dt 1 total 2000 njmp 10 xhi 2000 yhi 8 ylo 8 done The program is pretty self explanatory Since I only define the function f x on the interval 0 1 I must consider x modulo 1 to make it periodic Run this and you will see the discrete steps that x makes as it ratchets downward Note that this particular ratchet tends to move toward negative x Run this 100 times and compute the average trajectory Use the nUmerics stocHastic Compute command and then the Mean item as above Change a from 0 8 to 0 2 Run it again Change a to 0 5 and run it once more Note that by varying a you can make the ratchet go in any direction at a variety of velocities 5 2 Stochastic equations 113 A surprising game In Harmer and Abbott 1999 an intriguing paradox was displayed and subse quently analyzed In this paper two g
219. hile as you become used to XPPAUT you will use the keyboard shortcuts more often I will tell you the full commands and the keyboard shortcuts In general the 12 Chapter 2 A very brief tour of XPPAUT E Equations x Eose Ge iow 220 T A4 ERY T dT CA DY Figure 2 2 The Equation Window keyboard shortcut is the first letter of the command unless there is ambiguity such as Nullcline and nUmerics and then it is just the capitalized letter N and U respectively Unlike Windows keyboard shortcuts the letter key alone is sufficient and it is not necessary to press the Alt Ctrl key at the same time The top region of the Main Window is for typed input such as parameter values The bottom of the Main Window displays information about various things as well as a short description of the highlighted menu item The three little boxes with the words parameter are sliders to let you change parameters and initial data Across the top of the Main Window you will see a menu bar of turquoise buttons Clicking on these opens up a variety of windows In addition you may or may not have several other yellow buttons which are shortcuts to commands These are defined by the user in the ODE file as options see Appendix B or in the XPPAUT resource file see Chapt 1 Click on the button labeled Eqns and the Equation Window will appear as shown in figure 2 2 This allows you to see the differential equations that you are solving We will des
220. his is a kt order difference equation Furthermore you don t want to write all 17 equations to create a 17th order equation in normal form We can avoid this by using the delay operator Furthermore we can even let the order k be a parameter Recall that delay x k is z t k so that if t and k are integers then delay x k is just T k With these considerations the ODE file for this problem is cicada model here are the predators decay plus feeding on emerged adults pti nu pta mu k delay n k amount of food left after taking into acct of all the grubs c max d sum 1 k 1 of mu i delay n i 1 0 p t 1 pt1 new number is what s left after predation and what could have eaten n t 1 min f max mu k delay n k 1 pt1 0 c set initial delay to 100 n 0 100 0 t aux ct c par a 042 f 10 d 10000 nu 95 mu 95 k 7 note you must set delay initial data to 100 meth disc dt 1 total 200 bound 1000000 delay 20 xhi 200 yhi 10000 yp n ylo 0 done 5 1 Functional Equations 99 There are a few points to focus on here e I have defined the temporary variable pt1 since the predation depends on this year and not the previous year This way I don t have to compute this twice e I have used the summation operator along with the delay operator to compute the sum gt 444 5410 e I have set dt 1 so that when data points are stored for the delay operator they are properly referenced
221. his is a four dimensional system As with all mechanical systems you derive the differential equations by writing down the potential and kinetic energy They are complicated and instead of the equations I will just give you the ODE file doubpend ode 8 3 My favorites 211 the double pendulum doubpend ode thi is the angle between the pivot and the first mass th2 is the angle between the first mass and the second mass mu is friction L1 2 mi 2 are the length and mass thi thip th2 th2p th2p mu th2p 1 L2 mi m2 sin th2 th1 2 m1 m2 g sin th2 mi m2 L1 thip 2 sin th2 th1 cos th2 th1 Gn1 m2 g sin th1 m2 L2 th2p 2 sin th2 th1 thip mu thip 1 L1 m1 m2 sin th2 th1 2 mi m2 g sin th1 m2 L2 th2p 2 sin th2 th1 cos th2 th1 m2 g sin th2 m2 L1xth1p 2 sin th2 th1 parameters par Li 1 L2 1 m1 028 m2 04 g 9 8 mu 01 init thi 0 th2 0 th2p 31 th1p 0 total 30 dt 01 meth qualrk bound 100000 done This is set up to simulate two 10 cm pendula with a masses of 28 and 40 grams respectively I give the lower one a good spin Time units are in seconds so the whole simulation represents a half a minute Integrate these equations Before we get to the animator convince yourself that the solution is chaotic Compute the maximal Liapunov exponent nUmerics stocHastic Liapunov and also com pute the power spectrum of the angular velocity thip
222. his model We note that t t t t thus x t x t r where the delay r t t XPPAUT has a function called flr t which returns the largest integer less than t This will do the trick cook ode a delay equation due to Ken Cooke tx r x 1 x t where t integer part of t 94 Chapter 5 More advanced differential equations Figure 5 2 Solutions to Ken Cooke s delay equation frac t t flr t x r x 1 delay x frac t aux xd delay x frac t par r 2 2 init x 8 delay 2 total 100 done I have added the delayed variable as an extra plottable quantity The user defined function frac t takes the fractional part of t Since t is constant over n n 1 where n is an integer this means that x t is also constant say x n Thus x rx 1 x n in the interval and you can integrate this equation to obtain a map for n x n a n 1 x njer Uam This unimodal map goes through the usual period doubling cascade to chaos Thus the delay equation is also chaotic Try the delay equation for values of r between 2 and 3 Also write an ODE file for the map and iterate it for different values of r Figure 5 2 shows a solution to this equation when r 3 in the x t x t phase plane 5 1 2 Integral equations There are many physical
223. hole interval of values of c The only constraint is to recall that U is a population so that it can never fall below 0 Thus the trajectory into 0 0 must be monotone Since c lt 2 implies that the origin is a spiral this means that we better take c gt 2 Let s see how to use XPPAUT to compute these trajectories As we said above XPPAUT lets you compute one dimensional invariant manifolds so we will 132 Chapter 6 Spatial problems PDEs and boundary value problems work from the fixed point 1 0 which has a one dimensional unstable manifold Here is the ODE file fisher ode traveling waves in fisher s equation u_t u_xx u 1 u cu u u 1 u HHHH FH f u ux 1 u u v v cxv f u par c 2 init u 1 xp u yp v xlo 25 xhi 1 25 ylo 5 yhi 5 done I have set it up so that the phaseplane is the default plot I have set the velocity to the critical value c 2 How to shoot from a fixed point Here are the steps to shoot for a good traveling wave 1 Set the initial conditions near or at the fixed point from which you want to shoot 2 Click on Sing pts Go S G 3 Answer No to the Print eigenvalues question 4 Answer Yes to the Compute invariant sets question 5 Click Esc when the message boxes come up or when the computation of the trajectory seems to have have converged to a fixed point XPPAUT integrates the equations for arbitrarily long times so it is often necessar
224. how the DAE solver fails as it reaches a turning point The equation is day 2 ove 1 q z x with initial condition x 0 0 and 2 0 1 Write an ODE file for this and try to solve it using XPPAUT At about t 1 5 the solver will fail since v dx dt will be close to zero and so Og x v 0v will be small Newton s method will be unable to converge Note when the DAE solver fails and you want to change some numerical tolerances to perhaps make it work better you will often have to reset the starting guess Click on Initialconds DAE guess to reset the initial guess There is no cure for this failure a solver such as MANPAK has additional methods to deal with rank 1 Jacobian matrices 2 In this example we set the tolerances to be rather crude allowing for a jump to a new branch in order to solve a classic relaxation oscillator The van der Pol oscillator has the following form dV 3 dw an v v w ae v In the limit as e 0 it becomes a DAE dw a aT OU 0 v v w J v The problem is that for w 0 38 0 38 there are three roots to this equa tion The outer branches of the cubic are the relevant roots and when w reaches a maximum of the cubic V must jump to the other root With close tolerances the DAE solver will run into the same problems as in problem 1 However loosening the tolerances a bit will allow the Newton solver to find the other root and make the jump Try this ODE file dae_ex5 ode v
225. ics and change Par Min 6 Par Max 6 and then click OK Now click on Run and you will see a line that is almost diagonal When done click on Grab again and watch the bottom of the AUTO window until you see Per 35 and click Enter In the AUTO window Numerics menu change Ds 02 to change directions and click Ok Now click Run and there will be another diagonal line that is in the opposite direction Click 7 3 Boundary value problems 189 SIG y Figure 7 7 Left The continuation of the homoclinic orbit for the example problem Right Sample orbits computed for a 0 6 in the x y phaseplane on Grab and grab point number 7 corresponding to a 6 In the Main Window click on Init Conds Go and you will see a distorted homoclinic It is not that great and could be improved probably by continuing with Per some more Grab the point labeled 11 a 6 and the Main Window try to integrate it It doesn t look even close This is because the homoclinic orbit is unstable and shooting which is what we are doing when we integrate the equation is extremely sensitive to the stability of the orbits In the AUTO window click on File Import Orbit to get the orbit that AUTO computed using collocation In the Main Window click on Restore and you will see a much better version of the homoclinic orbit This is because collocation methods are not sensitive to the stability of orbits In fact you can verify that the
226. iddle The axes have labels in the figure To label axes click on View axes 3D and fill in the last three dialog box entries We can rotate this graph around the vertical axis in the form of a little movie to get a better view of the attractor Click on 3D params and fill in the right half of the dialog box as follows 82 Chapter 4 XPPAUT in the classroom Figure 4 12 The Lorenz attractor Movie Y N Y Vary theta phi theta Startangle 35 Increment 15 Number increments 23 This tells XPPAUT that to make a series of 23 screen captures in which the rotation angle theta is varied by 15 degrees at a time starting with 35 degrees These screen shots can be played back smoothly like a movie using the Kinescope command Click on Ok after you have filled in the dialog box Make sure that the window in which the graph is drawn is not obscured by any window other than at most the dialog box Otherwise the capture will not work properly A series of pictures will be drawn on the screen These are captured in a buffer and can be played back either one frame at a time or in smooth succession using the Kinescope command Click on Kinescope Autoplay to smoothly run through all the captured screens You will be prompted for the number of loops you would like choose 3 and the amount of time between snapshots in milliseconds 50 100 are reasonable values You will then see the three dimensional plot smoothly rotated and this is repeated 3 ti
227. ies emerging from the saddle points triangles and come into the the blue trajectories toward the saddle points Thus a point starting near one of the blue trajectories will go toward the saddle to which it is connected and then out along a yellow orange trajectory Since there are no stable fixed points and no limit cycles in this example integrating the equations forward in time will almost always result in solutions that go to infinity Try a few initial conditions to convince yourself of this fact We can study a different example either by writing a new ODE file or editing the right hand sides of the equation Let s examine the following equation x x 1 a y y x 4 Click on File Edit RHS to edit the right hand sides Enter these two equations carefully and then click on Ok Change the window to 0 25 1 25 X 0 25 1 25 by using the Window Zoom Window dialog Draw some direction fields Scaled or Unscaled Recall that scaled direction fields represent only the direction whereas unscaled also change the length as a function of the size of the vector field I prefer the former but this is a matter of taste Try either one Click on D S and use a grid size of 16 instead of 10 and you will get pure direction fields Click on N N to get the nullclines You will see a picture like Figure 4 9 There are clearly two fixed points and the third 0 0 is ambiguous the nullcline algorithm is not perfect Verify that 0 0 i
228. ike invisible parameters that are evaluated anytime the other parameters are changed Unlike fixed variables these are only evaluated once per integration of the equations The 10 auxiliary variables contain the initial data if they correspond to the time slice and the x coordinate is positive otherwise they are set to 100 100 I use the Dormand Prince 8 3 integrator as it is very fast especially when the output times are large e g dt 1 Run this file click on Initialconds 2 par range and fill in the resulting dialog box as 9 6 Arcana 245 Varyi dx Reset storage N Starti 0 Use old ios Y enai 1 cycle color N Movie N Movie N Start2 0 Crv 1 Array 2 2 Ena 1 steps 101 stepe E and click on Ok A series of small circles will be drawn at different points Since xp5 yp5 are in the plot window and the plots are at even times 2 4 etc this graph shows all the initial conditions which end up in the right half plane at t 10 Look at a plot of xp10 yp10 to see a big difference Make a movie You can make a quick flip book of the plotted sets at the different time slices as follows Click on Viewaxes 2D and change the plotted variables to xp1 yp1 the first in the set Then click on Kinescope Capture to grab the screen image Next change the view to xp2 yp2 and capture the screen again with the K C commands Repeat this until you have plotted all 10 sections Now they are stored in memor
229. ing However it is not very robust Consider the ratio In Rn n As n gt clearly Rn N M p This limiting value p is called the rotation number Now suppose that a 4 0 Then we can no longer explicitly write down the solution n However the rotation number still exists if f x gt 0 e g if ja lt 1 27 and in fact depends continuously on the parameter b This is a consequence of Denjoy s Theorem p 301 Guckenheimer amp Holmes Furthermore it is constant almost everywhere Thus if we plot p as a function of b we will get what is known as the Devil s Staircase We can use XPPAUT to plot the rotation number as a function of the parameter b to see what this looks like Here is the XPPAUT file rotnum ode the rotation number for the std map 56 Chapter 4 XPPAUT in the classroom f x x b a sin 2 pi x x f x b b par a 15 init b 0 init x 0 aux rho x max t 1 bound 100000 total 1000 trans 1000 meth disc xp b yp rho xlo 0 ylo 0 xhi 1 001 yhi 1 001 rangeover b rangestep 200 rangelow 0 rangehigh 1 rangereset no done NOTES As with the other bifurcation problems we have treated the parameter as a variable so that it is available for plotting We do not mod x by 1 here so that we can compute the total accumulation of phase The rotation number is approximated by the finite ratio x t where t is the iteration number As only the last value is of interest the trans
230. ing a line that starts with aux followed by the definition of the quantity E g aux stuff xty sin t stuff will now be plottable Let s take as an example the pendulum which satisfies ml 6 mglsin 0 where l is the length g is gravity and m is the mass of the pendulum 0 is the angle of the pendulum with respect to the vertical axis The kinetic energy is K m 10 2 and the potential energy is P mgl 1 cos0 Thus the total energy is E m 16 2 mgl 1 cos6 Let s write an ODE file for the pendulum and make the energy a plottable quantity As usual we want to first make the equation a system of two first order equations d v g l sin The ODE file is 3 4 Auxiliary and temporary quantities 33 the undamped pendulum theta v v g 1 sin theta par g 9 8 1 2 m 1 aux E m 1 v 2 2 g 1 1 cos theta done Now if you run this the energy will be accessible You will find that it is constant along solutions to the differential equation a hallmark of frictionless mechanical systems In XPPAUT auxiliary quantities that are made available for plotting etc are called auxiliary variables Rewrite the above ODE file to incorporate a linear friction term with magnitude f 0 v g l sind fv Run this plot the energy for f 0 and f 0 1 and notice how the energy dissipates to zero when there is friction 3 4 1 Fixed variables If you have a particularly complicated e
231. ing commands state commands or style commands Without further ado here are all the allowable commands e dimension xlo ylo xhi yho e speed delay e transient e permanent e line x1 y1 x2 y2 color thickness e rline x1 y1 color thickness e rect x1 y1 x2 y2 color thickness e frect x1 y1 x2 y2 color e circ x1 y1 rad color thickness e fcire x1 y1 rad color e ellip x1 y1 rx ry color thickness e fellip x1 y1 rx ry color e comet x1 y1 type n color e text x1 yl s e vtext x1l yl s z e settext size font color e end That s it In all commands except settext the color and thickness entries are optional All commands can be abbreviated to their first three letters and case is ignored At startup the dimension of the animation window in user coordinates is 0 0 at the bottom left and 1 1 at the top right Thus the point 0 5 0 5 is the center no matter what the actual size of the window on the screen The transient and permanent declarations tell the animator whether the coordinates have to be evaluated at every time or if they are fixed for all time The default when the file is loaded is transient Thus these are just toggles between the two different types of objects Color is described by either a floating point number between 0 and 1 with 0 corresponding to the lowest value in your color map blue in the default and 8 4 The animator scripting language 219 1 to the highest red in the default When described as a
232. ing options can only be set outside the program They are e MAXSTOR integer sets the total number of time steps that will be kept in memory The default is 5000 If you want to perform very long integrations change this to some large number e BACK Black White sets the background to black or white e SMALL fontname where fontname is some font available to your X server This sets the small font which is used in the Data Browser and in some other windows e BIG fontname sets the font for all the menus and popups e SMC 0 10 sets the stable manifold color e UMC 0 10 sets the unstable manifold color e XNC 0 10 sets the X nullcline color e YNC 0 10 sets the Y nullcline color e OUTPUT filename sets the filename to which you want to write for silent integration The default is output dat 257 258 Appendix B The options e BUT label sequence This option can be used repeatedly in the same file and allows you to define a clickable button on the top menu bar of the main window The label defines the label that will appear on the button and the sequence is a sequence of keys used for the command These are the keys you would press to get an item from the main menu see Appendix E For example if you want to have a one click command for computing the Liapunov exponent the sequence is uhl since you would press nUmerics stocHastic Liapunov Thus the option is BUT liap uhl A quit button would be B
233. is to the mean field equation ul u f 5ctu c u t T where the factor of 5 comes from the fact that each cell receives 10 inputs with mean value 0 5 A dendritic tree The bulk of most neurons consists of the dendritic tree of the neuron One approach to simulating the tree of the neuron is to break it into a series of small connected cylinders Each cylinder has a specified length and diameter as well as a specified axial resistance and transmembrane resistance The former depends on the diameter and length of the cylinder while the latter depends on the surface area excluding the faces Ohm s law then allows one to approximate the dynamics of the dendritic tree by an electrical circuit which in turn leads to a series of differential equations for the voltage of each cylinder Mark Fenner a student in one of my classes has written some C code which allows one to take a list of endpoints of the cylinders and their diameters and create an XPPAUT file that simulates the dendritic tree The C program produces four files two tables that describe the connectivity and the weights an animation file which draws the tree and the ODE file to drive the simulation The key utility of the program is the production of the connectivity and weight matrices as well as the animation file The ODE file produced is simple and assumes passive dendrites It is trivial to alter this by hand to make some of the compartments active Each cylinder can conne
234. is zero since I want the chain to be a finite line of oscillators rather than a ring Since the fconv function requires a function of two variables I have defined gam u v gamf u v I subtract off r 0 from each of the equations so that the resulting solutions are the phases relative to theta0 I have used the stiff integrator CVODE even though the problem is not stiff I experimented with several different methods and this was the fastest I have initialized the program so that there are four quantities plotted Up to 9 can be so initialized Spatial profiles Run this program and notice that all 4 plots seem to converge to a fixed value that is the relative phase Thus this particular equation system phaselocks To see a spatial profile get rid of all the plots by clicking Graphics Remove all G R Then transpose the data by clicking File Transpose and filling in the dialog box as Colum 1 theta Mois colskip E Row 1 o Rousi 1 and click on Ok Now in the main window click on X vs t and choose theta0 You will see that the oscillator in the middle is the most phase advanced and the two edge oscillators fire last Numerical simulations like this have suggested many theorems about coupled oscillator arrays Change some of the parameters in the models Can you break up the phase locked solutions Non convolution coupling The examples presented above assume convolution coupling between the units Suppose we wanted to sim
235. ith failure Boundary conditions in XPPAUT are input in the ODE file or in the Bound ary Cnd window shown in Figure 6 1 and obtained by clicking on the BCs button along the top of the Main Window Since XPPAUT only pays attention to con ditions at the endpoints of the integration you can only have conditions that are given at t to and t t where to t1 are the endpoints The conditions are given 6 1 Boundary value problems 125 Boundary Conds x bd E q DE Figure 6 1 The boundary condition window as a series of quantities that are set to zero Thus if you want the variable u to be 1 at t to the condition you would input is u 1 The variables are given different names at t t they are primed So that if you wanted u to 1 and u t 1 you would type the conditions in the Boundary Cnd window 0O u 1 0O w 1 The last condition tells XPPAUT to try to make u 1 at t t The syntax for these conditions in the ODE file is bndry u 1 bndry u 1i Note you do not have to write bndry rather b alone will suffice With these preliminary comments here is the ODE file eigen ode for our eigenvalue problem standard eigenvalue problem omega is unknown v vp vp omega omega v omega 0 bv b vp 1 b v init v 0 vp 1 omega 1 dt 002 total 1 0 xhi 1 done 126 Chapter 6 Spatial problems PDEs and boundary value problems We note the following i we rewrite the se
236. iting glass of water This action moistens the duck s head and also causes the freon to drop back to the duck s belly He swings back and the process repeats Model this as a pendulum with two weights The head weight grows slowly proportional to the angular velocity When the head drops the weight is reset to the bottom and the process begins anew 42 Chapter 3 Writing ODE files for differential equations 0 8 F 0 6 F 0 4 H 0 2 pp j L 1 1 L 1 L 80 82 84 86 88 90 92 94 96 98 100 Figure 3 2 A pair of integrate and fire neurons coupled with alpha function synapses With slow excitatory coupling synchrony is unstable with fast inhibition synchrony is also unstable with slow inhibition synchrony is stable 3 5 Discontinuous differential equations Figure 3 3 A kicked clock and its chaotic brother 43 44 Chapter 3 Writing ODE files for differential equations Chapter 4 XPPAUT in the classroom XPPAUT although developed as a research tool has been used by many people in a variety of courses in order to allow students to look at models and differential equations on the computer Since the program is free and runs on a variety of plat forms it is suitable for classroom use While the interface is not as simple as some dedicated systems such as Java applets or the Macintosh program MACPHASE it
237. ix E Complete command list Appendix F Error messages XPPAUT is full of obscure error messages Here I provide a list of common ones where they come from and what to do about them Out of bounds Some variable or auxiliary quantity went beyond the proscribed maximum Fix this by increasing the Bounds in the nUmerics menu But be careful some solutions to ODEs really do blow up Bad formula The parser was unable to figure out what you meant You have probably mistyped a name forgotten a parenthesis or used a name not known to XPPAUT Singular Jacobian In Newton s method the matrix was not invertible In boundary value problems this sometimes means that you have not written the boundary conditions correctly In integration sometimes setting the time step to be smaller works Maximum iterates exceeded Again in newtons method or in curve fitting the specified tolerances could not be achieved in the given number of iterates Either loosen tolerances or increase the maximal number of iterates Delay negative or too large Increase the maximal allowable delay If the delay is negative you have an ill posed problem Could not converge to root In solving a nonlinear equation the Newton solver failed to converge You should make a better guess Working too hard This error occurs when you have global flags It is a difficult error to fix It generally means that when condition A occurs it induces condition B which in turn induc
238. k You will then be asked for a filename The File Selector box is shown in the figure 2 5 You can move up or down directory trees by clicking on the lt gt choose files by clicking on them scroll up or down by clicking on the up down arrows on the left or using the arrow keys and the PageUp PageDown keys on the keyboard change the wild card or type in a filename For now you can just click on Ok and a PostScript plot will be created and saved The file will be called linear2d ode ps but you can call it anything you want Once you have the postscript file you can type lpr filename on UNIX In Windows if your computer is hooked up to a postscript printer then this usually does the trick copy filename 1pt1 2 3 Solving the equation graphing and plotting 15 Print postscript DIR home bard tex xppbook ps o E LON Da lt gt siam linear2d ode ps fhn ode ps dendtree ps ratchet ode ps junk ps bistable ode ps cable100 ode ps Cancel Figure 2 5 File selector In Windows if you don t have a postscript printer it is still easy to print the postscript files to your printer However you have to download a program called GhostView This allows you to print postscript onto non postscript printers Most Windows distributions of GhostView come with a command line utility called gsprint exe in the Ghostgum gsview directory Either of the following two lines work for me gsprint myfile ps gsprint colour m
239. k on nUmerics Dt and change it to 0 02 Draw the phase portrait for this system x y y 23 y 22 x 3x y Show that it is integrable How many fixed points are there and what is their nature Draw all the connecting trajectories from saddle to saddle al y 1 y y x 1 2x7 This system can be perturbed to produce a system with 11 limit cycles Here is a classic two neuron neural network TU u f w1101 wiu 01 T2ug u2 f wa181 wa2U9 02 where f u 1 1 exp u The four numbers w are the weights 7 are the time constants and 0 are the thresholds The function f u is the firing rate as a function of the inputs This system has an incredible range of different phase portraits Type this in as an ODE file a Using the following values for the parameters wi 12 wig 6 wo 13 ww 2 01 3 5 02 6 71 1 and 72 1 draw the nullclines There are three fixed points The middle fixed point is a saddle Draw the unstable and stable manifolds Note that the two branches of the unstable yellow manifolds form a loop which terminates on the left most fixed point Now decrease 01 to around 3 2 and draw the nullclines Note that the two lower left fixed points have disappeared Integrate the equations to see what happens There is a stable limit cycle that has emerged from the loop formed by the unstable manifolds of the saddle point which no longer exis
240. k one parameter which will be slaved to all the other ones you vary and let this satisfy a trivial differential equation a 0 This will be varied by AUTO to satisfy the conditions 7 10 7 11 Here is an example of continuing a homoclinic in two dimensions Y y y x 1 2x axr o0xy When a c 0 0 there is a homoclinic orbit Prove this by integrating the equations this is a conservative dynamical system For small a it is possible to prove that there is a homoclinic orbit for a particular choice of o a using Melnikov 7 3 Boundary value problems 187 methods see Guckenheimer and Holmes We now write the equations as a 5 dimensional system using o as the slaved parameter and introducing a parameter P for the period x Pf z y y Pg z y x 0 ye 0 o 0 where f x y y g x y x l x ax oxy and the following boundary conditions 0 f Xe Ye 0 g e Ye 0 s x 0 ze y 0 ye 0 Lu x 1 ze y 1 ye and the integral condition XPPAUT has a defined function for the projection boundary conditions called hom bcs k where k 0 1 n 1 corresponding to the total number required You do not need to be concerned with ordering at this point as long as you get them all and you give XPP the required information Here is the ODE file tsthomi ode f x y y g x y x 1 x axyt sigex y x f x y per y g x y per auxiliary ODE for fixed point xe 0 ye
241. k to the main menu Click on Restore to draw the variance It is almost a perfectly straight line with slope 1 Click on Graphics Add curve and add the curve with t on both the X and Y axes This represents a line with slope 1 and is quite close to your plot of the variance 5 2 1 Markov Processes XPPAUT is able to simulate continuous Markov processes with transition rates that depend on other state variables in the differential equations An obvious example is the simple two state voltage dependent channel model Suppose you have a patch of membrane with a single sodium channel that opens at a rate a V and closes at arate 3 V Let the open state have conductance 0 Then the equation has the form dV Ce gL V Vr I gz V Vina where z is a random variable that is 1 for the conducting state and 0 for the nonconducting state z flips from 0 to 1 in the interval t to t dt with probability a V dt and flips from 1 to 0 with probability 6 V dt XPPAUT simulates this by computing the probability of flipping at each time step and then changing the variable z z is always regarded as a nonnegative integer In order to define a Markov process you must define the transition matrix For the present model the matrix is defined in XPPAUT with the statement markov z 2 0 al v beta v 0 where al v beta v are the rate functions In XPPAUT each entry in the matrix is delimited by a pair of curly brackets The diagonal e
242. ke is clearly defined by some crossing event We can quantify this apparent periodicity by looking at the spike time auto correlation function This makes a histogram of the differences between spike times Click on nUmerics stocHast Data to get the list of spike times back into the Data Viewer Click on stocHast Stat autOcor to make an autocorrelation Choose 200 bins ranging from 50 to 50 with t as the variable and replot the first two columns You will see a clear periodicity You can see it clearer if you narrow the range of the histogram from 20 to 20 This is the spike time autocorrelation function There is a clear periodicity with a peak distance of about 3 A Poisson process with refractoriness Recall that a Poisson process can be modeled using Markov variables here we will simulate a neuron which spikes randomly with an exponential distribution of the interspike intervals but which has an exponentially decaying refractory period Thus the rate r is rmac 1 s where s 1 when there is a spike and s decays exponentially with a time constant 7 Here is the ODE file poisref ode 5 2 Stochastic equations 117 so l ln E i Y time count Figure 5 9 PSTH for the noisy theta neuron left and the spike count distributions for a Poisson process with rate 0 05 and the same process with a re fractory period poisref ode poisson process with refractory period s0 0 for no refractory
243. l lock to a periodic orbit and if so whether it will be synchronous or whether other more complex solutions are possible The file m110 o0de implements a coupled array of ten such neural oscillators m110 ode morris lecar dimensionless vtot 1 sum 0 9 of shift v0 i wtot 1 sum 0 9 of shift w0 i itot v w I gl el v gk w ek v gca minf v eca v g v w winf v w lamw v v 0 9 itot v j w j d vtot v j w o 91 8 v j w j par d 05 par 1 095 phi 1 par ek 7 eca 1 el 5 par gl 5 gk 2 gca 1 par va 01 vb 0 15 vc 0 1 vd 0 145 minf v 5x 1 tanh v va vb winf v 5x x 1 tanh v vc vd lamw v phi cosh v vc 2 vd total 200 nout 4 xlo 6 xhi 5 ylo 25 yhi 75 O xp v0 yp w0 done Notes The term sum 0 9 of shift v0 i sums up all the voltages similarly for the recovery variables we will use the latter in the animation I have set the output to every fourth point to make the animation go faster Run XPPAUT and use the Initialconds Formula command setting the variables v 0 9 to 5 5 ran 1 which assigns them random voltages between 0 5 and 0 5 Now fire up the animator and load the animation file m110 ani m110 ani use with m110 ode fcircle v 0 9 1 5 w j1 1 02 j 1 10 fcircle vtot 5 wtot 1 03 BLACK end This file draws a filled circle in the v w plane for each cell and colors it by its index so that it is easy to see each indivi
244. ld be set to the values that you want to send back to the fixed variables xp will have the value of out 0 etc The integers nin nout are just the dimensions of the arrays Finally two more arrays are passed which you can generally ignore These contain the complete list of all state and fixed variables as well all parameters They are organized as follows for the above example con 2 a con 3 b con 4 c con 5 d v 0 t v 1 x v 2 y v 3 xp v 4 yp That is the array con contains all your parameters in order starting with con 2 and the array var contains the time variable followed by the state variables and then the fixed variables Thus you could directly communicate with all the variables or parameters Once you have written your C file you should compile it and create a library Here is how to do that gcc shared fpic o libexample so funexample c This just compiles it as a relocatable object file and creates a shared library Now run XPPAUT with the file tstd11 ode Click on File Edit Load DLL Choose the library you have created libexample so and then choose one of the 252 Chapter 9 Tricks and advanced methods following three functions 1v vdp duff which are respectively the Lotka Volterra model the van der Pol oscillator and the forced Duffing equation respectively Now integrate the equations and you should see the Lotka Volterra or whatever solutions Change parameters etc and these will be reflected in
245. le one with one loop using the program xfig which produces text output that consists of the coordinates of the points on the coaster I then extracted from these points two tables of the x y values which I stuck in two tables xcoast tab and ycoast tab available on the web Here is the ODE file for the coaster coaster2 ode coaster2 ode this is a rather simple coaster tab xx xcoast tab tab yy ycoast tab numerically compute x x etc xp xx sth xx s h 2 h yp yy sth yy s h 2 h xpp xx sth 2 xx s xx s h h h ypp yy sth 2 yy s yy s h h h vVsq xp xptyp yp aa xp xpp yp ypp vsq now add a few parameters to scale time and gravity etc s v v r yp vsq q aa v v par r 175 h 02 q 1 give it a little push init v 1e 5 8 3 My favorites 213 here is the coaster aux x xx s aux y yy s bound 100000 total 40 xp x yp y xlo 0 ylo 0 xhi 12000 yhi 4500 1t 0 done I have used the table function from XPPAUT to load in the coordinates of the coaster I also numerically differentiate these two functions Note that I have set a flag in the table files that tells XPPAUT to use a cubic interpolation rather than the default linear interpolation for evaluating the tables This way the derivatives are a little smoother The flag is set in the first line of the table file as a letter preceding the number of entries of the table Here are the first few lines
246. left the voltage as a function of time during a burst top right the slow parame ter J and the voltage bottom superimposition of the bifurcation diagram in the J v plane 2 2 ooo oo o The Zipper carnival ride Left is the manufacturer s schematic right shows the Zipper as rendered by XPP Curve fit to experimental kinetics model Top show the initial guess and bottom shows the parameters found by curve fitting Top The PRC for the van der Pol oscillator with different ampli tudes Bottom The PRC for the Morris Lecar model The basin boundaries for the iteration which arises from the appli cation of Newton s method to the equation z 1 in the complex plane Each point in the plane is colored according to the root that it CONVEFSES TO e oy a a bee ae ee OA Ae a Available postscript linestyleS o o Error from Euler left and improved Euler right xi xii List of Figures Preface XPPAUT is a tool for simulating animating and analyzing dynamical sys tems The program evolved from a DOS program that was originally written so that John Rinzel and I could easily illustrate the dynamics of a simple model for an excitable membrane The DOS program PHASEPLANE became a commercial project and was used for many years by a number of patient folks In the early 1990 s I spent a month in a beautiful office at the Mathematical Sciences Research Institute
247. liders are described next There are many ways to change parameters as well Here are three of them e From the Main Window click on Parameters In the command line of the Main Window you will be prompted for a parameter name Type in the name of a parameter that you want to change Click on Enter to change the value and Enter again change another parameter Click on Enter a few times to get rid of the prompt e Bring up the Parameter Window by clicking on turquoise Param button at the top of the Main Window In the Parameter Window shown in figure 2 6 type in values next to the parameter you want to change Use the scroll buttons or the keyboard to scroll around As in the Initial Data Window there are four buttons across the top Click on Go to keep the values and run the simulation click on Ok to keep the parameters without running the simulation Click on Cancel to return to the values since you last pressed Go or Ok The Default button returns the parameters to the values when you started the program e Use the little sliders Fig 2 7 We will attach the parameter d to one of the sliders Click on one of the unused parameter sliders Fill in the dialog box as follows Paranster d Value 0 and click Ok You have assigned the parameter d to one of the sliders and allowed it to range between 1 and 1 Grab the little slider with the mouse and move it around Watch how d changes Now click on the tiny go button in th
248. llators say Xi and Xo Xi F X1 Gi X2 X1 X F X2 a eGa X1 X2 where G1 G2 are possibly different coupling functions and e is a small positive number One can apply successive changes of variables and use the method of averaging to show that for e sufficiently small X t Xo 0 Ole 228 Chapter 9 Tricks and advanced methods with 0 1 eH1 02 61 6 1 H2 01 02 and H are P periodic functions of their arguments This type of model is called a phase model and has been the subject of a great deal of research One of the key questions is what is the form of the functions H and how can one compute them This is where XPPAUT can be used The formula for H is 1 P H X0 GX1 0 Xo 0 at 9 1 which is the average of the coupling with a certain P periodic function X called the adjoint solution This equation satisfies the linear differential equation and normalization dX t TA Dx ROCA X t X4 1 Here Dx F is the derivative matrix of F with respect to X and A is the transpose of the matrix A Thus to compute the functions H the adjoint must be computed along with the integral XPPAUT implements a numerical method for computing the adjoint X t invented by Graham Bowtell The method only works for oscillations which are asymptotically stable Here is the idea Let B t Dx F Xo t Since the limit cycle is asymptotically stable if we integrate the equation Y
249. llespie method applied to the Brusselator left and to a membrane model e eee eee 109 An asymmetric periodic potential 111 Illustration of Parrondo s paradox games 1 and 2 are losers but played together become winners o o 115 PSTH for the noisy theta neuron left and the spike count distri butions for a Poisson process with rate 0 05 and the same process with a refractory period ooo 2 0 00 000 4 117 The boundary condition window 000 125 The connection of the unstable dashed and stable solid mani folds of a fixed point as the parameter varies 131 Stable and unstable manifolds for the fixed point 1 0 of the Fisher CGUAION 60 e andar de Aes aye Se a Ag ayn Aly ee ae 133 Establishing shooting sets for the bistable reaction diffusion equation 136 Steady state behavior of a nonlinear dendrite Top Some repre sentative solutions Bottom Flux at x 0 as a function of g 140 The array plot window showing the space time behavior of the cable equation eo a a RR ee as 142 Solutions to the cable equation at 4 different times 143 Solutions to the CGL equation 146 Solutions to the delayed neural net equation with a delay of 1 151 A dendritic tree rendered in the animation window 157 Cellular automata simulation 0 0 159 The AUTO window s s sere dh bod dee pat
250. lls XPP that the variable jnamej is to be considered modulo the period You can repeat this for many variables AUTO options The following AUTO specific variables can also be set NTST NMAX NPR DSMIN DSMAX DS EPSS EPSL EPSU PARMIN PARMAX NORMMIN NORMMAX AUTOXMIN AUTOXMAX AUTOYMIN AUTOYMAX AUTOVAR The last is the variable to plot on the y axis The x axis variable is always the first parameter in the ODE file unless you change it within AUTO Miscellaneous e AUTOEVAL 1 0 tells XPPAUT whether or not to recompute ta bles whenever parameters are changed If you have a table of random connectivity you want this turned off 0 e BELL 0 turns of the bell for XPPAUT e COLORMAP 0 1 2 3 4 5 switches the color map e 0 standard e 1 periodic hot cool red blue grey oF Ww N Appendix C Numerical methods Here I briefly describe the numerical methods used in XPPAUT C 1 Fixed points and stability XPPAUT can find fixed points of maps and differential equations Finding fixed points involves solving G X 0 where G X F X for differential equations X F X and G X X F X for maps Xn 1 F X Newton s method is iterative and satisfies the scheme Xk 1 Xk J7 G Xx where J is the matrix of partial derivatives of G evaluated at Xy XPPAUT s im plementation of Newton s method requires three parameters the maximum number of iterates the tolerance and a p
251. lly the only choice and then accept all the defaults except maxPoints Change this to 1000 or even 400 Then click on Esc to get back to the main menu Integrate this equation Now let s plot it in three dimensions Click on View axes 3D Fill the first three entries in putting X on the X axis and so on for Y Z Click on Ok and you will see an empty window Click on Window zoom Fit W F and the window is optimally chosen to include the entire graph You should see something like Fig 5 3 For fun you might want to look at the bifurcations of this equation as the parameter r varies There is a constant c1 7 such that if r lt ci r then u 0 is a stable fixed point At r c T there is a transcritical bifurcation to a nonzero positive fixed point This remains stable until r ca 7 where there is a Hopf bifurcation Small amplitude oscillations emerge and then go through the usual period doubling cascade to chaos You may want to prove some of these statements this problem has never been analyzed 5 1 3 Making 3D movies While you have this file running and a nice three d image you can try to make a little movie Click on 3d params Ignore the first column of the dialog box but fill in the second column as 5 1 Functional Equations 97 Movie Y Vary theta phi theta Start angle 15 Increment 15 Number increments 23 This tells XPPAUT to draw 23 frames of the screen image rotating around the theta axi
252. lot is scaled to a cube with vertices that are 1 and this cube is rotated and projected onto the plane so setting these to 2 works well for 3D plots F 1 Cheat sheet 285 e XMAX value XMIN value YMAX value YMIN value ZMA X value ZMIN value set the scaling for three d plots e OUTPUT filename sets the filename to which you want to write for silent integration The default is output dat e POIMAP section maxmin sets up a Poincare map for either sections of a variable or the extrema e POIVAR name sets the variable name whose section you are interested in finding POIPLN value is the value of the section it is a floating point POISGN 1 1 0 determines the direction of the section POISTOP 1 means to stop the integration when the section is reached RANGE 1 means that you want to run a range integration in batch mode RANGEOVER name RANGESTEP number RANGELOW number RANGEHIGH number RANGERESET Yes No RANGEOLDIC Yes No all correspond to the entries in the range integra tion option e TOR_ PER value defined the period for a toroidal phasespace and tellx XPP that there will be some variables on the circle e FOLD name tells XPP that the variable name is to be considered modulo the period You can repeat this for many variables e AUTO stuff The following AUTO specific variables can also be set NTST NMAX NPR DSMIN DSMAX DS PARMIN PARMAX NORMMIN NORMMAX AUTOXMIN AUTOXMAX AUTOY
253. luated at each step Thus we will put the string t on the screen followed by the value of t Finally let s make a permanent version of your animation There are two ways to do this an easy and a hard way The hard way is to first produce a sequence of ppm files and then combine them with mpeg encode This technique produces smaller files but requires an enormous amount of disk space The manual that comes with XPPAUT explains how to do this in detail Instead we will use the easy way make an animated GIF file Animated GIF is a standard used to produce those annoying animations that populate many web pages The files can be played using any browser Suppose you have an animation you are happy with Then click on the MPEG button in the animation window In the resulting dialog box change the 0 to a 1 in the entry labeled AniGif Close the dialog box Now click on Skip and change the Increment from 1 to 2 which plots every other frame Click on Reset to reset the animator to the beginning of the data file Then click on Go in the animation window and the animator will start up again It may pause on occasion as it writes to the disk After a while 200 frames will be written into a file called anim gif The file is always called this so if you want to have multiple animations make sure you rename it after you are done The file anim gif is about 225000 bytes which is pretty small for a 200 frame animation Try playing it by loading it into
254. lue sets the starting time default is 0 e TRANS value tells XPP to integrate until T TRANS and then start plotting solutions default is 0 NMESH integer sets the mesh size for computing nullclines default is 40 e METH discrete euler modeuler rungekutta adams gear volterra backeul qualrk stiff cvode 5dp 83dp 2rb ymp sets the integra tion method default is Runge Kutta DTMIN value sets the minimum allowable timestep for the Gear integrator e DTMAX value sets the maximum allowable timestep for the Gear integrator e VMAXPTS value sets the number of points maintained in for the Volterra integral solver The default is 4000 JAC_EPS value NEWT_TOL value NEWT_ITER value set parameters for the root finders e ATOLER value sets the absolute tolerance for CVODE e TOLER value sets the error tolerance for the Gear adaptive RK and stiff integrators It is the relative tolerance for CVODE BOUND value sets the maximum bound any plotted variable can reach in magnitude If any plottable quantity exceeds this the integrator will halt with a warning The program will not stop however default is 100 DELAY value sets the maximum delay allowed in the integration default is 0 e PHI value THETA value set the angles for the three dimensional plots e XLO value YLO value XHI value YHI value set the limits for two dimensional plots defaults are 0 2 20 2 respectively Note that for three dimensional plots the p
255. lving differential equations Furthermore there is not much flexibility in the choice of integration methods and the integration is not done interactively That is you cannot see the progress of the solution until it is computed Standard qualitative tools such as direction fields and nullclines require additional packages or writing by hand MATLAB has great flexibility and can even integrate differential equations with discontinuities such as the integrate and fire equations However the numerical integration is generally slower than can be achieved with XPPAUT None of the packages offers an interface to AUTO the main reason that some people use XPPAUT The syntax for setting up differential equations is pretty simple compared to the other programs Finally XPPAUT is xiii xiv Preface free no license demons crashing once a year no guilt copying to another computer and the source code is always there for the taking How to use this book This book is written to be used by either a researcher or modeler who wants to simulate and analyze her system or by students as an adjunct to a modeling class or a class in differential equations I have used it in many such courses both at the level of sophomore engineering students up through graduate students in a dynamical systems class I have used XPPAUT in applied courses for students in neuroscience and physiology The present book contains many examples and many exercises Along the way I h
256. lyze a real world example that we have already studied the Morris Lecar equations dV Coe I gil Ei V gw Ex V JCaMx V Eca V dw dt We considered a dimensionless version of these and will continue to do so here Here is the ODE file for a single neuron m1l ode Woo V w Aw V 174 Chapter 7 Using AUTO bifurcation and continuation ml ode morris lecar dimensionless v I g1 el v gk w ek v t gca minf v eca v w winf v w lamw v par I 0 phi 333 par ek 7 eca 1 el 5 par gl 5 gk 2 gca 1 par vi 01 v2 0 15 v3 0 1 v4 0 145 minf v 5x x 1 tanh v v1 v2 winf v 5x x 1 tanh v v3 v4 lamw v phi cosh v v3 2 v4 aux ica gca minf v v eca aux ik gk w v ek total 50 xlo 6 xhi 5 ylo 25 yhi 75 xplot v yplot w done Note that the first two parameters the current I and the overall rate of the potas sium current phi are the ones we want to vary I have added a pair of auxiliary variables which track the active currents Start up XPPAUT and integrate this equation using Initconds Go followed by Initconds Last to get rid of all the transients Click on File Auto to bring up AUTO Since the parameters that we want to vary were the first defined they will be among the list of usable parameters Click on Parameter in the AUTO window just to check and see that I and phi are the first two entries Click on Ok to close the dialog box From here on in th
257. m aw lt 00 r gt 0 r r 0 r A Taylor series expansion near the origin reveals that in fact k r r 0 as r 0 This is a third order boundary value problem but there are 4 conditions that must be met However the parameter Q is free so that we actually have four degrees of freedom there is hope In Paullet et al the existence of solutions to this problem was proven for sufficiently small values of g d however the form of the solutions and the extent of the existence is not evident from the proof In order to solve this using XPPAUT we want to cast it as a system of first order differential equations Since we are not starting at the origin it is best to introduce an equation for the independent variable r The following ODE file will do the trick tt greenberg problem set up for r0 1 r0 gberg_auto ode 6 1 Boundary value problems 129 init a 0 00118 ap 1 18 k 0 omeg 0 19 r 001 domain is reasonably small sqrt 1 d par d 0 177 q 0 5 sig 3 r0 001 the odes a ap ap a k k ap rta r r ax 1 axa d k k r 2 k ap a omeg q rara d extras to make it autonomous omeg 0 r the boundary conditions at r 0 bndry a r0 ap bndry k bndry r r0 at r 1 bndry ap bndry k set it up for the user xhi 1 001 dt 01 total 1 001 ylo 0 done The file is pretty self explanatory Note that I have included a differential equation for r the radial coordinate which is the independent
258. m called shooting For each free parameter there should be a differential equation so that the number of differential equations equals the number of conditions that must be satisfied The eigenvalue problem has only two differential equations However we can add the trivial differential equation Wz 0 which just says that w is a constant Now given that there are N differential equations and N conditions how does XPPAUT solve the equations Suppose that t Xo is the solution to the differential equation X F X t X 0 Xo We have k conditions at t 0 and N k conditions at t L the end of the interval Thus there are N k initial conditions that are free to be chosen This means that we have to solve a subset of N k nonlinear equations from the conditions L Xo XL The only general way to solve nonlinear equations is Newton s method This is exactly what XPPAUT does Here is the algorithm 1 Guess a set of initial conditions Solve over the given interval Compute the error If it is small enough then exit with success 2 Numerically compute the Jacobian from the differential equations by making small perturbations of the initial conditions 3 Use the error in step 1 and the Jacobian in step 2 to improve the initial guess via a Newton iterate x pew age as J E where E is the error from step 1 4 If the Jacobi matrix is uninvertible or too many iterates have been run without convergence then exit w
259. m ode animation for the pendulum the frame line 5 1 9 1 GREEN 2 8 2 Some first examples 201 The pendulum Figure 8 1 The animation window line 5 5 7 5 GREEN 2 line 7 5 7 1 GREEN 2 the pendulum line 5 5 5 3 sin theta 5 3 cos theta BLUE 3 fcircle 5 3 sin theta 5 3 cos theta 04 RED done For the green frame we have drawn a series of three line segments which are two pixels wide The new command starts fcircle x y r and draws a filled circle centered at x y with radius r The last option is just the color Thus we draw a bob that is a filled red circle centered at the end of the rod and with a radius of 0 04 Edit your old pendulum ani file and change it as above In the animation window click on File and Ok since this file should be the default now as you 202 Chapter 8 Animation already loaded it Click on Go and you will see the full animation Note that we really don t have to draw the frame every time as it never changes it should be drawn only once The animator has a control for that PERM that says that every drawing command that follows only has to be calculated at the beginning and doesn t change thereafter The opposite command is TRANS which tells the animator to recompute every time So we could improve this slightly by adding PERM and TRANS to the file We will also add a nice title in black pendulum ani goes with pendulum ode animation for th
260. m of two first order equations xt v v av sinz Look at the phaseplane plot x along the horizontal and v along the vertical axis for this in the case of no damping a 0 and small positive damping A good window to use is 8 8 x 3 3 Draw the direction field and classify all of the fixed points with and without damping In any message boxes related to the computation of fixed points answer No in every case If by mistake you click on Yes then repeatedly tap the Esc key Draw some representative trajectories with and without damping Plot your work in the damped and undamped case as a Postscript file Fire up the Fitzhugh Nagumo equation Increase the parameter I from zero and determine the value at which the fixed point becomes unstable Continue to increase J and figure out the value at which the fixed point again becomes stable Next set J 0 and change y to 10 erase the screen and recompute the nullclines How many fixed points are there Determine their stability Choose a bunch of initial conditions and see if you an detect a difference between the long time behavior of the trajectory 26 2 3 2 4 Chapter 2 A very brief tour of XPPAUT Write a differential equations file for the Lorenz equations x s x y y rxz y rz z bz xy with initial conditions x 7 5 y 3 6 z 30 and starting parameter values of r 27 s 10 b 2 66 Don t forget to put the multiplica
261. maps Each map consists of first dividing the coordinate in half Then with probability 1 3 1 2 is added to the x coordinate 1 2 is added to the y coordinate or nothing is added Here is an ODE file simple iterated function system sierpinsky triangle par c0 0 c1 5 c2 0 par d0 0 d1 0 d2 5 p f1lr ran 1 3 x 5x x shift c0 p y 5 y shift d0 p meth discrete total 20000 maxstor 100000 xp x yp y xlo 0 xhi 1 ylo 0 yhi 1 1t 0 aux ppp done 64 Chapter 4 XPPAUT in the classroom Notes I define 3 values for constants that are added after multiplying by 1 2 c0 d0 c1 d1 c2 d2 Then I flip a three sided coin p which takes values of 0 1 or 2 The function shift c0 p will be c0 c1 c2 according as whether p is 0 1 or 2 respectively Run this and you will see the Sierpinsky triangle appear on the screen Zoom in to see similar versions of it We close this section by writing a simple IFS that has 4 general probabilities and 4 general affine maps It should be clear from this how to generalize to more maps ifs4 ode general code for ifs with 4 choices par ai 0 a2 85 a3 2 a4 15 par b1 0 b2 04 b3 26 b4 28 par c1 0 c2 04 c3 23 c4 26 par d1 16 d2 85 d3 22 d4 24 par e1 0 e2 0 e3 0 e4 0 par f1 0 f2 1 6 f3 1 6 f4 44 par p1 0 01 p2 0 85 p3 0 07 p4 0 07 s1 p1 s2 51 p2 s3 s2 p3 z ran 1 i1 if z gt s1 then 1 else 0 i2 if z gt s2 then 2 else i1 i if z gt s3 then 3
262. merics Colorcode Another quant U C A Accept the default of T Click on Choose and choose 2 for Min and 12 for Max Click on Esc to get to the main menu and redraw the graph by clicking R You should see the familiar albeit somewhat crude colorized Mandelbrot set To get a nice picture of it just create a new smaller window which is 200 x 200 Click on Makewindow Create and resize it You should see something like Figure 4 6 Exercise Make a Julia set animation with the function f z 22 c and then make a Mandelbrot set for this function Note that for cube roots you want to randomly choose between three possible roots A good range for the cubic Mandelbrot set is Cy between 0 8 and 0 8 and cy between 1 5 and 1 5 Hint 2 a 3y x ya y Create some art by changing some of the coefficients in the nonlinearities obtained above Add some quadratic terms change a few signs whatever To speed things up change the 200 to 100 in the two parameter range When you find something good then do a finer version You don t have to create a new XPPAUT file for this exercise Just click on File Edit RHS s to edit the right hand sides of the equations 62 Chapter 4 XPPAUT in the classroom Figure 4 6 Mandelbrot set 4 2 6 Iterated function systems A variety of interesting fractals can be created by using randomly chosen affine maps in the plane An affine map is just a transformation of the form zo Ar b If you tak
263. mes If you want to manually step through the screen shots choose Kinescope Playback Clicking any mouse button advances to the next frame Clicking Esc gets you out of the loop You can save these as an animated GIF file also by using the Kinescope Make Anigif command The file is always saved as anim gif 4 6 1 Poincare maps FFTs and chaos The Lorenz equation is famous as it was one of the first numerical examples of chaotic behavior There are many ways to characterize chaos in a deterministic 4 6 Three and higher dimensions 83 dynamical system Two of the most common characteristics are the existence of infinitely many unstable periodic orbits There is a classic mathematical theorem due to Sarkovskii who showed that for discrete dynamical systems if there is period 3 orbit then there are infinitely many periodic orbits This was later reproved by Li and Yorke in a classic paper Period three implies chaos Recall that we found all the period 3 orbits in the logistic equation in section 4 2 2 Another way and better known to characterize chaos is through sensitive dependence on initial conditions That is if one starts with two initial conditions close to each other the solutions will diverge from each other at an exponential rate Sensitive dependence Delete the graphs other than the original plot of the Lorenz equation by clicking Graphic stuff Remove all G R Then plot X as a function of time in your window by clicking X
264. mple consider the two text strings ZI iapp and I iapp Both will appear the same However if you change the parameter iapp then the second one will reflect this change while the first remains frozen with the original value 9 2 Graphics tricks 223 The Arrow option lets you draw little arrowheads in the direction that you define with the mouse The tip of the arrow head is the first point and the direction is defined by the release point of the mouse You will have to play around with this until you are happy with it as the relationship between the size and what is drawn is obscure The Pointer option lets you draw straight lines with arrow heads Choose size 0 to put on no arrow heads The Marker option lets you draw a variety of symbols all centered at the position of the mouse when you click it The markerS option lets you mark points along the trajectory you have drawn In addition to size shape and color you should choose the starting row row 0 starts at the beginning of the trajectory row 1 is at the next output point end so on the number of markers and the number of rows to skip between each You can alter the little graphics objects or delete them all from this menu item Line type In the Graphic stuff Add curve or Edit curve dialog box you can choose both a color and a linetype Linetype 1 the default is a solid line Linetype 0 draws a single point instead of a line However a negative line type say 3 will dra
265. n Window click on View axes Toon and a new window will pop up see Fig 8 1 It has as its title some random cartoon that I like In the animation window click on the File button Choose the animation file pendulum ani assuming you have either downloaded or typed it in and saved it If you made errors an error message will pop up Otherwise it is probably OK In the animation window click on the Go button and it should start to swing sort of like England Click on Pause to stop it and Reset to rewind it to the beginning The left and right buttons advance by a single frame at a time The Skip button lets you change the frame rate so e g only one out of every 3 frames are drawn MPEG allows you to create mpeg movies with an additional program called mpeg encode or animated gifs I describe how to do this later Finally there is a little checkbox in the upper right corner If you click on this then the animation is run simultaneously while you solve the ODE This slows the integration of the equations down dramatically as drawing the animation is quite CPU intensive Check the box and then change parameters or initial data and then integrate You can watch the animation on the fly Change the initial conditions and re integrate the equations Click on Reset Go inthe animation window and see how different it looks Choose theta 3 v 5 and watch the pendulum It windmills Now lets add the bob and the frame pendulum ani goes with pendulu
266. n increase by one of Y and reactions 2 and 4 lead to a loss of Y If you run this in XPPAUT and plot the Y1 Y2 phase plane you will get a figure something like the left side of Figure 5 6 As a final example we consider a simple model neuron which has three chan nels a leak channel a potassium channel and a persistent sodium channel The leak channel has a fixed conductance the potassium channel has the usual voltage gated conductance as does the sodium channel However here we will assume that the number of sodium channels is finite so that the random opening and closing of them becomes important Here is the system of equations as I g1 V Er gkw V Ex gnat V Ena 5 2 dw _ Wo V w de aV 5 3 and z is the fraction of open sodium channels Let o be the number of open channels and c N o be the number of closed channels where N is the total number of channels The transition between open and closed states is governed by voltage dependent rates C O Oo C with rates a V and b V respectively We can implement the Gillespie method on the transitions between open and closed just as we have in the previous examples 110 Chapter 5 More advanced differential equations However we now have in addition a pair of ordinary differential equations to deal with Thus as each reaction advances the clock by an amount At we must integrate the equations by this amount of time XPPAUT does not
267. n of the people in a car aux xpeople xp aux ypeople yp now for the animation we will make the track xt 0 17 zx j xb zy j yb yt o 17 1 zx 3 yb zy 3 xb O xp x yp y x10 30 yl0 30 xhi 30 yhi 30 dt 05 total 240 bound 10000 done It looks more complicated than it really is most of the file is stuff to compute the second derivative of the forcing function The animation file is real simple as we have already done all the work in the ODE file Here is zipper ode animation file for the zipper perm line 5 0 5 5 GREEN 5 trans here is the track line 5 xt 0 15 s 5 yt j s 5 xt j 1 s 5 tyt j 1 s BLACK 2 fcircle 5 xc s b yc s 015 RED fcircle 5 xp s 5 yp s 02 BLUE line 5 xc s 5tyc s 5 xp s 5 yp s end The first line draws the stand for the zipper The next line is the track A red circle marks the pivot point on the track and a blue one the car A line joins them This company has many interesting rides which await analysis For example try to model their ride called CHAOS and see if it is appropriately named It consists of a rotating tilted wheel with freely swinging cars Thus it is essentially a periodically forced pendulum Who ever thought that Melnikov and transverse homoclinics could be so fun 8 3 4 The Lorenz equations Strogatz 1990 derives the famous Lorenz equations by looking at the leaky water wheel Consider an array of small cups placed ar
268. n you plot the results a line from the end of one run is drawn to the beginning of the next This can look quite ugly There are two ways to avoid this The first is to make the linetype 0 or negative 224 Chapter 9 Tricks and advanced methods But this is not completely satisfactory since you just get points instead of nice solid lines The second is to take advantage of a plotting trick XPPAUT uses When XPPAUT plots solutions to equations defined on a torus it knows that there is a jump from T to 0 where T is the period of the torus See section 9 5 4 As a consequence it looks for jumps and will not plot lines joining points that are more than some fraction of this apart in distance Thus look at your graph and estimate how big the jump from the end of one run to the beginning of the next is Say for example you are plotting many runs against time and the time interval is 20 AFTER you have done your integration click on the phAsespace and choose All For the Period choose a number slightly less than this jump XPPAUT will see that the distance between these two points is larger than the period and will not plot line between them Furthermore the PostScript plots will also not have this jump 9 3 Fitting a simulation to external data XPPAUT is able to import external data solve an initial value problem and used a least squares approach to find the best values of initial data and parameters to match the data The method
269. nd of this section The minimal steps are e Use an editor to open up a text file Write the differential equations in the file one per line Use the par statement to declare all the parameters in your system Optionally define initial conditions with the init statement e End the file with the statement done e Save and close the file NOTE The equation reader is case insensitive so that AbC and abC are treated as identical WARNING In statements declaring initial conditions and parameters do not ever put spaces between the variable and the sign and the number XP PAUT uses spaces as a delimiter Always write a 2 5 and never write a 2 5 2 2 Running the program 11 ParsYar ParsYar Par Var L T TF EAT L TTE Figure 2 1 The main XPPAUT window 2 2 Running the program Run XPPAUT by typing xpp linear2d ode Replace xpp with whatever you have decided to call the executable with all the desired command line options If you are using winpp click on the winpp icon then choose the file from the file selection dialog box A single window will appear unless you start XPPAUT with all the windows visible xppaut allwin see Chapt 1 2 2 1 The main window The Main Window contains a large region for graphics menus and various other gadgets It is illustrated in Figure 2 1 Commands are given either by clicking on the menu items in the left column with the mouse or tapping keyboard shortcuts After a w
270. nd sample trajectories for the equation x a 1 zx you can plot or view later Delete the saved trajectories if you want by clicking on Graphic stuff Freeze Remove all Figure 4 7 shows an example Exercises Draw direction fields for the following functions f x t i t 10 ii sin 4x cos t 2 iii t x iv t 10 z v sin xt Try to guess what solutions look like based on the direction fields and verify your guess by computing a number of trajectories Hint To change the function f x t just click on File Edit Function and change f z t 4 4 Planar dynamical systems Two dimensional differential equations play an important role in many undergradu ate differential equations courses This is because of the special topological proper ties of the plane in particular the Jordan Curve Theorem This constraint implies that the only attractors in the plane are fixed points and limit cycles Most text books that emphasize the qualitative aspects of differential equations include a chapter or two on planar ODEs XPPAUT has many features that are useful in the analysis of these systems such as the ability to draw direction fields nullclines and separatrices for saddle points I have used XPPAUT with several undergraduate texts in ordinary differential equations Used intelligently one can automate the analysis of planar systems and teach some complicated dynamical systems concepts I will start this chapter with the an
271. ndary 158 Chapter 6 Spatial problems PDEs and boundary value problems conditions are periodic Fix m and let f be a map from 0 1 2m 1 to 0 1 Then we can view a one dimensional cellular automata as the discrete dynamical system oss va d Since the domain of f is generally a small number of discrete points we will read in f as a lookup table We will define a weight matrix of all 1 s which we convolve with u from m to m and then apply f to the resulting sum Here then is an XPPAUT file for a CA of 101 cells which looks at 2 neighbors to the left and right as well as the value of the cell itself The rules are in the form of a table which the user can modify externally and read in ca100 ode 101 cell CA model here is the function for the new value table f ca5 tab add up with equal weight 5 cells table wgt 5 2 2 1 The convolution is created special k conv periodic 100 2 wgt u0 here is the update u o 100 f k Lj make sure XPP knows it is discrete time total 200 meth discrete done NOTES The table that defines f the rule is read in as a file Here is the table file PRRPROPOOCIO OD The first line tells XPPAUT how many points Since the sum of neighbors can be 0 1 5 there are 6 possible sums The next two lines tell XPPAUT the domain of the function The last 6 lines are the rule itself Thus for example if the sum of neighbors is 3 then the value is f 3 0 while if the
272. ng 100 1000 and 4000 channels 3 A recent PRL paper showed that you could get directed motion with mul tiplicative noise if it is correlated with additive noise Here is an example system a sin x 2 Ji V Di A1y1 sin x 20 J2 Y D2 y2A2 VD Ary Aya Y 1 AZ A3 w where y1 ya w are normally distributed properly scaled random variables Note that the additive noise is correlated with the multiplicative noise if A 0 Write an ODE file and simulate it up to t 500 200 times and plot the mean value of x as a function of time for D 3 J 0 7 j2 1 D Do 0 3 o 7 2 and Ay A2 0 3 Repeat this with A A2 0 Note the lack of any drift in the latter case 5 3 Differential algebraic equations 119 5 3 Differential algebraic equations XPPAUT has some simple code to solve differential algebraic equations DAEs These are equations that have algebraic relationships between variables that are in turn used as the right hand sides of other variables They are commonly used in chemical reactor theory A typical DAE has the form du dt f u v 0 g u v Programs such as DASSL Brenan et al 1989 and MANPAK Rheinboldt handle equations like this fairly easily but use very different methods XPPAUT uses a rather simple scheme which is related to the methods used in MANPAK but not nearly as sophisticated Newton s method is used to solve the algebraic part and then the ODE part is
273. ng backwards Change 91 0 02 3 55 and w21 7 and show that there is a stable limit cycle a stable fixed point and an unstable limit cycle This bistability occurs for a completely different mechanism from the bistability induced by the homoclinic bifurcation above c Find some parameters such that there are 9 fixed points Hint try 111 W22 large and positive say around 10 01 62 positive and wi2 wai small and negative 4 6 Three and higher dimensions XPPAUT is able to create three dimensional plots of higher dimensional dynamical systems that are occasionally encountered in textbooks Here we will examine a few classic examples and then show some fancy projection plots Consider the Lorenz equations which are an approximation of a large scale atmosphere model Here are the equations Z sly a dy q TE Y UR e Oe dt The standard parameters are s 10 r 27 b 8 3 The XPPAUT file that I show below is rather complicated since 1 will do some fancy graphics tricks Here is the idea I will draw a three dimensional plot and then on one or more of the coordinate planes plot the corresponding two dimensional projections as in Figure 4 1 We will also explore Poincare maps Fourier spectra and Liapunov exponents with this example Here is the ODE file Lorenz ode the lorenz equations 4 6 Three and higher dimensions 81 with some fancy graphics options x s x y y r x y xX Z Z b Z x y pa
274. ng the y axis To do this in XPPAUT you want to make the parameter a variable and then use the Range Integration option to draw the diagram Here is the XPPAUT file logbif ode the logistic map bifurcation diagram f x a x 1 x x f x a a init x 1 init a 2 maxstor 100000 total 500 trans 350 meth discrete xlo 2 xhi 4 001 ylo 0 yhi 1 001 xp a yp x done NOTES I have set maxstor 100000 so that XPPAUT will store many points I throw away the first 350 iterates to avoid transients I treat the parameter as a variable so it can be plotted parameters are not generally allowed to lie on axes Run this file and then do the following First click on Graphics Edit and select 0 Then in the resulting dialog box change the line type from 1 to 0 This makes XPPAUT just plot points rather than lines Now we can draw the diagram Click on Initialconds Range and fill in the dialog box as follows 4 2 Discrete dynamics in one dimension 51 Figure 4 3 The logistic map showing an orbit diagram on the left and a plot of the periodic orbits of period 3 and period 5 on the right Range over a Steps 20 Sa E and click on Ok You should see the bifurcation diagram appear You can put more points in it by changing the number of steps from 200 to 500 The Range command allows you to range over either parameters or initial data and save the results Figure 4 3 shows the orbit diagram
275. nt this means crossing from above to below When this happens x is kicked backwards by the amount k Run this program Change the initial conditions note that no matter what you choose the solutions always converge to a stable periodic solution Let s add another window and plot x and y as a function of t Click on Makewindow Create M C The main window will be repeated Note the little rectangle in the upper left corner This indicates that this view is the active view All new plotting and changes in the graphics will occur in this window If you want the main window to be active simply click in it For now make the little window active and stretch it out a bit In the Initial Data Window click on the little boxes next to the x y variables and then click on the button xvst The result of this is that x and y will be plotted with x white and y red Click on Window zoom Zoom in W Z to zoom in on this and pick a few cycles so that you can see it better by clicking with the mouse on the little window and releasing the mouse after choosing a suitable area If you mess up click on Window zoom Fit W F which will resize the window to contain all the data The period of this clock is about 6 3 or so 3 5 Discontinuous differential equations 39 A chaotic clock We can make this little clock chaotic and in so doing introduce some additional features of XPPAUT In fact it is easy to prove that the resulting system is chaotic Here is wha
276. nted Graphics stuff Edit and choose 0 as the graph to edit Finally turn off the axes by clicking Graphics stuff aXes opt and set change both X org and Y org from 1 to zero Now run the simulation You will see a beautiful folded map This structure arises due to something called a Smale horseshoe that can be found in this system The horseshoe was one of the first examples of rigorously proven chaos in a dynamical system If you let the friction f and the forcing a be small in the forced Duffing equation it is possible to rigorously prove that there is a Smale horseshoe for the system by simply finding some zeros of a certain integral This is one of the few general methods we have of rigorously proving chaos in a dynamical system A simple map that has a similar folded structure is the Henon map Eny l ata Yn Yat jin which is a two dimensional map Here is the ODE file henon ode the henon map x 1 a x 2 y y j x par a 1 4 j 3 init x 316 y 206 xp x yp y xlo 1 3 xhi 1 3 ylo 4 yhi 4 total 20000 meth discrete 1t 0 maxstor 40000 done It is set up to iterate 20000 times I have to tell XPPAUT to increase the storage maxstor 40000 so that we can keep all the points I have also told XPPAUT to use linetype 0 Try this and then zoom into the folded parts See that it is like pastry all folded up tighter and tighter The Field Worfolk equations arise as a four dimensional normal form for cer tain symmetri
277. nter Oo rr er ore O cr jj f D 0 then there is a line of fixed points We will illustrate these with a generic two dimensional linear differential equa tion with parameters twodl ode simple linear planar dynamical system x a x bx y y c x dx y par a 1 b 2 c 3 d 1 x1o 2 y10 2 xhi 2 yhi 2 xp Xx yp y total 30 nmesh 100 done Note The file is straightforward the statement nmesh 100 tells XPPAUT to make a finer mesh for drawing nullclines which are explained below Run this file Click on Dir field flow Direct field D D to draw unscaled direction fields Since T 0 and D gt 0 it follows that the origin is a center Try d 2 This changes the center into a stable spiral Try d 3 What is the origin At what value of d does it become a node At what value of d does it become a line of equilibria Draw the direction fields and some trajectories when d 6 You should be able to make out the line of fixed points y x 2 Change d back to 2 Click on Dir field flow Flow D F and change the grid to 5 4 4 Planar dynamical systems 71 to draw a flow diagram of the system You will see a nice vortex Unfortunately XPPAUT does not save these trajectories but they do give a nice picture of what is going on Hardcopy hint Click on Kinescope Capture To grab the screen image Now click on Kinescope Save and choose any name you want for the base name e g twodl and then choose 2 GIF as the
278. ntries are always 0 since they are never actually computed they are just the probability of not changing and equal 1 minus the rates of the rest of the row The complete XPPAUT file for simulating the single channel is channel ode random channel model using HH alpha and beta v I g1 v v1 gna z v vna c par I 0 gl 1 gna 001 vna 55 v1 65 c 1 phi 1 init v 65 z 0 markov z 2 0 al v beta v 0 al v phi 1 v 40 1 exp v 40 10 beta v phi 4 exp v 65 18 aux ina gna zx v vna total 100 meth euler xlo 0 xhi 100 ylo 66 yhi 64 done 5 2 Stochastic equations 105 The file is pretty self explanatory I have added the current ina as something extra to plot The method of integration is Euler s method If you want to simulate thousands of channels it is much better to use a method due to Gillespie However for small numbers of channels one can define a bunch of Markov variables Another example of a continuous Markov model is due to Tom Kepler Here we take the standard logistic equation for growth of a single species We assume that there is a random mutation whose probability depends on the population of the first species This mutation competes with the first and eventually dominates The equations that Kepler proposes have the form x x1 1 21 before the new mutant appears and xi x1 l1 z1 2 2 axz l 21 22 e21 after it appears XPPAUT does not
279. o create a table from the the data in one of the columns Back to convolutions Suppose you have the array u 0 100 of 101 elements and you want to convolve it with the weights w 25 w 25 Then you want to form the sums 25 k 3 wuli J l 25 Since u is defined only for 0 to 100 we need a way to handle this sum when 1 is not between 0 and 100 XPPAUT provides three possibilities zero even periodic zero means that we define u to be zero outside the points O to 100 even means that the values are reflected across the boundaries For example u 2 ul2 and u 105 u 95 Finally periodic means that u j N u The special statement in an ODE file allows us to define the function k j For this particular example we would write table w 51 25 25 exp abs t 5 special k conv zero 101 25 w u0 where we have arbitrarily chosen an exponential decay for the weight function The conv function takes five arguments special k conv type n m w u0 6 2 Partial differential equations and arrays 149 As described above type is either even zero periodic depending on how you want to set the edges of the array The parameter n is the length of the array ula b and 10 is the name of the first element in the array n is also the length of the return values for k m is the halfwidth of the weight that is w j is defined for m lt j lt m and wis the name of the weight table As a warm up example we will
280. o know the name of your browser since it calls the browser to display the help Thus in my bashrc file I have the two lines export XPPHELP home bard xppnew help xpphelp html export BROWSER usr bin netscape The same should work for OSX since this new Mac operating system is Unix In Windows you can do the same thing using the set command set BROWSER c Program Files Netscape Communicator Program netscape exe set XPPHELP c xpp help xpphelp html I usually include all this in a batch file which sets up other parameters as well See below 1 5 Environment variables 7 1 5 1 Resource file The other environment variable that XPPAUT makes use of is the HOME directory XPPAUT looks here for the file xpprc Each time XPPAUT is run it loads the options defined in xpprc These are described in appendix B The resource file xpprc just contains global options that you might want to have for every ODE that you run For example here is a short one my xpprc file but quit fq maxstor 50000 be11 0 meth qualrk tol 1e 6 atol 1e 6 thats it 000 This automatically puts a quit button on the top bar allocates 50000 storage points instead of the default 5000 makes the default integrator the adaptive Runge Kutta and turns off the bell In Windows you set the HOME directory and in that directory create a file called xpprc It will be the same form as the UNIX version I usually make a batch file which does e
281. odel is the Fitzhugh Nagumo equation which is used as a model for nerve conduction The equations are dV dw with parameters f a e y Typical values are a 1 J 0 1 and y 0 25 Let s write an ODE file for this Fitzhugh Nagumo equations v I v 1 v v a w w eps v gamma w par 1 0 a 1 eps 1 gamma 25 O xp V yp w xlo 25 xhi 1 25 ylo 5 yhi 1 total 100 maxstor 10000 done 2 7 A nonlinear equation 21 We have already seen the first four lines i lines beginning with a are comments ii the next two lines define the differential equations and iii the line beginning with par defines the parameters and their default values The penultimate line beginning with the sign is a directive to set some of the options in XPPAUT These could all be done within the program but this way everything is all set up for you Details of these options are found in the appendix For the curious these options set the x axis xp to be the V variable the y axis yp to be the w variable the plot range to be 25 1 25 x 5 1 and the total amount of integration time to be 100 The last option maxstor 10000 is a very useful one XPPAUT allocates enough storage to keep 4000 time points You can make it allocate as much as you want with this option Here I have told XPPAUT to allocate storage for 10000 points Type this in or download it all of the files are available for downloading at the author s w
282. of dirent h not the one in usr include e In your local copy of dirent h change the include lt sys dirent h gt statement to call your local copy of sysdirent h e In your local copy of sysdirent h change the lines 6 Chapter 1 Installation u_int32_t d_fileno file number of entry u_int16_t d_reclen length of this record u_int8_t d_type file type see below u_int8_t d_namlen length of string in d_name to the new lines unsigned long d_fileno file number of entry unsigned short d_reclen length of this record unsigned char d_type file type see below unsigned char d_namlen length of string in d_name These occur in the struct dirent declaration and save the file 5 In the Makefile use the following options CC cc CFLAGS 0 DAUTO DCVODE_YES I usr X11R6 include LDFLAGS L usr X11R6 lib AUTLIBS 1f2c 1X11 1m LIBS 1X11 1m OTHERLIBS libcvode a libf2cm a Note that in the subdirectories cvodesrc and libI77 make sure that CC cc Then in the main directory type make and everything should go fine The rest of the story is like the UNIX installation 1 5 Environment variables XPPAUT makes use of certain environment variables In UNIX these are set in files such as bashrc In order to use the online help XPPAUT needs to know the start ing help file For example on my Linux computer this is home bard xpppaut help xpphelp html XPPAUT also needs t
283. of the table s35 0 1 300 600 900 The letter is s for spline the tables xcoast tab and ycoast tab are available on the web I have also set the plotting window to show the actual shape of the coaster as well as the velocity Since the coaster was defined with more grid points along the loops it is not really completely physically correct and things slow down along the curves more than they should 8 3 3 The Zipper The Zipper Chance Rides is a carnival ride that consists of a large ovoid track along which freely hanging cars move In addition the whole track moves in a circle The ride is full of action with the chairs spinning randomly as it makes it way around the track Figure 8 3 shows a sketch of the Zipper The freely hanging car with riders is essentially an underdamped pendulum If all the Zipper did was move the cars around the track it would be a periodically forced pendulum However the additional rotation about the central axis makes this a quasi periodically forced pendulum Thus we can expect the motion to be quite complex Conveniently one can find the manufacturer of this ride www rides com and get all the specifications such as the rate of rotation of the track and the dimensions of the ride see the figure which includes the manufacturer s diagram The hardest part of this simulation is to create the track I use a trick to do this I create an integrable dynamical system of the form a afyly y ful
284. of the number of games played Use ca as the variable to vary and set the maximum and minimum to 0 Game 2 is more subtle Here is the ODE file game2 ode this plays game 2 game p e if ran 1 lt p e then 1 else 1 par eps 005 p2 1 p3 75 m 3 p if mod ca m 0 then p2 else p3 ca catgame p eps total 100 meth discrete dt 1 done As above we play 100 games and look at the amount of cash we have Simulate this 1000 times and you will see that it is clearly a losing game You lose about 1 5 per 100 games Now finally we write a simulation for alternating between the two games Let Pa be the probability from game 1 and let p1 p2 be the two probabilities from game 2 Let y be the probability of choosing game 1 at any point in time Let e be a small parameter that makes the games unfair It is elementary to show that game 1 is a loser if 1 pa gt pa A less routine calculation shows that game 2 is a 114 Chapter 5 More advanced differential equations loser if 1 p2 1 ps gt p2 p3 e You can check that for the parameters chosen both games are losers However game 3 is a winner and thus illustrates what is called Parrondo s paradox if the following conditions are met 1 q2 1 q3 lt q243 q2 Pa 1 7 p2 q3 Pa 1 y p3 game3 ode this plays the combined game with probability 1 2 each
285. om 1 to 2 As an example consider the complex Ginzburg Landau equation CGL zi z 1 i z Z 1 ie zzz where z u iv For certain ranges of the parameters P and e this model ex hibits spatio temporal chaos We write it in real form and solve it with a small perturbation of u in the middle of the medium Here is the ODE file coupled Ginsberg Landau equation cgl ode d 1 h h r 0 127 u j u j v j v j u0 u0 u0 beta v0 r0 d u1 u0 eps v1 v0 vO v0 vO betax u0 r0 d vi v0 eps u1 u0 1 126 u j u j Culj betaxv 3 r 3 d ulj 1 u 5 1 2xu 3 eps v 5 1 v 5 1 2 v 31 v j v 3 v j betaxu j r 3 dx v 3 1 v 5 1 2xv 5 eps ulj 1 u 5 1 2xu 31 u127 u127 u127 beta v127 r127 d u126 u127 eps v126 v127 v127 v127 v127 beta ul27 r127 d v126 v127 eps u126 u127 parameters par beta 1 8 eps 8 h 1 aux rr 0 127 r j init u 50 55 1 dt 25 total 200 plot the modulus rr0 with 128 columns in the array plot done 146 Chapter 6 Spatial problems PDEs and boundary value problems 0 lt t lt 199 Figure 6 8 Solutions to the CGL equation I use the default Runge Kutta integrator since the problem as written is not partic ularly stiff The domain size is 128 units and the spatial grid h 1 Two lines bear mention d 1 h h defines d to be a derived parameter It is hidden from the user and its value dep
286. om 3 to 3 5 This system is very similar to the Rossler attractor 90 Chapter 4 XPPAUT in the classroom Chapter 5 More advanced differential equations In this chapter and the next we will go beyond the usual simple ordinary differential equations that arise in simple applications Many physical problems do not have a local dependence on time rather the state of the system depends on either the complete history of what has already happened or on the state some finite time earlier XPPAUT has many tools to let the user solve and analyze such functional equations Many biological systems are not deterministic but instead have stochastic components These can take many forms such as thermal noise Brownian motion and random switching of states e g the random opening and closing of channels XPPAUT has a variety of tools that let you study and simulate these random systems Finally in chemical and mechanical systems there are often algebraic con straints which relate variables This leads to so called differential algebraic systems DAE s XPPAUT is able to solve a restricted class of such models This chapter shows you how to write ODE files for these more complicated model systems 5 1 Functional Equations A large class of interesting dynamics are described by functional equations such as delay equations and Volterra type integral equations XPPAUT allows you to numerically solve delay equations as well as solve a variety o
287. om the Hopf bifurcation Plot the frequency of the periodic orbits as a func tion of the current J For a bonus problem compute the two parameter curve of Hopf bifurcation points using phi as the second parameter window the two parameter graph with O lt phi lt 2 Thus show that if lt there are two Hopf bifurcations Find Compute the one parameter bifurcation diagram for the Brusselator ul a b 1 u vu v bu vu witha 1 b 1 5 u 1 v 1 5 and treating b as the bifurcation parameter In Numerics set Par max to be 4 Window the diagram 0 lt u lt 6 and 0 lt b lt 4 This one is for bifurcation jockey s only Explore the bifurcation diagram of the coupled Brusselators uy f ui 1 7 4 vi g ur v1 d v2 v1 7 5 uz f uz va 7 6 va g u2 v2 d v1 v2 7 7 f u v a b 1 u vu 7 8 glu v bu vu 7 9 with d 0 2 a 1 and letting b be the bifurcation parameter In order to do this one completely you should follow all special points In particular branch points of periodic orbits labeled BP For these you may have to grab them twice and change the direction of the continuation Ds in the Numerics menu A number of other parameters have to be changed in the Numerics menu for AUTO namely the number of collocation points used to track peri odic orbits Ntst 30 the minimum step size Dsmin 1e 5 and the maximum step size Dsmax 0 1 The diagram should be drawn
288. on happen at shorter time intervals if you double the number of channels you want to double the number of reaction steps In the numerics menu change Total and also change Nout doubling that as well The right panel of figure 5 6 shows the result of one such calculation 112 Chapter 5 More advanced differential equations 5 2 3 Ratchets and games Brownian ratchets of various sorts have been the subject of much recent interest Many molecular motors work on this principle The idea is that an asymmetric potential is periodically varied and combined with Brownian motion The net effect of these two random processes is the performance of work Let V x be a potential function Assume that it is periodic on the line with period 1 and that it is asym metric see the figure 5 7 Let z be a two state Markov process that determines the magnitude of the potential when z 0 the potential has zero magnitude and when z 1 it is maximal Finally assume a certain amount of Brownian motion Then when the potential is on the probability distribution builds up proportionally to P x exp V 1 0 When the potential is turned off there is diffusional drift but it favors one direction more than the other because of the asymmetric potential The result is that there is a net flux in one direction Many molecules work in this fashion by changing configuration due to the hydrolysis of ATP to produce in effect a molecular ratchet driven by the noise T
289. onlinear example Below the stable and unstable manifolds of the fixed points This is not simple to analyze by hand That s why we use the computer to complete the picture The first thing to do is to put it into XPPAUT Here is the ODE file ex2d ode example from Golubitsky x x y x 2 y72 0 1 y y 2 x y x 2 2 y72 O xp x yp y xlo 6 xhi 6 ylo 6 yhi 6 dt 02 done I have taken the liberty of setting it up as a phaseplane in a 12x12 window centered at the origin To get a general picture of what it looks like plot the direction field Dir field flow Direct Field and accept the defaults Another quick way to get a picture of the behavior is to make a flow picture which draws a bunch of trajectories both forward and backward in time click on Dir field flow Flow and accept the default A nice picture will emerge It is not clear how many fixed points there are but there are certainly more than one To get a feel for the number of fixed points and where they are click on Nullcline New N N The x axis nullcline is red and the y axis nullcline is green Note Sometimes the nullclines look quite choppy You can make them much smoother by increasing the numerical mesh size click on nUmerics Ncline ctl and change the mesh from 40 to 100 Escape back to the main menu erase the screen and recompute the nullclines It looks like there are either 2 or 4 fixed points We can blow up the region where the fixed points are a bit to r
290. ons are almost nonexistent for I This is because e is small and so the J equation is essentially satisfying dl 25s e vo lt v T where lt v gt is the average of v t Since I is slowly varying this suggests treating J as a parameter and studying the behavior of the v w system as I varies The file m12dhom ode will do the trick ml2dhom ode dv dt I gca minf V V Vca gk w V VK g1 V V1 c dw dt phi winf V w tauw V v 0 41 84 w 0 0 002 minf v 5x x 1 tanh v v1 v2 winf v 5x 1 tanh v v3 v4 tauw v 1 cosh v v3 2 v4 param i 30 vk 84 v1l 60 vca 120 param gk 8 g1 2 c 20 param vi 1 2 v2 18 param v3 12 v4 17 4 phi 23 gca 4 total 150 dt 25 x10 60 xhi 60 yl0 125 yhi 6 xp v yp w dsmax 2 parmin 30 parmax 60 autoxmin 10 autoxmax 60 autoymin 60 autoymax 40 done 7 5 Importing the diagram into XPPAUT 197 I have set this up specifically for AUTO so that it is ready to analyze the system The virtues of hindsight The last two lines of options set up the AUTO window and the numerical continuation Run XPPAUT and integrate this a few times using the Initialconds Go and then the Initialconds Last commands Then click on File Auto and click on Run Steady state in the AUTO window You should see a cubic like curve of steady states Click on Grab and follow the branch until you find the Hopf bifurcation labeled HB at the bottom of the AUTO window Click
291. onsider the normal form for a cusp bifurcation al a bx a Let s pick a b 1 and let XPPAUT find the root by integrating the equation Make an ODE file for this problem and run it Integrate it and then integrate it again using the Initialconds Last I L command to pick up where you left off You should have settled onto a fixed point of about 1 3247 Click on File Auto to bring up the AUTO window See figure 7 1 In this window there are additional menu items and several windows The lower left shows a circle The number of crosses inside outside the circle corresponds to the number of stable unstable eigenvalues for a solution The two windows at the bottom are respectively information about the computed points and the hints or tips window This bottom window also tells you the coordinates of the main graphics window The main window is where the relevant bifurcation diagrams are drawn Before you can use AUTO you must prepare your system for it You must start your bifurcation analysis from either a fixed point of your model a periodic orbit or a solution to a boundary value problem We have already done this for the present example Once you have brought up the AUTO window you should do the following steps e Use the Parameter item to tell AUTO the list of 5 or fewer parameters that 168 Chapter 7 Using AUTO bifurcation and continuation It s AUTO man arameter xes rics Figure 7 1
292. onstant The statement 1t 0 is to set the line type to zero which plots single dots Run this Then use the Window zoom Zoom in command to zoom into the left or right halves Notice you get the same sort of set Continue zooming in like this At each step you will see the same set The Cantor set is self similar that is magnified regions of it look the same as the whole set Here is another cool trick Make a histogram for the Cantor set with 100 bins Click on Numerics Stochastic Histogram U H H Choose 100 for the number of bins and accept the defaults Exit the numerics menu by clicking on Esc Next change the line style click on Graphic stuff Edit curve Choose 0 as the curve to edit change the linetype to 1 and click on Ok Finally click on X vs t and choose X as the plot variable You will see the histogram for the Cantor set Go back into the Numerics menu and make another histogram this time with 1000 bins and replot it The histogram has many more dips than before If you zoom in you will find the histogram also is fractal Now let s make a two dimensional iterated function system We will start with a classic the Sierpinsky triangle This fractal is obtained by dividing a right triangle into 4 equal pieces and removing the middle piece This is repeated in each of the three smaller triangles which remain and so on The result is like a two dimensional version of the Cantor set The iterated function system consist of three
293. oose a very small step size in order to avoid numerical problems If you use one of the standard XPPAUT adaptive integrators like Qualrk or the Dormand Prince integrators it is also possible to run into problems since these are meant for ODEs which don t have large amplitude right hand sides Instead it is useful to use one of the stiff integrators in the XPPAUT tool kit These converge to a solution even if you use a fairly large stepsize The recommended stiff integrator is CVODE The disadvantage of stiff integrators is that they must invert a large matrix possibly many times per time step By default XPPAUT assumes that the matrix that must be inverted is full That is most of the entries are nonzero However in PDEs the equations can often be written so that only a few entries near the main diagonal are nonzero XPPAUT can implement a banded form of CVODE which can be hundreds of times faster For example consider the reaction diffusion equation ut Duter f u v 6 3 Ut DyVex glu v If we discretize this into N points to get 2N ODEs we would have to invert a 2N x 2N matrix If we defined all the u s first and then all the v s then there would be interactions between j 1 and j N Thus the resulting matrix is not banded However if we organize the discretization as U1 U1 U2 V2 UN UN then the resulting system is banded with a bandwidth of 2 XPPAUT has a way to write the equations so th
294. ope that it can aid in teaching certain concepts in the analysis of the behavior of differential equations Most of the problems and examples are taken from research papers The emphasis I am afraid is skewed toward biological applications as that is what I do If all you want to do is solve differential equations and graph the solutions then most of what you want can be found in chapter 3 Suggestions for how you can use XPPAUT in a classroom setting are found in chapter 4 Research problems involve a more complete set of tools Chapter 5 introduces functional and stochastic differential equations while chapter 6 shows you how to discretize partial differential equations and solve these Boundary value problems are also covered in this chapter Tricks and special classes of differential equations are described in 9 Chapter 7 introduces bifurcation theory and the use of the AUTO interface in XPPAUT Chapter 8 shows you how to make animations with the built in animator and 9 shows other ways to make animations Acknowledgments I have benefitted a great deal from the many users of various versions of the program To those one or two of you who sent me a note about how useful the program is rather than what new features you wished I d put into it or which ones don t work I salute you To the others well I reluctantly thank you as your comments motivated new features and bugs Mostly I would thank John Rinzel and Artie Sherman for being guinea pi
295. ormula or name formula Define a differential difference equation dependent variable name and the formula defining its right hand side For example 269 270 Appendix D Structure of ODE files x x sin t z t 1 z 4 z dw dt 1 cos w 1 cos w x a name t formula or volterra name formula Define a Volterra integral equation for the variable name For example y t int exp t x y y a intlexp t t max y 1 1 t gt 72 0 y t int 5 1 y The symbol means a convolution and a multiplies the integrand by 1 t t where 0 lt a lt 1 markov name nstates Poo Poa lt 2 Pon tt P10 Pi ai Pinty Pcia oi 7 Poel This defines a Markov variable name which has nstates states Then fol lowing is the transition table as a series of formulas that are delimited by the curly brackets The diagonal entries are ignored but should still contain a number or formula For example markov z 2 0 alpha v beta v 0 aux name formula defines a named quantity name which appears in the data browser and is available for plotting Note that auxiliary variables are not known internally to XPPAUT so that you can t use them in formulas name formula defines an internal or quantity which can be used in other for mulas This fixed variable serves the same function as temporary quantities in C code par name1 value1 name2 value2 defines named parameters with default values Thes
296. oscillators J Math Biology 29 195 217 Fife Paul C McLeod J B The approach of solutions of nonlinear diffusion equations to travelling front solutions Arch Ration Mech Anal 65 1977 no 4 335 361 C W Gear Numerical Initial Value Problems in Ordinary Differential Equa tions Pretice Hall 1975 Daniel T Gillespie Exact Stochastic Simulation of Coupled Chemical Reac tions J Phys Chem Vol 81 pp 2340 2361 1977 L Glass and MC Mackey From Clocks to Chaos Princeton Univ Press 1988 J Guckenheimer dstool tp cam cornell edu pub dstoo1 287 288 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Bibliography J Guckenheimer and P Holmes Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields Springer 1983 New York E Hairer S P Norsett and G Wanner Solving Ordinary Differential Equa tions I Nonstiff Problems 2nd ed Springer 1993 G P Harmer D Abbott Game theory Losing strategies can win by Par rondo s paradox Nature 402 864 1999 P E Kloeden and E Platen Numerical Solution of Stochastic Differential Equations Springer 1992 H Kocak Differential and Difference Equations through Computer Experi ments PHASER Springer 1986 C Koch and I Segev Methods of Neuronal Modeling MIT Press 1989 Y Kuznetsov Elements of Applied Bifurcation Theory Second Edition Springer 1998 L
297. other is a stable node Then the existence of trajectory joining the two fixed points is generic However if the sum of the the dimensions is less than or equal to the dimension of the system then we cannot expect this to generically happen For example suppose that you want to find a homoclinic in two dimensions Then the one dimensional unstable manifold must intersect the one dimensional stable manifold of the fixed point This would not be expected unless there was some free parameter A This is illustrated in figure 6 2 As some parameter changes the unstable and stable manifolds switch positions At a critical value of the parameter 0 they intersect resulting in the homoclinic orbit XPPAUT allows you to compute one dimensional stable and unstable man ifolds to an equilibrium point for an ordinary differential equation This is done by linearizing about the fixed point and then computing the eigenvector associated with the single real positive or negative eigenvalue This eigenvector is tangent to the manifold so that it serves as an approximation If the manifold is the stable manifold then the system is integrated backwards in time to take it away from the fixed point If the manifold is an unstable one then the system is integrated forward in time Many problems such as traveling waves to reaction diffusion equa tions lead to systems in which there is a one dimensional invariant manifold so 6 1 Boundary value problems 1
298. ou want to include this you must include the color as well thickness is any nonnegative integer and will result in a thicker line e rline x1 y1 color thickness is similar to the line command but a line is drawn from the endpoints of the last line drawn to xold x1 yold y1 which becomes then new last point All other options are the same This is thus a relative line e rect x1 y1 x2 y2 color thickness draws a rectangle with lower corner x1 y1 to upper corner x2 y2 with optional color and thickness e frect x1 y1 x2 y2 color draws a filled rectangle with lower corner x1 y1 to upper corner x2 y2 with optional color e circ xl yl rad color thick draws a circle with radius rad centered at x1 y1 with optional color and thickness e fcirc xl yl rad color draws a filled circle with radius rad centered at x1 y1 with optional color e ellip x1 y1 rx ry color draws an ellipse with radii rx ry centered at x1 y1 with optional color and thickness e fellip x1 y1 rx ry color draws a filled ellipse with radii rx ry centered at x1 y1 with optional color 220 Chapter 8 Animation e comet x1 yl type n color keeps a history of the last n points drawn and renders them in the optional color If type is non negative then the last n points are drawn as a line with thickness in pixels of the magnitude of type If type is negative filled circles are drawn with a radius of thick in pixels e text x1 y1 s draws a string s at posi
299. ound the rim of a bicycle wheel Suppose that each cup has a small hole in it There is a source of water at 12 o clock that fills the cups As they fill the waterwheel becomes unbalanced and rotates allowing another cup to take its place In the meantime the previously filled cup drains slowly because of the hole Strogatz shows that in the limit as the number of 8 3 My favorites 217 cups becomes a continuum the resulting dynamics are completely described by the Lorenz equations The angular velocity is the x coordinate and the y z coordinates are the cosine and sine components of the mass of the system which is distributed periodically on the wheel So with this as a motivation I present a waterwheel with 10 cups Water flows in at a rate flow Each cup has a mass so that like everything else we have looked at this waterwheel is essentially 10 rigid pendula aligned in a circle There are 10 2 differential equations the first 10 are for the amount of water in each cup and the other two are the dynamics of the wheel rotation angle and angular velocity The ODE file waterwheel ode is the waterwheel ala Lorenz but discrete there are 10 cups and each leaks here we figure out which cup is under the spigot ff u heav cos u cos pi n flow into cup i i 0 9 flow 0 9 flow ff theta 2 pi j n input output for each cup cp O 9 f10w j muxc j tt mass of the cups m 0 9 c j mc n ODE for the cup c
300. oundary condition r 0 ro As before Q is a free parameter so we write this as an ODE also Here is the new ODE file which is autonomous and thus suitable for AUTO gberg_auto ode init a 0 00118 ap 1 18 k O omeg 0 19 r 001 domain is reasonably small sqrt 1 d par d 0 177 q 0 5 r0 001 the odes a ap ap a k k ap rta r r ax 1 axra d k k r 2 k ap a omegtqxaxa d 7 3 Boundary value problems 185 extras to make it autonomous omeg 0 r the boundary conditions at r 0 bndry a r ap bndry k bndry r r0 at r 1 bndry ap bndry k set it up for the user xhi 1 001 dt 01 total 1 001 ylo 0 parmax 1 autoxmax 1 autoxmin 0 autoymax 1 autoymin 0 done We have added a few setup parameters for AUTO I have also computed a good starting point through trial and error The parameter of interest is d which is like the diffusion constant is d increases the effective domain size decreases and as d 0 the domain becomes infinite XPPAUT can only follow a branch with d as a parameter in a limited range thus we will use AUTO to get a fuller picture Run XPPAUT with this ODE file and integrate it once to load the initial solution Click on File Auto to get the AUTO window In the AUTO window click on Run Bdry value and then when the curve hits the horizontal axis click on ABORT to stop it Click on Numerics and change Ds to 0 02 and click Ok so that we can continue in the opposit
301. own x 1 x aux 1 exp t aux err abs 1 exp t x total 2 done C 2 Integrators 263 err 0 018 p eer h 0 05 n 0 0006 sano h 0 05 0 016 i eat 0 h 0 025 h 0 025 0 014 ri 0 0005 0 012 0 0004 0 01 S A ater 0 0003 0 006 2 ee 0 0002 0 004 Ea AS E fot 0 0001 0 002 Ff ok o ke 0 02 04 0 6 08 1 0 Figure C 1 Error from Euler left and improved Euler right The classical Runge Kutta method is a four step method that gives O h accuracy so that halving the step size results in a 16 fold decrease in error ky f t n kg f t h 2 n hk 2 kg f t h 2 a_ hka 2 ka f t h an hkg h Intl Un e 2k3 2k3 ka These methods are called explicit since the evaluation of the next term depends only on the previous values Explicit methods all suffer from stability problems This is nicely illustrated by the following nonlinear equation x ax 1 u x 0 0 1 C 1 For all a gt 0 solutions to this converge monotonically to x 1 Consider Euler s method applied to this Lny1 Zn har 1 zn hazn 1 Tn The latter will I hope be recognized as a variant of the famous logistic map which as all sorts of chaotic behavior That is fix the step size and let the parameter a get larger Then clearly once a goes beyond about 2 h we can expect to see the fixed point n 1 become unstable Integrate C 1 over the inter
302. p yddot f p 1 xdot abs x p odes for the track x xdot y ydot omega and f are chosen so that they agree with the real machine one is the angular frequency and the other the track g is gravity mu is friction 1 feet is the effective length of the cars a is aspect ratio for the track so that it is about 8 feet wide s is a scaling factor for the animation par p 1 a 005 1 4 5 f 6 g8 32 mu 01 par s 60 q 1 the initial condition here scales the length of the track 2 28 56 feet init x 0 y 28 par omega 8 the endpoints of the cars now the angular variable for the zipper xbdot yb omega ybdot xb omega xbddot omega omega xb ybddot omega omega yb simple harmonic oscillator for the rotation of the zipper track xb xbdot HOH H FH yb ybdot init xb 1 now put into a rotating frame xc x xb y yb yc x ybty xb xcdot xdot xb x xbdot ydot yb y ybdot xcddot xddot xb 2 xdot xbdot x xbddot yddot yb 2 ydot ybdot y ybddot ycdot xdot yb x ybdot ydot xb y xbdot ycddot xddot yb 2 xdot ybdot x ybddot yddot xb 2x ydot xbdot y xbddot this is car s dynamics now we add the pendulum with friction to the whole thing th v v g sin th mu v xcddot cos th ycddot sin th 1 position of people in the car scaled a bit for the animation xp xc q 1 sin th 216 Chapter 8 Animation yp yc q 1 cos th position of a point on the track aux xcar xc aux ycar yc positio
303. p b t r xint exp t tau 2 x 1 x par tau 3 b 8 c 1 r 2 4 par d1 1 8 d2 3 6 I add two more variables so we can make a cool 3D plot aux y delay x d1 aux z delay x d2 delay 4 total 300 trans 100 dt 0 05 done Here are some comments on the file e The construction int f t g t performs the convolution J t H t FE s als ds and creates a lookup table for f t so that it is evaluated very rapidly e I have added two additional variables y z so that I can plot x y z in three dimensions y z are just delayed versions of x t We could have done this inside XPPAUT with the rUelle plot option under the nUmerics menu which allows you to plot delayed versions of the variables on different axes e also skip the initial 100 time units of transients with the statement trans 100 If you run this you will notice a big slowdown as time goes on The reason for this is that at t progresses the domain of the integral gets larger and larger XPPAUT truncates this limit to T 4000 dt For the present problem since the kernel is 96 Chapter 5 More advanced differential equations Figure 5 3 Three dimensional reconstruction of the chaotic integral equation a Gaussian this amount is a gross over estimation For the computer exp 400 is zero Thus a dramatic increase in speed can be attained by setting the maxi mum number of points to 1000 To do this click on nUmerics Method Choose Volterra which is rea
304. p B Open a MS DOS prompt from the START menu Change to the xpp directory In Windows 2000 this is called Command Prompt It is available off the Start Applications menu If you cannot find it click on Run and type in command com Step C Now you have to tell X where to send the display If you are on a network Type set DISPLAY mypc 0 0 where mypc is the name of your PC on the network If you are not on a network Type set DISPLAY 127 0 0 1 0 0 Note that even on a network the second command usually works Step D You are now ready to run Type xppaut lecar ode and XPPAUT should fire up in the X window If not then check that you have started the X server and set the DISPLAY correctly Note that if you get an error Can t open display then you should try to find out the name of your PC as that is probably the problem Another possibility is that your X server won t let your PC host the display Look for something that allows you to set HOSTS in your X server and set the host to your display name Step E If successful exit XPPAUT by clicking on the File and then the Quit entry and answer Yes 1 4 Installation on MacOSX 5 NOTE My home computer is not on a network so I have just created a batch file xpp bat and included in it a line that sets the DISPLAY for me set BROWSER c Program Files Netscape Communicator Program netscape exe set XPPHELP c xpp help xpphelp html set DISPLAY 127 0 0 1 0 0 set HOME c xpp C xpp xppau
305. pace constant for the dendrite When b 0 a 0 the voltage at x 1 is held at 0 When b 0 a 0 there is no leak from the cable and the conditions are for sealed end Since this is a second order equation and XPPAUT can only handle first order we write it as a system of two first order equations in the file which is Linear Cable Model cable ode define the 4 parameters a b v0 lambda 2 par a 1 b 0 v0 1 lam2 1 equations v vx vx v lam2 The initial data v 0 1 vx 0 0 and finally boundary conditions First we want V 0 VO 0 bndry v v0 We also want aV 1 bVX 1 0 bndry a v b vx xlo 0 xhi 1 ylo 0 yhi 1 2 total 1 done Note that I have initialized V to be 1 which is the appropriate value to agree with the first boundary condition With the choice of a 1 and b 0 the end of the dendrite is pinned to 0 Run this and you will see that the solution grows exponentially and is not even close to the correct value Click on Bndryval Show B S and the solution will be computed and drawn Note that this is a linear equation so Newton s method will get the right answer after one iteration Change b 1 and a 0 and rerun Notice that big difference in the resulting steady state potential Try the following Set a 1 and b 0 Click on Bndryval Range to vary over a 128 Chapter 6 Spatial problems PDEs and boundary value problems range of parameters Choose b as the parameter to vary sta
306. parts Finally the value of the fixed points is shown under the line As can be seen from this example there are two complex eigenvalues with negative real parts the fixed point is 0 0 XPPAUT reports a very small nonzero fixed point due to numerical 2 7 A nonlinear equation 23 0 4 L L a E 0 2 0 0 2 0 4 0 6 0 8 1 12 Figure 2 10 Nullclines direction fields trajectories for I 0 4 in the Fitzhugh Nagumo equations error Integrate the system using the mouse starting with initial conditions near the fixed point In the Main Window tap I I Note how solutions spiral into the origin as is expected when there are complex eigenvalues with negative real parts For nonplanar systems of differential equations you must provide a direct guess Bring up the Initial Data Window type in your guess and click on Ok in the Initial Data Window Then from the Main Window click on Sing Pts Go S G Another way to get fixed points is to use the Sing Pts monte C ar com mand which randomly picks starting guesses in a range and then tries to find them with Newton s method You have to provide the ranges for starting guesses as well as the number of guesses The fixed points found are listed in the Data Viewer You should use this method only if you have no clue where the fixed points lie Bring up the Parameter Window Change the parameter I from 0 to 0 4 in the Parameter Window and click on Ok in
307. ple The left figure shows the maximum and minimum of the stable filled circles and unstable empty circles limit cycles The right figure shows the norm of the limit cycles and the stable fixed point at 0 XPPAUT lets you pick up other branches of solutions as and put them on the same diagram Click on File Clear grab This clears any grabbed points and thus allows you to start from a completely different point From the Axes Norm menu change Ymin 5 Now go back into the Main Window of XPPAUT Change a to 5 and change the initial conditions to X 0 and Y 0 This is a fixed point Go back to the AUTO window and click on Run Steady state When prompted to destroy the diagram click on No This way you don t delete the part of the diagram that was already computed You should see a thick horizontal line appear indicating the existence of a stable fixed point You should see something like the right panel of Figure 7 3 If you want a postscript printout of this click on File Postscript and save it with a name of your choice Click on File Reset diagram to get rid of the diagram and all the AUTO junk left on your disk This is not necessary these files are deleted when you exit XPPAUT but it is always a good idea Resetting a diagram is also useful if you want to make changes in the parameters and start fresh 7 1 2 A real example The examples presented above are pretty lame we knew the answers already Instead lets ana
308. pressions to get an answer Even sums can be evaluated such as sum 0 100 of exp i You can set any variable or parameter to an expression by typing e g x 1 1 pi 2 which would set x to 0 091999 Tap Esc to exit the calculator 3 You can run XPPAUT over a modem line in silent mode That is set up your ODE using the options Then type xpp file ode silent and it will run without ever invoking the interface The data will be saved in a file output dat This is also a great way to program in batch mode 4 Parameter sets are a very useful way to set up problems so that different groups of parameters are associated with nice names that can be invoked by using the File Get par set command For example consider the 3 different ways to integrate the harmonic oscillator harmonic oscillator x y y x dt 2 total 10 set euler meth euler set backeul meth backeul set rk4 meth rk4 done 9 7 Don t forget 247 Use it with the silent mode of XPPAUT and XPPAUT will dump out 3 data files called euler dat backeul dat rk4 dat To go one step better use the parameter sets with the array trick to go through a range of sets Here is a stupid example where I integrate x for 11 different initial conditions I use many of the tricks integrate over a range of initial conditions silently x x global 0 t x a la f index par index 0 f x 1 x set x 0 10 index j total 5 done
309. quation with quantities that are repeatedly used then it is often useful for both computational and readability issues to define them as temporary quantities Say the quantity is called z then you just include a statement z x y sin t and that quantity is defined and can be used in the ODE For example consider the nonlinear oscillator dx dy q a R yl qR q TCE 2 qR where R z y We should define R in the ODE file and then write the equations standard nonlinear oscillator R x 2 y 2 x x a R y 1 q R y yx a R x 1 q R par a 1 q 1 done You can have many such temporary quantities and they can depend on each other In XPPAUT these are called fixed variables IMPORTANT NOTES e Auxiliary variables are not known to XPPAUT and so you cannot use them in any formulas 34 Chapter 3 Writing ODE files for differential equations e Fixed variables are not accessible to the user they cannot be plotted but can be used in formulas e The order of definition for fixed variables is important as they are evaluated in the order in which they are defined Thus if you define a quantity x that depends on another quantity y then you should define y first or you will get unintended results Just remember if you want to use it in a formula make it a fixed variable and if it is just an auxiliary quantity you want to plot make it an auxiliary variable You can use fixed variables in definitions of auxiliary
310. r Biol 16 15 42
311. r r 27 s 10 b 2 66 init x 7 5 y 3 6 z 30 now add some projection planes for 2D plots aux x2 xplane aux y2 yplane aux z2 zplane par xplane 35 yplane 60 zplane 10 set up the numerics total 50 dt 02 set up 3D plot xplot x yplot y zplot z axes 3d tell XPP there are 4 plots altogether nplot 4 here are the 3 projections xp2 x yp2 y zp2 z2 xp3 x yp3 y2 zp3 z xp4 x2 yp4 y zp4 z set up the 3D window xmin 40 xmax 18 ymin 24 ymax 64 zmin 12 zmax 45 xlo 1 4 ylo 1 7 xhi 1 7 yhi 1 7 and rotate the plot a bit so it looks nice theta 35 done OHFHFOOHOOOHO HO HF OC Notes As I mentioned this looks quite complicated for a simple ODE The first few lines are the actual equations Then I add three auxiliary variables which are just constants So if I plot x y z2 for example in three dimensions the z coordinate is held constant This results in a projection in on the x y plane The choice of planes was done after some experimentation in order to give the best view The rest of the file is simply setting XPPAUT s plot parameters all of which could be done within the program The comments should make it clear exactly what I am doing The parameters phi theta control the view angle for three dimensional graphs Run XPPAUT with this file and integrate the equations You should see something like Figure 4 12 There are three two dimensional projections in various Fall colors and the big white Lorenz attractor in the m
312. rN is the rate Thus to compute when the next reaction takes place we merely draw a uniform random number s between 0 1 and take 1 tnext Stio In 1 s a a 5 2 Stochastic equations 107 The reaction reduces the number of molecules of X by 1 If there are several reactions then the total rate ay is the sum of all their rates A random number sy is drawn to determine the next time tneat z In s1 0 Then another random number is chosen to determine which reaction takes place Let s2 be the second random number uniformly drawn from the interval 0 ao Let ak Mes Pj k 1 where a are the m reaction rates Reaction 1 occurs if sz lt P4 reaction 2 occurs if Pi lt s2 lt Pa and so on This method is exact there is no inherent time step the reactions occur when they naturally would Thus the iteration for this process counts reaction steps rather than time We first show the ODE file for the simple degradation reaction since there is no need to determine the reaction there is only one gillespie ode this implements gillespies algorithm X gt Z at rate c par c 5 xin 1000 init X 1000 tr 0 a0 c X tr tr log 1 ran 1 a0 max X 1 1 aux cts xin exp c tr bound 100000000 meth discrete total 1000 xl1o 0 xhi 10 yl10 0 yhi 1000 xp tr yp x done We use the discrete integrator since the reactions are discrete events We let X be the number of molecules of X The total rate is ag cX and t
313. rab and Tab to the second limit point LP labeled 3 Click on Run again and you will see another vertical line corresponding to the left most limit point Within the two vertical lines there are 3 fixed points and outside there is one Next click on Grab and Tab to the Hopf point HB labeled 4 Click on Run Two param and you will see a diagonal line go down and right This is the curve of Hopf bifurcation points Finally we want to extend the curve of Hopf points to the left To do this we must tell AUTO to 176 Chapter 7 Using AUTO bifurcation and continuation 04 H 018 F 02 F z 0 14 F Eras 006 H bp pe i S o bid i e A by gt a K E e oot L os r 00 L i o 0 1 0 0 02 03 04 0 05 0 1 0 15 02 0 25 03 0 1 0 05 o 0 05 01 0 15 02 0 25 03 1 Figure 7 4 Various diagrams associated with the Morris Lecar equation Top left One parameter diagram showing fixed points and limit cycles Top right Frequency of the periodic orbits Bottom Two parameter diagram showing the curve of Hopf points and straight line the lines of saddle node points These are independent of the parameter phi reverse direction Click on Numerics and change Ds 0 02 to Ds 0 02 and close the dialog box Recall that AUTO uses the sign of Ds to determine a direction to go Finally click on Grab and get that Hopf bifurcation point again Click on Run Two param and you will s
314. rd and invent MATHEMATICA just like the guy who did this research 160 Chapter 6 Spatial problems PDEs and boundary value problems 6 2 4 Exercises 1 Try initial data such as u x 0 cos nrx for higher values of n and verify that the steady state is reached much more quickly for higher values of n in the cable equation Eq 6 2 2 The steady state for 6 2 is du Co apu 0 t bous 0 t cy apu L t bru ZL t which is just a second order equation with boundary conditions Use XPPAUT to solve this boundary value problem for a variety of boundary conditions including the one we used above Figure out what values to use for L and D in order to quantitatively compare it to the steady states of the PDE If you get stuck here is the ODE file but try to create it before looking cable boundary value problem 0 Du u cO a0 u 0 bO u 0 cl al u 1 bl u 1 u up up u D par D 1 c0 1 a0 1 b0 0 c1 0 al 0 b1 1 bndry a0x u b0x up c0 bndry al u bl up cl total 10 001 dt 01 xhi 10 001 yhi 1 00 ylo 0 click on Bndryvalue Show to solve this try different values for the parameter done 3 Solve the PDE g Ou Oru with D 1 on a domain of length 20 with 100 grid points a time step of dt 0 025 with Neumann boundary conditions u 0 t uz L t 0 for 0 lt t lt 20 Initialize the first grid point to 1 and all the rest to 0 Hint copy and modify the cable PDE
315. re attracted to the fixed point 0 from those that are attracted to the fixed point at infinity The set of points which are not attracted to these fixed points is called the Julia set Thus the Julia set of the map z 2 is the unit circle Now consider the general quadratic map with a complex parameter c Zn 1 oe c If the parameter c is small then the Julia set of this map should be close to the unit circle It is a connected set and there is an inside and an outside However if c is large then the Julia set will not look at all like the circle In particular there is no separation between initial conditions that stay bounded and those which go off to infinity The famous Mandelbrot set is the set of all points c for which the Julia set is connected Now that we have connected the logistic map to the Mandelbrot set through the Julia set how can we compute the Julia sets Since the Julia set is a separatrix between fixed points we can compute it by starting inside the unit circle and iterating backwards The inverse of the map 2 c is z c The way we will compute it is to randomly pick between the two square roots at each iterate Note the idea of randomly choosing one of the square roots makes this computation related to so called iterated function systems which we visit below To compute the square roots we first write the number z c in polar form rexp i and then take the square root of r and divide 0 by two Recall that
316. relative tolerance for CVODE and the Dormand Prince integrators BOUND value sets the maximum bound any plotted variable can reach in magnitude If any plottable quantity exceeds this the integrator will halt with a warning The program will not stop however default is 100 DELAY value sets the maximum delay allowed in the integration de fault is 0 AUTOEVAL 0 1 tells XPP whether or not to automatically re evaluate tables every time a parameter is changed The default is to do this How ever for random tables you may want this off Each table can be flagged individually within XPP Poincare map POIMAP section maxmin period sets up a Poincare map for ei ther sections of a variable the extrema or period between events POIVAR name sets the variable name whose section you are interested in finding POIPLN value is the value of the section it is a floating point POISGN 1 1 0 determines the direction of the section POISTOP 1 means to stop the integration when the section is reached 260 Appendix B The options Range integration e RANGE 1 means that you want to run a range integration in batch mode e RANGEOVER name RANGESTEP RANGELOW RANGEHIGH RAN GERESET Yes No RANGEOLDIC Yes No all correspond to the en tries in the range integration option Phasespace e TOR PER value defined the period for a toroidal phasespace and tells XPP that there will be some variables on the circle e FOLD name te
317. renz s seminal paper he chose to plot successive maxima of the variable z We will do this now and will see that the map turns out to be essentially one dimensional Note that since we are fixing one variable in a three dimensional system we expect to see a two dimensional map This computation may take a few seconds as we have to make a small time step for accuracy in finding the maximum and also we should integrate 4 6 Three and higher dimensions 85 for a long time Click on nUmerics to get the numerics menu Click on Total and change the total to 200 Click on Dt and change the time step to 0 01 Now click on Poincare map and Max min to select maxima or minima of a variable Fill in the resulting dialog box as follows Variante 7 Section 0 PirectionG1 1 01 Stop on sect g m N and click on Ok What you have done is tell XPPAUT that you want to plot all the variables every time Z has a local maximum the 1 direction is for local maxima 1 for local minima 0 for both and to not stop when this maximum is crossed Escape to the main menu Esc Now integrate the equations This could take several seconds Now we have a series of values tn 2n Yn 2n that are the values of the variables at every time that z t has a local maximum Lorenz plotted Zn41 against Zn to obtain a nice one dimensional map XPPAUT allows you to do this very easily We will first set up a view with z plotted against itself Click on Viewaxes 2D and
318. roblems f urA 0 ufeRr AER This is done by parameterizing the solutions by a parameter s so that solutions of the form u s A s must be found Differentiate the equation f u s A s 0 with respect to s and we obtain an n x n 1 dimensional matrix M falfa which is assumed to have a one dimensional nullspace The nullvector is normalized as cyt tte 1 where cu C are constants Suppose that we have already found wj 1 A 1 and j 1 43 1 Then we must solve f uj 0 0 As Cu Uy uj 1 031 cal j Aj 1 Aj 1 where As is the desired step size This is now an n 1 x n 1 system and the Jacobian matrix is f u f A Ma which is not singular even if f is Thus AUTO can track around fold points Limit or fold points are tracked with two parameters A 4 by applying the same techniques to the 2n 1 dimensional system f u A 11 0 fulu A u v 0 vly 1 0 Continuation of Hopf bifurcation points is a bit more complicated but depends on the fact that at a Hopf bifurcation the linearized problem V t Tf V T will have a nonconstant solution satisfying V 0 V 1 0 for some value of T Since solutions will be proportional to sine and cosine at the Hopf point V t sin 2rt y cos 271t The result is the two algebraic equations 1 20 fu 9 0 E 1 27 fun 0 and the normalization ETE n 1 Due to translation invariance one more normalization con
319. rovide 2 tables of length nm which give the connectivity matrix and the strength of the connections As with the above example two dimensional arrays are stored in row by row That is the rows of the matrix are appended in order to form a one dimensional array For example instead of 6 4 consider the random network ul uj HF jaa eu t 7 j 0 99 1 0 where Cy is a randomly chosen set of 10 indices and wj is a random number between 0 and 1 In order to make the XPPAUT file for this we will need to define two tables One of them is the connectivity table which gives the indices of the cells that cell j is connected to The other table contains the weights The connection matrix is 10 x 100 1000 entries whose elements are the indices 0 99 since there are 100 units labeled u0 through u99 The following does the trick table con 1000 0 999 flr ran 1 100 The function flr x returns the greatest integer less than or equal to x The actual weights are randomly distributed between 0 and 1 The special function k sparse n m w c u0 returns n values m k j X w i m i shift u0 e j m 1 1 0 where c is the connectivity array Thus the ODE file is written sparse ode a random sparse network with delays each is connected to 10 others table con 1000 O 999 flr ran 1 x 100 table w 1000 O 999 ran 1 par cp 1 cm 6 tau 3 r 3 ut 25 special k sparse 100 10 w con u0 f u 1 1 exp r
320. rs and initial conditions in XPPAUT We have already seen how to change the initial data using the mouse This method works for any n dimensional system as long as the current view is a phaseplane of two variables Here are two other ways to change the initial data e From the main menu click on Init Conds New and manually put them in at the prompts You will be prompted for each variable in order For systems with hundreds of variables this is not a very good way to change the data However if you get tired just tap Esc and the remaining variables will assume their current initial conditions e In the Initial Data Window you can edit the particular variable you want to change Just click in the window next to the variable and edit the value Then click on the Go button in the Initial Data Window If there are many variables you can use the little scroll buttons on the right to go up and down a line or page at a time If you click the mouse in the text entry region for a variable you can use the PageUp etc keys to move around Clicking Enter rolls around in the displayed list of initial conditions The Default button returns the initial data to those with with the program started If you 2 5 Looking at the numbers the Data Viewer 17 don t want to run the simulation but have set the initial data you must click on the 0k button in the Initial Data Window for the new initial data to be recognized e Attach the variable to a slider S
321. rs that you can set These will be dealt with in subsequent sections of the book where necessary However the most common things you will want to change are the total amount of time to integrate and the step size for integration You may also want to change the method of integration from the default fixed step Exercises 25 Runge Kutta algorithm To alter the numerical parameters click on nUmerics U which produces a new menu This is a top level menu so you can change many things before going back to the main menu To go back to the main menu just click on the Esc exit or tap Esc There are many entries on the numerics menu The following four are the most commonly used Total sets the total amount of time to integrate the equations Shortcut T Dt sets the size of the timestep for the fixed step size integration methods and sets the output times for the adaptive integrators Shortcut D Nout sets the number of steps to take before plotting an output point Thus to plot every fourth point change Nout to 4 For the variable step size integra tors this should be set to 1 Method sets the integration method There are currently 15 available Shortcut M They are described in the appendix When you are done setting the numerical parameters just click on Esc exit or tap the Esc key Exercises 2 1 2 2 Write a differential equation file for the damped pendulum T ar sinx 0 Hint Rewrite this as a syste
322. rt at 0 end at 10 and use 20 steps Set the Cycle color value to 1 You will get a range of curves with different colors converging close to the sealed end curve Note that these are not permanently stored in memory You could turn on the Freeze curve option and run each of these individually Sometimes it is necessary to resort to tricks in order to numerically solve a BVP For example in some BVPs that arise from partial differential equations in the unit ball or disk there are singularities at the origin Thus to solve them numer ically you must often move away from the origin This can be done by computing a local series expansion near the origin Such is the case in the following prob lem which arises in the search for rotating waves in a two dimensional oscillatory medium The equations for the PDE are 2 z 1 1 ig zz dV z Ot which is the normal form for a Hopf bifurcation in a two dimensional diffusive medium The equations are defined on a unit disk with Neumann boundary condi tions We look for rotating waves in this equation that take the form z x y t A r exp i Qt 0 a k s ds Here r are the usual polar coordinates Substituting this form into the equations reduces the existence of rotating waves in the unit disk to a BVP problem 0 A 1 A d A AJr Ak Q qA d k k r 2kA A with boundary conditions A 1 k 1 0 and lim Aw A 0 lt co li
323. ry axis then the behavior of 7 1 near the fixed point o is the same as the behavior near the origin of the linear differential equation In particular the stability of the two systems is the same This is why we can determine stability of fixed points by linearizing However if one or more eigenvalues of Ag have a zero real part then the local behavior of the nonlinear system is not necessarily the same as the linear system Convince yourself of this fact by looking at the following three systems which all have the same linearization g y y y r al y r y r g yt e y r Thus if we vary u and one or more eigenvalues of A u crosses the imaginary axis we expect that there will be a qualitative change in the behavior of 7 1 The analysis of this change in behavior near the fixed point is the goal of local bifurcation theory Similar points can be made for discrete dynamical systems Ln 1 F n H 7 3 but the critical set for the eigenvalues of A u is the unit circle rather than the imag inary axis Consider a limit cycle solution to 7 1 and take a local Poincare section Then this defines a map and the limit cycle is a fixed point of the map Thus the local bifurcations of limit cycles are found by studying the local bifurcations of the associated Poincare map One of the beauties of local bifurcation theory is that the behavior of the systems 7 1 and 7 3 is the same as certain low dimensional polynomial syst
324. s around the z axis at 15 degree increments starting at 45 degrees Before clicking OK make sure that no other windows obscure the picture Now click on Ok and sequence of 23 images will be drawn These images have been stored in memory and can be played back Click on Kinescope Autoplay Choose 5 for the number of cycles and accept the default of 50 for the millisecond between frames Your picture will be nicely rotated 5 times Click on Kinescope Make AniGif and XPPAUT will create an animated GIF file called anim gif You can play this back using your web browser or using other software such as xanim Repeat this exercise varying phi the other angle 5 1 4 Singular integral equations XPPAUT can solve integral equations with removable singularities e g t I t O 4 Vena 5 1 Since u lt 1 we can solve this equation by integrating by parts See Linz 1985 for a discussion of numerical methods that are used to solve singular and nonsingular Volterra equations In the appendix we describe the methods used in XPPAUT in more detail The format for the expression 5 1 is int mu k t R The singular part is enclosed in the brackets Here is an example ODE file demonstrating the syntax The equation is t 4 t exp t s t Vt 0 Vi 2 ds ult p War TE T us ds As can be easily verified ur t exp t 2 vt is the true solution Here is the ODE file a singular integral equation u t exp t 2 sqrt
325. s a saddle point and that the stable and unstable manifolds are the y and x axes respectively Examine the fixed point x 1 y 0 Show it is a saddle point as well In this case note that one branch of the unstable manifold in yellow appears to have a limit cycle as its limit You will have to click on Esc to stop the integration of this trajectory Finally note that the fixed point x 0 4 y 0 24 is an unstable vortex Choose some initial data in the positive quadrant to verify that solutions all tend to a stable limit cycle Recall that a limit cycle is an isolated periodic orbit for a differential equation 4 5 Nonlinear systems 75 S x N N J a s w x RS 08 N S N E 0 6 Figure 4 9 Direction fields and nullclines of a predator prey example 4 5 1 Conservative dynamical systems in the plane The equations for a frictionless particle with mass m operating under the influence of a potential function P x have the following form d dz OP E yasa 4 4 br oe x 4 4 where f x is the force and mdx dt is the momentum If we write this as a system of equations for the position x and the velocity v we obtain dx du dt U m f z where we have assumed that the mass is constant If we multiply 4 4 by dx dt we can integrate the equation revealing that the total energy the kinetic plus the potential is conserved 1 E mo F a that is dE dt 0 In general we say the system 4 2 has an
326. s and the second the stable Whenever there is a one dimensional unstable manifold XPPAUT will draw this first If you erase 72 Chapter 4 XPPAUT in the classroom the screen you will find that you have lost these manifolds However XPPAUT keeps the initial data required to recompute them To draw the unstable manifolds click on Initialconds sHoot I H You will be prompted to choose a number from 1 4 Since the first two trajectories were the unstable manifold pick either 1 or 2 and one or the other branches will be drawn To get both of the branches pick the second of the two after freezing the first trajectory To get the stable manifolds you must change the direction of integration that is you must solve the equations backward in time To solve ODEs backwards in time in XPPAUT click on nUmerics Dt and change it to a negative value e g 0 05 then exit the numerics menu Click on Initial conds sHoot and choose either 3 or 4 as the stable manifolds are always the last two Make sure that you change the time step Dt back to a positive value if you want to explore the system more Note as with the flow fields you can use the Kinescope to capture the screen with the manifolds as a quick and dirty way of getting a permanent record 4 5 Nonlinear systems Since XPPAUT uses numerical methods it doesn t matter if the system is linear or not We now turn to the analysis of two dimensional nonlinear systems Z flav 4 2 Y
327. s is just shorthand for the scaled noise For example consider the case of pure Brownian motion f x 0 dx od The ODE file below is sufficient brown ode the wiener process wiener w x sig w par sig 1 meth euler ylo 20 yhi 20 done Note that whenever you do simulations with this type of noise it is important that you use Euler s method as XPPAUT does not have any special integrators for noisy simulations You can use other integrators like Runge Kutta but the noise is held constant for the entire interval from t to t dt Try this ODE file Here is an experiment to do Look at the mean and variance of 1000 sample paths To do this click on nUmerics stocHastic Compute U H C and fill in the first 4 lines of the dialog box as follows Range over X Steps 1000 start 0 ao and click on Ok Once the simulation is done it will take a few seconds or more depending on the computer speed click on stocHast Mean H M and exit back to the main menu Click on Erase and Restore E R to erase the screen and redraw the mean The mean will hover very close to 0 with only a small amount of deviation Save this curve in memory using the Graphics Freeze Freeze command Choose 1 for the color red and click Ok As is well known for brownian motion the variance grows linearly with time To see this click on 104 Chapter 5 More advanced differential equations nUmerics stocHast Variance and Exit bac
328. s the name of a variable and wgt is a table produces an array zip with npts ncon zipli 5 wet j nconjroot i j j ncon The sparse network has the syntax special zip sparse npts ncon wgt index root F 1 Cheat sheet 283 where wgt and index are tables with at least npts ncon entries The array index returns the indices of the offsets to with which to connect and the array wgt is the coupling strength The return is zipli sum j 0 j lt ncon wli ncon j root k k index i ncontj The other two types of networks allow more complicated interactions special zip fconv type npts ncon wgt root1 root2 f evaluates as ziplil sum j ncon j ncon wgt ncon j f root 1 i j root2 i and special zip fsparse npts ncon wgt index root1 root2 f evaluates as ziplil sum j 0 j lt ncon wgt ncon i 3 f root1 k root2 i k index ix ncon j OPTIONS The format for changing the options is namel valuel name2 value2 where name is one of the following and value is either an integer floating point or string All names can be upper or lower case The first four options can only be set outside the program They are e MAXSTOR integer sets the total number of time steps that will be kept in memory The default is 5000 If you want to perform very long integrations change this to some large number e BACK Black White sets the background to black or white e SMALL fontname where fontname
329. s then x0 else 100 aux yp 1 31 if dd x r jl y s j1 lt eps then y0 else 100 iteration x x f gy g fy det y y g fx f gx det initial data x0 x0 y0 y0 set initial data as parameters glob O t x0 1 1 dx 2 2 yO 1 1 dy 2 2 x 1 1 dx 2 2 y 1 1 dy 2 2 plot all three sets of points in lousy colors meth discrete total 20 bound 1000 maxstor 100000 trans 20 O xp xp1 yp yp1 xp2 xp2 yp2 yp2 xp3 xp3 yp3 yp3 nplot 3 1t 1 9 6 Arcana 243 xlo 1 1 ylo 1 1 xhi 1 1 yhi 1 1 done This rather lengthy ODE file is actually pretty simple and works as follows We draw 3 different sets of data corresponding to the three different roots We compute 20 iterations and only keep the last iterate by setting the total and the transient to 20 If 220 y20 is close the real root then we plot the initial conditions to yo other wise we plot 100 100 which is out of the plotting range Similarly if the twentieth iterate is close to the first complex root then we plot the initial condition otherwise we plot the out of bounds point We treat the data for each root as three different sets of points I hope the large number of comments make it self explanatory I point out the use of the global statement to set the initial conditions according to the values of the parameters dx dy In the options statements I tell XPPAUT to plot all three pairs of points xp1 yp1 xp2 yp2 and xp3 yp3 which are ei
330. sa 168 One and two parameter bifurcation diagrams for the cusp ODE x a bx 2x The one parameter diagram has b 1 The two parameter diagram also shows the artifactual line at b 1 from the one parameter continuation 4 171 The diagram for the isola example The left figure shows the maximum and minimum of the stable filled circles and unsta ble empty circles limit cycles The right figure shows the norm of the limit cycles and the stable fixed point at O 173 List of Figures 7 4 7 5 7 6 7 7 7 8 9 1 9 2 9 3 A l C 1 Various diagrams associated with the Morris Lecar equation Top left One parameter diagram showing fixed points and limit cy cles Top right Frequency of the periodic orbits Bottom Two parameter diagram showing the curve of Hopf points and straight line the lines of saddle node points These are independent of the parameter phi eV ogee ee ee i ee AP a A Bifurcation diagram for period 8 orbits of the logistic map left and period 7 orbits for the delayed logistic map right Solution to the boundary value problems from example 1 left and example 2 MSM is rt a A as Left The continuation of the homoclinic orbit for the example problem Right Sample orbits computed for a 0 6 in the 19 phaseplanes otc ee ha ae a eee oy ge a Bifurcation diagram for the periodically driven bistable system Bursting in a modified membrane model Top
331. set initial data that depend on a parameter Here is the ODE file cannon ode with linear friction X VxX vx f Vx y vy vy f vy g global 0 t vx v0 cos pi theta 180 vy v0 sin pi theta 180 par theta 30 par f 01 g 1 par v0 2 5 init x 0 y 0 xlo 0 xhi 5 ylo 0 yhi 3 xp x yp y bound 100000 done I have set up the problem in the x y plane so that you can see the trajectory of the cannon I have also scaled the angle so that it is in degrees rather than radians Try to hit the number 4 on the x axis Computing basin boundaries The ability to parametrically control initial data allows us to perform some inter esting computations we can compute the basins of attraction of systems with multiple attractors Let s first consider a classic example the complex cube roots of 1 2 1 9 6 Arcana 241 If we rewrite this in terms of the real and imaginary parts of z x iy then we want to solve xr 3ay 1 3a7y y 0 for x y This represents two nonlinear equations in two unknowns The way to solve these is to use Newton s method which we write as an iteration In41 Tn 3x 3y 6zxy T a yout nj Gay 3y 3a y y Here n Yn is the nt approximation to the root The matrix above is the Jacobi matrix for the two functions The point is that Newton s method produces a discrete dynamical system There are three cube roots to 1 in the complex plane the r
332. sk space Many universities have site licenses for X servers such as EXCEED see their site http www hummingbird com products nc exceed Here are the steps to install XPPAUT in Windows Chapter 1 Installation Step 1 Create a folder tmp that will be a temporary directory Create another folder called xpp Step 2 If you have an X windows emulator already then skip this step Other wise you should install one of the above Xservers from the tmp directory or the desktop I have an old version of the MI X server available to download If you want to try it here is how e Download the two files into the tmp directory runme1st exeand file000 bin e Run the program runme1st exe to install an X windows server onto your computer This server only needs a few megabytes of disk space so it is pretty small Test the installation by clicking on the START menu and running the program found under TNT Lite Step 3 Download the file xpp4win zip into the folder xpp Unzip this with the Winzip utility There will be a number of files including xppaut exe Note that there are two dynamic link libraries DLL s in the zipped file so if you want to move xppaut exe to a different directory you should move the DLLs there as well To make it available everywher you can copy xppaut exe cygwin1 d11 and 1ibX11 d11 into your Programs directory or any other directory in your path Step 4 Test your download Step A Start your X server Ste
333. so increase the number of steps for the initial guesses Steps2 as there are many such points and you want to get them all To save these points and superimpose a diagram with other periodic points click on Graphics Freeze Freeze G F F and choose 1 2 9 for the color entry to make XPPAUT use points instead of lines Figure 4 3 shows the period three and five points 4 2 3 Liapunov exponents for 1 d maps One of the hallmarks of chaos is the local exponential divergence of nearby points under the dynamics For one dimensional maps this is easy to measure One needs to look at the average of the logarithm of the absolute value of the derivative of f evaluated at a point on the attractor Thus for a one dimensional map the maximal Liapunov exponent is 1 N A lim In f x5 l j 1 Noo N gt If A lt 0 then the attractor is asymptotically stable If A 0 the attractor is a periodic orbit Finally if the attractor is chaotic then A gt 0 For the logistic map with a 4 it is possible to show that A In2 0 693147 The following XPPAUT file illustrates how to compute this exponent as a function of the parameter I keep a running sum of the log of the derivative after throwing out the first 100 I compute the running average of this sum and keep only the final value computed after 2000 iterates Here is the file logliap ode 54 Chapter 4 XPPAUT in the classroom the liapunov exponent of the logistic map f x a x
334. some controls total 30 dt 025 bound 1000000 done The first part of the file contains all the equations we discussed above We have scaled the a y coordinates so that they don t get too big The last part of the ODE file contains information to pass on to the animator I first mod out the coordinates by 1 so that when the glider passes out one boundary of the animator it comes back in the other the world is a torus These new coordinates form the 206 Chapter 8 Animation center of the glider Then I construct the left and right endpoints of the glider The net size of the glider is 2x1g Run XPPAUT with the file glider ode and then run the animation Change the drag d making it lower and see if you can execute two loops A fancy glider Now we will build a fancier glider that looks more like a plane Well maybe only a little more This time I will use the lookup tables in XPPAUT for the coordinates of the fancy glider The tables are called xglid tab and yglid tab Here are the 14 values of the two coordinates xglid tab 0 2 5 2 5 1 1 75 5 22 258 200 1 2 1 28 0 yglid tab 0011221 1200 1 10 Here is the ODE file fglider ode fancy glider tab xg xglid tab tab yg yglid tab v sin th d v 2 th v72 cos th v x v cos th scale y v sin th scale par d 0 25 scale 2 1g 04 these initial data correspond to a nice loop init th 2 v 8 init x 4 y 0 these are points for the fanc
335. t 4 3 exp t sqrt t 3 int 5 exp t u 2 and the true solution aux utrue exp t 2 sqrt t done Run this and add the true solution to the graph to see that it does quite well 5 1 5 Some amusing tricks Periodic cicadas In his excellent book Mathematical Biology Murray describes a model for periodic emergence of cicadas I won t go through the derivation of 98 Chapter 5 More advanced differential equations the model but will briefly describe it Certain types of insects seemingly emerge synchronously as adults after spending many years underground in a larval stage Synchronous emergence means that all the insects come out in the same year 13 and 17 year cicadas come out synchronously while 7 year cicadas emerge every year The idea is that the grubs remain underground eating the roots of trees When they emerge predators are there to eat them The underground roots have a limited carrying capacity Here is the model Let n be the number of adults emerging each year At each year a fraction u survive The amount of food that is remains to feed emerging adults is k 1 c d gt ni 1 j l where d is the maximum amount of food The number of predators that are around at year t is Pi Vpi1 ap ree The number of new grubs is the fecundity times the number that are left after predation ni min f max p n _ p41 p 0 What makes this problem tricky is that k is the parameter of interest So t
336. t 1 2 3 Note this also sets some other environmental variables 1 4 Installation on MacOSX Installation on Macintosh computers running OSX is possible by downloading the source code for XPPAUT and then compiling it using the software development tools provided for the new OS In addition you will need to download the X de velopment libraries to compile it The following steps were helpfully provided to me by Chris Fall and James Sneyd I have managed to test this on one laptop and everything seems to work A Mac OSX binary can be found on the web site if you don t want to compile it yourself 1 Make sure you get and install the full Developer Kit for Mac OSX This is how you get the cc compiler 2 Install XFree86 on OSX Download from ftp ftp xfree86 org pub XFree86 4 1 0 binaries Darwin ppc Make sure no matter what the Install file says that you also get the Xprog tgz bundle You need it 3 Get the xpp source code and put it in a directory of your choice Pl assume you ve called it xpp Untar the archive 4 Make the following changes in the MAC system directories I think this is a bug in their header files e copy usr include dirent h to your xpp directory Pll assume you ve called it dirent h locally e copy usr include sys dirent h to your xpp directory giving it a new name obviously I called it sysdirent h e In the file read_dir c change the include lt dirent h gt statement to call your local copy
337. t 1000 characters Line continuation is done with the character x y z W This is interpreted as x y z w Every ODE file consists of declarations of the equations that you want to solve the parameters involved and any user defined functions etc that you will need Every ODE file must end with the done statement Other than that the form of the files is pretty flexible The name of anything that is defined in XPPAUT must be fewer than 10 characters The allowable characters are the letters of the alphabet numbers the underscore symbol Other symbols are not recommended since they may be confused with symbols that have meaning to XPPAUT Do not start names with a number Most of the input that you type into an ODE file is pretty much as you d expect and similar to how you would write it on paper The main exceptions to this are i you must use the symbol between items that are to be multiplied and ii XPPAUT is case insensitive All of the examples here and in the rest of the book are available online so that you need not type them in I also encourage you to run the example files and 27 28 Chapter 3 Writing ODE files for differential equations put in some initial conditions or parameters so that you get used to simulation In many of the examples I suggest some interesting things to explore 3 2 Ordinary differential equations and maps The easiest types of equations to type in are ordinary differential
338. t and end Click on Ok and after a short wait the calculation will be done Click on Escape to go to the main menu and then click on Xi vs t and hit Enter You will get a nice plot of the maximal exponent as a function of the parameter a Note that when a 4 A Maz 0 6911265 which is close to the actual value of In 2 0 69314 The difference is due to the finite number of points computed See figure 4 4 4 2 4 The Devil s staircase If you periodically force an oscillator then a number of interesting phenomena can occur One of these is phase locking in which the resulting behavior is periodic If 4 2 Discrete dynamics in one dimension 55 0 5 F eva A ry r 0 ral L 4 Al y l 0 5 F 4 al J 1 5 J 2 4 3 3 2 3 4 3 6 3 8 a Figure 4 4 Maximal Liapunov exponent as a function of the parameter a for the logistic map the forced oscillator cycles N times while the forcing function cycles M times this is called N M locking One way to model a periodically forced oscillator is to look at a one dimensional map Inti f n where the function f takes the unit circle back to itself x is the phase of the oscillator after the n stimulus The standard map has the form f z x b asin 27x mod 1 Consider a 0 for a moment Suppose that b N M Then after M stimuli In N n m that is x has traversed N cycles while the stimulus has traversed M This is N M lock
339. t is the output from some computer algebra system Most of these systems e g Maple or MATHEMATICA have a command that allows you to output the algebra in C or FORTRAN code It is then a simple matter to edit this code compile it write an ODE file which exploits this and then run XPPAUT Note that you do not have to recompile XPPAUT at all The dynamic linking is done inside XPPAUT You should make sure that you are running a version of XPPAUT with dynamic linking enabled The default is to not use it Another example is when you just cannot figure out how to implement the right hand side in XPPAUT 250 Chapter 9 Tricks and advanced methods This example is taken directly out of the users manual for XPPAUT I will write a C file that has three different right hand sides 1 will then switch between them on the fly while in XPPAUT In order to allow XPPAUT to communicate directly with the C file you have to include a line in the ODE file of the form export x y a b t xp yp where the first group of variables and parameters are values that you want to pass to the external routine The second group usually consists of FIXED variables that you want the routine to pass back to you So here is an ODE file for a two dimensional system with a bunch of parameters tstdll ode test of dll In XPP click on File Edit Load Library and choose libexample so Then pick either lv vdp duff as the function x xp y yp xp 0
340. t out that the basic idea is to take a Runge Kutta step of size h and then two steps of size h 2 and compare the results The difference gives a good estimate of the error This is used to adjust the time step The advantage of adaptive techniques is that it may be possible to take big steps sometimes and switch to small steps only when necessary so that time is not wasted C 2 1 Delay Equations Delay equations are solved by using cubic polynomial interpolation to obtain the delayed value XPPAUT stores output from the integrators on a circular stack at regular output intervals The size of the stack is determined by the parameter Maximum Delay When u to is needed for some value of to the values of u at nearby points are used to interpolate the new value If the delay is a multiple of the output times no interpolation is needed Thus if you are not using the delay as a parameter it is most accurate to choose Delta T accordingly Stability of fixed points for delay equations is done as follows The linearization results in a system of the form dy q 7 Aoylt Y Ajylt 13 j 1 where A are matrices Substitution of y t Bexp At yields the transcendental 266 Appendix C Numerical methods characteristic equation A A det Ao Y Aye AI j 1 We are interested only in positive eigenvalues Bounds on the largest positive real part are readily obtained XPPAUT then computes an approximate contour inte gral of the
341. t should take a second or two Plot the auxiliary quantity PRC against the variable tau Click on Viewaxes 2D and put tau on the X axis and PRC on the Y axis Click OK and then Window Fit to let XPPAUT figure out the window You should see something the the first curve in Figure 9 2 Freeze this curve and try using a different value of the amplitude a 9 5 Oscillators phase models and averaging 233 PRC 0 15 T T 0 1 F 0 05 0 15 F 0 1 F 0 05 1 1 i 0 5 10 15 20 tau Figure 9 2 Top The PRC for the van der Pol oscillator with different amplitudes Bottom The PRC for the Morris Lecar model Exercise Compute the PRC for the Morris Lecar oscillator by adding the required parts to the file m1 ode Define a square wave pulse just like above with a width of 0 25 and a magnitude of 0 1 If you are stuck download the file mlprc ode which sets up the ODE for you You should get something like the bottom plot of figure 9 2 9 5 4 Phase models In the previous section we saw how to reduce a pair of coupled oscillators to a pair of scalar models in which each variable lies on a circle XPPAUT has a way of dealing with flows on systems of scalar variables each of which lie on a circle The 234 Chapter 9 Tricks and advanced methods phase space of a system of say two such variables is the two torus Let s start with the simplest phase model 0 FSWT asin 2 61 0 Wa asin 1
342. t to do Change a from 0 1 to 0 1 and change the kick k from 0 5 to 0 5 Using the initial conditions x 0 5 and y 0 you should see a chaotic solution as in Fig 3 3 You can prove this is chaotic this is left as an exercise the key point is that since a lt 0 the damping is negative so that nearby trajectories will exponentially diverge The kick to the right keeps the solutions bounded However if you start with x large enough then the solutions will grow without bound You can also prove that there is an unstable periodic solution with initial conditions at about x y 0 7831 0 Any initial data inside this unstable periodic will converge to the chaotic attractor Integrate using as initial conditions the values at the end of the last integration click on Initialconds Last orI L Now let s look at the maximal Liapunov exponent a measure of chaos We haven t gotten much data but can try it anyway Click on nUmerics stocHastic Liapunov U H L and after a brief moment a window will come up telling you an approximation of the maximal exponent The approximation is not very good the actual value is 0 1 but if you integrate longer you will get a better value Try this in the numerics menu change Total to 1000 Change nout to 5 Integrate Recompute the Liapunov exponent and it will be much closer to 0 1 Another quantity that is indicative of chaos is the power spectrum To get this click on nUmerics stocHastics Power U H P
343. t y z y2 x and y3 2 Try to write an equation for yz x Write an ODE file for this with parameters a 1 b 2 c 3 7 and initial data x 1 v 0 and x 0 Integrate this for a long time to see the chaotic orbit Set the maxstor option to something like 20000 by adding the line maxstor 20000 somewhere in your ODE file and integrate for a total of 400 You may want to look at x t versus 2 t y versus ya 5 Write an ODE file for the Rossler attractor Y y z y xu ay z br cz rz with parameters a 36 b 0 4 and c 4 5 and initial data x 0 y 4 3 and z 0 Run this file for a total of 200 point Make a three dimensional plot by clicking on the little boxes next to the three variables in the Initial Data Window and then click on the xvsy button in the Initial Data Window 6 Write and simulate the following differential equation which has 3 limit cycles a xf r y y yf r 2 f r r 2 r 3 r r y 2 y Window it as 4 4 x 4 4 Can you create a differential equation with 5 limit cycles 7 As a bonus problem run some of the examples in the text and that you have written yourself to see what they do 3 5 Discontinuous differential equations 3 5 1 Integrate and fire models Consider the integrate and fire model for a neuron dV f dt 36 Chapter 3 Writing ODE files for differential equations with the reset condition that each time V hits some
344. th by integrating the discretized partial differential equation and by shooting in the ODE arising from the traveling wave assumption The equations are cu u u 1 u u a which we rewrite as a system ul Up f u up Up CUy u 1 u u a g u up The fixed point 1 0 has a one dimensional unstable manifold and 0 0 as a one dimensional stable manifold Recall that we found that when a 0 25 the velocity c was approximately c 0 34375 We will now use a different method to compute the solution solve the approximate boundary value problem We seek a solution heteroclinic from 1 0 to 0 0 For a 0 5 and c 0 there is an exact solution joining the two saddle points Prove this by showing that u u 2 2u 3 u4 2 is constant along solutions when a 0 5 c 0 We will use this as a starting point in our calculation Here is the ODE file tstheti ode a heteroclinic orbit unstable at u 1 stable at u 0 f u up up g u up c up u 1 u u a the dynamics u f u up per up g u up per dummy equations for the fixed points uleft 0 upleft 0 uright 0 upright 0 the velocity parameter c 0 fixed points b f uleft upleft b g uleft upleft b f uright upright 7 3 Boundary value problems 191 b g uright upright projection conditions b hom_bcs 0 b hom_bcs 1 parameters par per 6 67 a 5 initial data init u 918
345. the constant planar vector field a 1 y a Tf this is integrated on a torus that is each time x 1 respectively y 1 x respectively y is reset to 0 then if a is irrational the trajectories densely fill the unit square representing the unfolded torus For the first part of this exercise verify this with a numerical simulation Plot the phase plane for the unit square and use the Graphics Edit curve 0 to change the Linetype to zero so that just dots are drawn Use the global declaration to reset x and y Curiously enough if you solve this equation on a Klein bottle instead of a torus all solutions are periodic How do you make a Klein bottle phase 3 5 Discontinuous differential equations 41 space Recall that to create a Klein bottle you must reverse the orientation along one of the edges of the square Thus when y 1 reset y to zero and set x 1 x which flips its orientation Make an XPPAUT file that does this and verify that all solutions are periodic Prove your answer Put a lemon seed in a glass of soda It will begin to sink but as it sinks bubbles accumulate on it making it more buoyant It will slow down and float toward the surface When it hits the surface the bubbles pop and it begins to sink again We can model this in a manner similar to the dripping faucet As the seed sinks it accumulates bubbles which have considerably less density than the soda Eventually the volume fraction of the bubbles
346. the sinusoidal forcing and w is the frequency Here is the corresponding ODE file forcpend ode parametrically forced pendulum forcpend ode yo t a sin w t yopp t a w w sin w t th thp thp f thp m g sin th m 1 1 yopp t sin th 1 par 1 2 g 9 8 m 1 par a 0 3 w 50 f 25 init th 2 meth qualrk total 100 bound 10000 tol 1e 5 atol 1e 4 done Start up XPPAUT with this file and run the integration Here is the animation file forcpend ani forcpend ani force pendulum use with forcpend ode the oscillating base line 4 5 2 yo t 6 5 2 yo t BLUE 2 the pendulum line 5 5 2 yo t 5 2 1l sin th 2 1 cos th 5 2 yo t BLACK 4 fcircle 5 2 1 sin th 2 1 cos th 5 2 yo t 05 RED end The base is just a horizontal blue line which oscillates up and down in the vertical direction The pivot point of the pendulum is centered on this point and the bob end of the pendulum is determined by the angle 6 The scaling factor 0 2 which multiplies all the dimensions is just to keep everything in the window Load this into the animator and let it rip Notice how the pendulum goes right up Try this experiment set the frequency to a low value w 4 the friction f 03 to a low value and the amplitude a 3 and integrate the equations The pendulum chaotically switches back and forth Animate it The next example is another classical mechanics standard the double pen dulum T
347. the solutions to the ODE 9 8 1 An array example The export directive is useful for small numbers of variables or parameters but is not so good if you are exporting a big array of values In this example I consider an array of 51 coupled oscillators with nearest neighbor coupling I only need equations for 50 oscillators since the relevant variables are the relative phases The differential equations are x 2541 25 5 1 ae 25 To where H u ao a1 cos u az cos 2u b sin u bz sin 2u bs sin 3u I have subtracted off xj so that this represents the relative phases The key point to remember is that the ordering for the storage of the time variable the dependent variables and the fixed variables is t v1 v2 f1 f2 where v are the variables and f are the fixed variables Here is the ODE file chain ode this is a chain of 50 oscillators using dll s to speed up the right hand sides x 1 50 xp j xp 1 50 0 par n 50 a0 0 a1 25 a2 0 b1 1 b2 0 b3 0 export a0 a1 a2 b1 b2 b3 n total 100 done Notice that I have defined 50 fixed variables which will contain the right hand sides They are set to 0 but will be altered by the external library routines I pass the number of oscillators and the parameters for the interaction function I do not need to import anything since this will all be done via the fixed variables Here is the C code for the right hand sides include lt math h gt define H
348. the time step for the integrator default is 0 05 e NJMP integer or NOUT integer tells XPPAUT how frequently to output the solution to the ODE The default is 1 which means at each integration step This is also used to specify a the period for maps in the continuation package AUTO e T0 value sets the starting time default is 0 259 TRANS value tells XPP to integrate until T TRANS and then start plot ting solutions default is 0 NMESH integer sets the mesh size for computing nullclines default is 40 BANDUP int BANDLO int sets the upper and lower limits for banded systems which use the banded version of the CVODE integrator METH discrete euler modeuler rungekutta adams gear volterra backeul qualrk stiff cvode 5dp 83dp 2rb ymp sets the integra tion method see below default is Runge Kutta The latter four are the Dormand Prince integrators and a Rosenbrock 2 3 integrator and a symplectic integrator DTMIN value sets the minimum allowable timestep for the Gear inte grator DTMAX value sets the maximum allowable timestep for the Gear inte grator VMAXPTS value sets the number of points maintained in for the Volterra integral solver The default is 4000 JAC_EPS value NEWT_TOL value NEWT ITER value set pa rameters for the root finders ATOLER value sets the absolute tolerance for several of the integrators TOLER value sets the error tolerance for the Gear adaptive RK and stiff integrators It is the
349. ther the initial conditions of the iteration or the points 100 100 By choosing linetype 1 a series of small circles will be drawn instead of dots or lines Run this file Click on Initialcond 2 par range I 2 and fill in the dialog box as follows Vary dx Reset storage N Stari 0 Use old ic s Y Enar 1 Cycle color N Vary dy_ WovieN Crv 1 Array 2 2 Steps2 101 Steps TO and click on Ok A series of colored points will be drawn To better distinguish the points you should click on Graphics Edit crv select curve 2 and change the color from 2 which is red orange to 7 which is green Exercises e Compute the basin boundaries for the complex roots of z 1 and z4 1 Note that the first is equivalent to x y 1 2xy 0 and the second is equivalent to xf 6x y yt 1 4 2 y xy 0 For the fourth roots problem you should set the total number of iterates to 50 as well as the transient e Consider the discrete bistable system ae 2 2 T 1 Yn Un 1 QYb bx e Ta in the same square as the above examples There are two stable fixed points for this system c 0 Choose a 6 b 3 c 15 and compute the basins of attraction of the two roots You should set the total time and the transient to 100 244 Chapter 9 Tricks and advanced methods Invariant sets revisited Recall that the periodically forced Duffing equation is chaotic and there can be transverse intersections o
350. this file The axes are set up so that the time like variable s is on the horizontal axis and x is on the vertical axis Click on Dir field flow Scaled dir field D S and change 10 to 15 at the prompt You will see a direction field appear For fun try the nonscaled direction fields D D and see why I used the scaled ones In the unscaled fields the length is proportional to the magnitude of the vector 1 f Based on the direction fields you should have no trouble figuring out the trajectories starting at any point To check if you are right click on Initialconds Mice I I and use the mouse to click in a bunch of different points in the window Click outside the view window to quit inputting initial data Hardcopy hint XPPAUT only keeps the latest integration in memory so that if you want hardcopy of all the trajectories you computed along with the direction fields you won t be able to get it However there is a way to keep up to 26 trajectories Click on Graphic stuff Freeze On freeze G F O to automatically freeze the trajectories onto the screen Now redraw the direction fields and compute a bunch of trajectories Finally keep your work Click on Graphic stuff Postscript and accept the defaults choose the default file name oned ode ps and XPPAUT will create a postscript file in your directory which 4 4 Planar dynamical systems 69 NIN l 1 4 2 0 2 4 6 8 10 Figure 4 7 Direction fields a
351. this using the two parameters I phi Do this by adding an equation for the applied current PEN and treating phi as a parameter You should start at the long period point that terminates the branch of periodic orbits This is a hard exercise 7 4 Periodically forced equations To solve periodically forced equations one can always use the same trick we used in the previous section and increase the dimension by one E g x f x t where f is periodic in t with period 1 We want to look for periodic solutions so we solve x f x s s 1 0 2 1 0 s 0 1 However this method does not say anything about the stability of the solutions Instead what I do is introduce the two dimensional dynamical system u ul w v w v v u v wu This admits an asymptotically stable periodic orbit u t v t cos wt 0 sin wt where is an arbitrary phase shift Thus to solve a periodically driven system I merely append this new system and couple u t to it Here is an example of a periodically forced bistable system x x 1 x x a ccos wt For use with AUTO we create the following ODE file 194 Chapter 7 Using AUTO bifurcation and continuation ee oe 0 o 33 o Boot t os 1 o8 06 04 02 0 02 o4 o6 og Figure 7 8 Bifurcation diagram for the periodically driven bistable system bistabfor ode periodically forced bistable system x xx 1 x x x a cxu u ux 1 u7
352. tial conds mIce Let s colorize the plane according to the log of the integral XPPAUT allows you to colorize trajectories and the phase plane according to the values of the vector field at each point along the trajectory or in the phaseplane You choose to color according to the current time along the trajectory the speed or some other quantity like the integral This other quantity must be a single variable or plottable quantity Click on nUmerics to go to the numerics menu Then click on Colorize Another quantity and select 1i the log of the integral as the quantity Then for the color scaling selectChoose Choose 7 5 as the minimum and 1 as the maximum Escape to the main menu Click on Dir field flow and Colorize and choose a grid of 100 You should see the plane fill with color Red dominates as away from the origin I is large Notice that near origin all the colors change rapidly You should see something like Figure 4 10 Now let s turn to the double well potential P x 2 4 x 2 The differ ential equation is m x r which we write as a system t y y 2 2 m 4 5 Nonlinear systems T7 Figure 4 10 Colorized conservative system Greyscale indicates log of the integral The total energy is E my 2 zt 4 x 2 so that you can either write a new ODE file or edit the right hand sides of the previous one Here is my ODE file double well ode double well potential x y y x x7
353. tion x1 y1 with the current color and text properties Only the coordinates can depend on the current values e vtext x1l yl s z draws a string s followed by the floating point value z at position x1 y1 with the current color and text properties Thus you can print out the current time or value of a variable at any given time As with ODE files it is possible to create arrays of commands using the combination of the i1 i2 construction For example fcircle 1 9 10 5 02 j 10 will create 9 circles centered at 0 1 0 2 and so on with radius 0 02 and color 0 1 0 2 respectively Chapter 9 Tricks and advanced methods 9 1 Introduction In this chapter I will describe a bunch of tricks and other advanced features that are not used that often but can be a great deal of help on those occasions when they are needed 9 2 Graphics tricks 9 2 1 Better plots Exporting data If you want to get real high quality graphics say for publication with multiple graphs and figures within figures then you should probably export the data to an ASCII file and use a plotting program such as xmgr or excel Most of these programs will read a series of x y values that are in space delimited format in a file For example if you have an ODE file with two curves plotted on it say V t w t as functions of t then click on Graphic stuff Export and give a file name The resulting file will have three columns t V w separated by spaces
354. tion sign wherever there are products Run XPPAUT and plot x against time Then plot x and z together Hint click on the variables x and y in the Initial Data Window and then click the xvsy button in the Initial Data Window You might want to go into the numerics menu and make Dt smaller and the total amount of time longer e g set Dt 025 and Total 40 Sketch phaseplanes for the following systems in the window 3 3 x 3 3 1 a y 2x y 1 yY 2 x y 2 y 1 3 Y x 1 2 2 y y y x 1 y 2 4 y y z 5 a 2 2 y y 2 y 3z 1 Hint Only make one ODE file Within XPPAUT click on File Edit RHS s to edit the right hand sides Click on OK when you are done Make sure you include the for multiplication Chapter 3 Writing ODE files for differential equations 3 1 Introduction XPPAUT is equipped with a powerful parser that has many useful functions and tools that make it easy to solve problems that are related to dynamical systems Furthermore there are many tricks that you can use to solve very complicated problems In the last chapter of this book 1 will provide a survey of the more obscure tricks Perhaps the best way to describe the range of ODE files is to give you a number of examples sorted by problem type Appendix D gives a complete description of the language for the creation of ODE files Every ODE file is a plain text file The total length of lines must be less tha
355. tions the wilson cowan equations u u f axu b v p v v f cxu d v q f u 1 1 exp u par a 16 b 12 c 16 d 5 p 1 q 4 xp u yp v xlo 125 ylo 125 xhi 1 yhi 1 done Note that in the ODE file I have defined the logistic function f x and also included a line so that when the program is run the graphics window displays the u v phaseplane This is done on line beginning with the symbol Here the statement xp u says that the variable to be plotted on the x axis is u yp v says to plot v on the y axis and the remaining statements set up the window size Fire this up and look at the phaseplane as you change parameters For example draw the nullclines by clicking Nullclines New Try choosing a bunch of different initial conditions with the Initialconds Mice command Vary the parameter a by making it lower Figure out at what point the limit cycle disappears To automate this attach the parameter a to one of the parameter sliders Give it a range between 0 and 20 3 4 Auxiliary and temporary quantities In XPPAUT the values of every one of the variables that define the dynamical system are available for plotting storing and manipulating However sometimes there are other quantities that you would like to see such as the total energy of a system or one of the currents in a model for a cell membrane XPPAUT allows you to define these within an ODE file so that they are accessible for plotting etc This is done by writ
356. tories See figure 2 9 2 1 2 Nullclines and fixed points A powerful technique for the analysis of planar differential equations and related to the direction fields is the use of nullclines Nullclines are curves in the plane along which the rate of change of one or the other variable is zero The x nullcline is the curve where dx dt 0 that is f x y 0 Similarly the y nullcline is the curve where g x y 0 The usefulness of these curves is that they break the plane up into regions along which the derivatives of each variable have a constant sign Thus 22 Chapter 2 A very brief tour of XPPAUT 0 8 F 0 6 F 0 4 F L i L a jo jo e 0 2 0 0 2 0 4 0 6 0 8 1 1 2 Figure 2 9 Direction fields and some trajectories for the Fitzhugh Nagumo equations the general direction of the flow is easy to determine Furthermore any place that they intersect represents a fixed point of the differential equation XPPAUT can compute the nullclines for planar systems To do this just click on Nullcline New N N You should see two curves appear a red one representing the V nullcline and a green one representing the W nullcline The green one is a straight line and the red is a cubic They intersect just once there is a single fixed point Move the mouse into the phaseplane area and hold it down as you move it At the bottom of the Main Window you will see the x and y coordinates of the mouse The intersection of the nullclines
357. ts This is a classical bifurcation of a large amplitude periodic orbit from a saddle node on a circle There is a value of 6 in which the saddle point and the stable node coalesce into one point Try to find this point Hint Tt is between 3 2 and 3 5 Change T2 to 0 9 and 01 back to 3 5 Again look at the stable and unstable manifolds for the saddle point Change 01 to 3 4 and redo the manifold calculation Note that the right branch of the unstable manifold now 80 Chapter 4 XPPAUT in the classroom terminates on a limit cycle For 01 between 3 4 and 3 5 the right branch of the unstable manifold rather than terminating on the stable node terminates back in the saddle along the stable manifold This is called a homoclinic loop For 6 slightly smaller than this critical value a stable limit cycle emerges from this homoclinic In this parameter regime there are three fixed points the leftmost stable and a stable limit cycle The limit cycle that emerges from the homoclinic orbit is stable because the so called saddle quantity A1 A2 is negative where A are the eigenvalues for the saddle point Check this is true b More fun with limit cycles Verify that there are 3 limit cycles for the neural network model and one fixed point for the following parameters Wit 8 Wa 6 Wa 16 Was 2 0i 0 34 A 2 5 T 1 and T2 6 Determine their stability Hint Unstable limit cycles in planar systems can be found by integrati
358. ty Look at the console window if you stated XPPAUT from a command line You will see a list of complex numbers Each is the eigenvalue with maximal real part for the particular parameter Note the change in sign of the real part 2 Solve the state dependent delay equation a t x t 2 x t 0 z t s where s max 0 px where p is a positive parameter Choose x 0 1 and vary p between 1 and 5 Make sure you choose a large value for the maximal delay say 10 For what values of p is the fixed point stable 3 Solve the neural network model r u t 1 S u s ds f u t where f u 1 1 exp 8 u 333 and r is a parameter Hint Write t t u s ds heav 1 t s u s ds for t gt 1 t 1 0 so that it is a convolution of u with the Heaviside function Try r 3 and then r 6 For what values is the rest state apparently stable 102 Chapter 5 More advanced differential equations 4 Solve T t uls ult vi Ze f ds 2 0 t s and compare the solution to the analytic solution u t vt 5 Solve another integral equation due to Mike Mackey T Teo pogl max u t A t e de where soe _ heav t t heav tmaz t _ uv O t ES t E t su bras 1 g v 1 y3 The parameters are e 001 IT 10 e 1 6 Bo 600 fo 8 tmax 1 625 Integrate between 0 and 20 with dt 0 01 Here is the ODE file mackey2 ode xi t heav t 1 heav tmax t tmax 1
359. u a0 al cos u a2 cos 2 u b1 sin u b2 sin 2 u b3 sin 3 u f double in double out int nin int nout double var double con int i double a0 in 0 a1i in 1 a2 in 2 double bi in 3 b2 in 4 b3 in 5 int n int in 6 double x var 1 9 8 Dynamic linking with external C routines 253 double xdot x n double x0dot xOdot H x 0 xdot 0 H x 0 H x 1 x 0 x0Odot for i 1 i lt n 1 i xdot i H x i 1 x i H x i 1 x i x0Odot xdot n 1 H x n 2 x n 1 x0dot This code is very flexible in that I could use it for any size array since the number of oscillators is exported The time variable dependent variables and fixed variables are sent in the array var Thus var 0 t var 1 x1 and so on The fixed variables are var 1 50 xp1 etc These are sent back to XPP to use as the right hand sides I define some pointers and some macros to make the file easier to write and read The pointer x var 1 points to the first variable and xp x n var n 1 points to the first fixed variable To run this fire up XPPAUT on the file chain ode Compile the C file with the line cc shared o chain so fpic chain c where I have called it chain c In XPPAUT click on File Edit Load DLL Choose chain so from the list and for the function name type in f Now let it rip Change parameters and see what happens 254 Chapter 9 Tricks and advanced methods Appendix A Colors and linestyles
360. u as well boundary value problem 0 u_xxxx pi 4 au_xx pi 2 utr tanh u u u_xx O at x 0 1 u up up upp upp uppp uppp pi 4 r tanh u u axupp pixpi par r 1 a 2 total 1 01 xhi 1 meth cvode bound 1000 dsmax 2 parmax 20 autoxmin 0 autoxmax 20 autoymin 1 autoymax 4 Run this in XPPAUT integrate once Then look at the bifurcation diagram with r as the parameter Follow any branches of solutions that arise Grab a point on a nontrivial branch not too far away from the branch point and back in the Main Window of XPPAUT integrate the solution showing that it is nonconstant 2 Here is another nonlinear boundary value problem 1 u rexp u u 0 u 1 0 7 4 Periodically forced equations 193 where r is a parameter Write this as a pair of first order equations u v v rexp u Starting with r 0 use AUTO to continue the solution u 0 for 0 lt r lt 10 Window the Y axis in the bifurcation diagram between 0 and 20 Show that if 0 lt r lt r there are two solutions to the boundary value problem and if r gt r there appear to be none 3 Consider the nonautonomous equation u x rexp uz 0 lt xr lt 1 u 1 u 0 0 By introducing the additional equation x 1 x 0 0 solve this using AUTO over the interval 0 lt r lt 20 4 Use the file m1 ode that we studied exercise 1 in the real example section find the homoclinic orbit and continue
361. uates lt exp1 gt If it is nonzero it evaluates to lt exp2 gt otherwise it is lt exp3 gt E g if x gt 1 then 1n x else x 1 will lead to 1n 2 if x 2 and 1 if x 0 e delay lt var gt lt exp gt returns variable lt var gt delayed by the result of evalu ating lt exp gt In order to use the delay you must inform the program of the maximal possible delay so it can allocate storage e del_shft lt var gt lt shft gt lt delay gt This operator combines the delay and the shift operators and returns the value of the variable lt var gt shifted by lt shft gt at the delayed time given by lt delay gt e ceil arg flr arg are the integer parts of lt arg gt returning the smallest integer greater than and the largest integer less than lt arg gt t is the current time in the integration of the differential equation pi is 7 argl arg9 are the formal arguments for functions int concern Volterra equations shift lt var gt lt exp gt This operator evaluates the expression lt exp gt converts it to an integer and then uses this to indirectly address a variable whose address is that of lt var gt plus the integer value of the expression This is a way to imitate arrays in XPP For example if you defined the sequence of 5 variables u0 u1 u2 u3 u4 one right after another then shift u0 2 would return the value of u2 e sum lt ex1 gt lt ex2 gt of lt ex3 gt is a way of summing up things The expressions lt
362. ulate a population of 30 excitatory and 10 inhibitory cells Let u v be the activities of the respective excitatory and inhibitory cells Then the equations are 29 9 1 ee te e uj uj FulCee Wik Uk Cie gt Wik Uk j 0 29 k 0 k 0 154 Chapter 6 Spatial problems PDEs and boundary value problems 29 9 ei ii E TU 07 Fy Cei Wik Uk Cii Weve J 0 9 k 0 k 0 Note that here the matrices that multiply the variables are not square XPPAUT has a way of dealing with this type of interaction Once again the weights are defined by lookup tables Since w is 30 x 30 the lookup table will be 900 elements The first 30 are row 1 the next are row 2 etc The matrix wt is 30x 10 so it requires a table of length 300 the first 10 elements are row 1 the next row two and so one For the present example we will assume that wj are uniformly distributed on 0 r such that the mean sum of any row is 1 Given the table w that corresponds to a matrix n x m and an array of variables of length m starting at z0 then the declaration special q mmult m n w z0 produces a function q such that q j sum 0 m 1 of w jmti shift z0 i Thus the ODE file for this problem is randnet ode table wee 900 O 899 ran 1 15 table wie 300 0 299 ran 1 5 table wei 300 0 299 ran 1 15 table wii 100 0 99 ran 1 5 special see mmult 30 30 wee u0 special sie mmult 10 30 wie v0 special sei mmult 30
363. umber of iterations to 200 instead of the default of 20 The t 1 tells XPPAUT that the file should be considered a model with discrete dynamics so that the method of solving the equations is automatically set to discrete I could also save typing by also writing this file as delayed logistic x y y r y 1 x par r 2 1 init y 25 meth discrete total 200 done Here since I don t use the construction t 1 I have to tell XPPAUT that the model is discrete XPPAUT defaults to a continuous differential equation Start up XPPAUT with this file e g xpp delaylog ode Solve the equations by clicking Initialconds Go I G Then expand the viewing window by clicking on Window Window and changing X Hi to 200 Change the parameter r making it smaller or bigger and watch what happens If r is too big show that solutions blow up i e they go out of bounds 3 2 1 Nonautonomous systems XPPAUT uses t to denote the independent variable For differential equations and other continuous dynamical systems t is continuous time For maps it is always a natural number 0 1 2 Consider the forced Duffing equation dx dx r de a ax x2 1 ccos wt 30 Chapter 3 Writing ODE files for differential equations Figure 3 1 Phase plane of the forced Duffing equation As above we write this as a pair of first order equations The resulting ODE file duffing ode is forced duffing equation x v v
364. used is the Marquardt Levenberg algorithm see Nu merical Recipes in C The implementation I have used here is somewhat restrictive and every parameter and initial condition is given equal weight Furthermore there are no constraints on the parameters or initial data I will give an example of how to use this on some model data from an enzymatic reaction The differential equations are d k ao C 2d bo Cc 2d koe 2 k3c kad d k3c gt kad and the data present 11 equally spaced values of c d at t 0 70 The data file is called mod dat and here it is 0 0 0 7 1 065 0 0058 14 1 383 0 2203 21 0 9793 0 4019 28 1 107 0 3638 35 0 7289 0 456 42 0 7236 0 5014 49 0 4674 0 715 56 0 6031 0 4723 63 0 6149 0 7219 70 0 3369 0 7294 Suppose that you have a vague idea of what the parameters are but want to do better Here is the ODE file for this problem kinetics ode kinetics ode 9 3 Fitting a simulation to external data 225 simple enzyme model to fit some data c k1 a0 c 2 d b0 c 2 d k2 c 2 k3 c c k4 d d k3x cx c k4x d init c 0 d 0 par a0 2 b0 3 k1 02 k2 002 k3 02 k4 004 total 70 done Let s first see how the data and the model compare before running the least squares Run XPPAUT on this file Don t integrate the equations yet Click on Load in the Data Viewer and load the data mod dat into XPPAUT Plot d vs time and Freeze this in color 1 Plot c vs time and Freeze this in color
365. ute the average Click on nUmerics Averaging Make H Then type in the first component of the coupling v v and the second 0 Then let it rip In a few moments the calculation will be done If your system has more than two columns in addition to time t as this example does v w ica ik then the first column contains the average function H the second column contains the odd part of the interaction function and the third column contains the even part Plot the function H by escaping back to the main menu and plotting v versus time remember v is the first column in the browser after the t column This is a periodic function For later purposes we want to approximate this periodic function Click on nUmerics stocHastic Fourier and choose v as the column to transform Look at the data browser and observe the first three cosine terms column 1 after the t column and the first three sine terms column 2 They are 3 34 3 05 29 and 0 3 61 0 33 respectively Thus to this approximation H 3 34 3 05 cos 0 29 cos2 3 61 sin 0 33 sin 2 9 2 To get the interaction function adjoint or original orbit back into the data browser click on Averaging Hfun etc from the nUmerics menu Summarizing 1 Compute exactly one period of the oscillation you may want to phase shift it to the peak of the dominant coupling component 2 Compute the adjoint from the nUmeric Averaging menu 3 Compute the interaction
366. v dx dt Thus m t gt 1 hv t m t The ODE 40 Chapter 3 Writing ODE files for differential equations file for this model is faucet ode x v v g k x f mu v m m f global 1 x 1 m max m h m v m0 par f 4 g 32 h 4 m0 01 mu 6 k 1 init x 0 v 0 m 1 total 200 done The first three equations are straightforward enough we have just rewritten the second order system into a pair of first order systems and used the fact that mv m v mv The global declaration states that when x crosses 1 from below we decrement m by hmv The additional function max x m0 makes sure that the mass never falls below some minimum value or becomes negative Run this and look in the x v phaseplane to see something very similar to the Rossler attractor see the exercises in the previous section Play with the parameter f and see a variety of different solutions and bifurcations 3 5 4 Exercises 1 Write a differential equation for John Tyson s cell growth model ul ka v u at u kgu v k m kgu m bm where k4 200 k 0 015 kg 2 a 0 0001 b 0 005 m is the mass and when the variable u drops below 0 2 the cell divides and m m 2 Start with u 0 0075 v 0 48 m 1 Integrate this for a total of 1000 with a time step of 0 25 and using the DoPri 8 integrator Plot the mass as a function of time Notice that after a few transients it goes to a periodic solution 2 Consider
367. val 0 1 with a step size of 0 05 using Euler s method for the following values of a 20 30 40 60 80 Even the classic Runge Kutta goes bad Try using Runge Kutta with the default stepsize and a 90 100 101 102 The solution to this problem is to use a backward or implicit method Consider the simple linear differential equation ve ar x 0 1 264 Appendix C Numerical methods where a is a parameter The Euler iteration with stepsize h for this is 141 1 ah z with a solution n 1 ah Clearly if ah gt 1 then the solution is not monotonic but instead undergoes oscillations If ah gt 2 then solutions grow in an oscillatory manner We saw this behavior in the nonlinear logistic model Implicit methods avoid some of these problems The simplest method is called implicit Euler and the scheme is n41 Tn thf t h en41 The big difference between this and Euler is that the right hand side depends on Ln 1 SO that at each time step we have to solve a nonlinear equation for 2 1 However the advantage of this method becomes obvious if we apply it to the linear equation above Tn41 Tn han 1 and solve for 2 41 to get n 1 In 1 ha Thus the solution is 1 n which is monotonically decreasing Thus unlike the explicit Euler method back ward Euler behaves like the differential equation no matter how large a is This idea can be applied to any linear system and the result is Eng I hA tEn
368. value Vy it is reset to 0 This leads to an oscillation when I gt Vr Often these models are coupled to each other through alpha functions a t b texp bt Thus the pair satisfies MA LT gsal 1 V2 gsl where each time t Vj crosses Vr it is reset to 0 and the quantity a t t is added to s XPPAUT has a mechanism for incorporating such discontinuities global flags These are quantities that the integrator checks for zero crossings If a zero crossing occurs then the variables can be updated and thus discontinuously changed The integrators are restarted so that even the adaptive step solvers will work fine Here is the ODE file for a single integrate and fire oscillator iandf1 ode integrate and fire model v v I global 1 v vt v 0 par I 1 2 vt 1 dt 01 ylo 0 done The new line is global 1 v vt v 0 This asserts that there is a global flag to check The general syntax for flags is global sign condition event1 event2 The condition is evaluated at each time step If the sign is 1 and the condition changes sign from negative to positive or if the sign is 1 and the condition changes sign from positive to negative then each of the events is done If the sign is 0 then the event occurs only if the condition is identically zero Events always have the form x expr where x is one of the variables and expr is some expression Thus in the above ODE file if V Vr changes from negative to positiv
369. verything for me automatically I repeat my batchfile xpp bat here set BROWSER c Program Files Netscape Communicator Program netscape exe set XPPHELP c xpp help xpphelp html set DISPLAY 127 0 0 1 0 0 set HOME c xpp c xpp xppaut 1 2 3 This sets the display for X and also tells XPPAUT where to look for the resource file the browser and the first page of the help file Then it calls XPPAUT Chapter 1 Installation Chapter 2 A very brief tour of XPPAUT Touring can make you crazy Frank Zappa In this chapter I will show you the way to get up and running Many of the types of problems most people need to solve can be done after going through this short session with the program I will assume that you are at least somewhat familiar with differential equations I will run through two quick tutorials on using XPPAUT for a linear differ ential equation and a nonlinear differential equations This should be sufficient for anyone to setup differential equations and solve them For more advanced topics the user should look at the rest of the book NOTES Menu commands will appear like this Command and single letter keyboard shortcuts will appear like this A Do not use the CapsLock key all shortcuts are lower case Every command can be accessed by a series of keystrokes To make sure key clicks are interpreted correctly click on the title bar of the window for which the shortcut is intended 2 1 Creating the ODE file
370. w a small circle of radius 3 points at each point in the trajectory Negative linetypes are useful for displaying periodic orbits of maps since they are easier to see than single points on the screen Problems with three d plots Sometimes three d plots seem to cut off the axes You can rectify this by clicking on the Window Window command and making the window for xlo xhi ylo yhi a bit bigger 9 2 2 Plotting results from Range Integration XPPAUT is a computer program It is thus pretty dumb and can be tricked into doing what you want it to even though it was not meant to Suppose you have a system of equations and you want to keep the results from a range integration Initialconds Range If you have fewer than 26 different curves then you can click on Graphic stuff Freeze On freeze and then do the range integration Each curve will be kept and can be plotted or output to Postscript This is probably the best way to keep the results There is another way which doesn t limit the number of curves and does not use up the frozen curves Here is the trick In the ODE file make the maximum amount of storage large e g in the ODE file add the line maxstor 20000 so that 20000 points will be stored rather than the default of 4000 Run the file In the Initialconds Range dialog box type in No in the Reset storage box and run the range integration All the data from each of the runs is concatenated into one long stream The problem is that whe
371. we just computed 4 2 2 Periodic points The bifurcation diagram constructed above is actually more correctly an orbit map That is it simply is a record of the stable solutions for any particular choice of parameters A true bifurcation diagram should show solutions which are both stable and unstable For example one can ask where all the period three points are for the map Period three is notable since it implies that there are periodic points for every possible period This is one of the standard measures of chaotic behavior How can one find the periodic points of a map as a function of a parameter Consider a fixed value of the parameter and the third iterate of the map Since the periodic points are not necessarily stable and in fact are unstable we need a method of finding them that is independent of stability Newton s method provides a good way to search for them as the convergence of the iterates is independent of the stability Thus for a fixed parameter value we could sweep through a range of starting values and plot those for which the Newton s iterates converge This shotgun approach has the advantage that we can pick up new branches of orbits as the parameter changes In XPPAUT there is a way to run a two parameter range of simulations This allows us to sweep through both the parameter and the state space Given the map f x a point of period p satisfies fP 1 x Here f x means f f x and so on Let g a x
372. when the magnitude of zn is 2 The statements poivar amp poipln 0 poisgn 1 tell XPPAUT to check the quantity amp hitting 0 with a positive derivative The statement poistop 1 means to stop the simulation when this happens Start XPPAUT with this input file Turn off the coordinate axes by clicking on Graphic stuff aXes opts and setting X org etc to 0 Next we will do a two parameter range integration like in the calculation of the bifurcation diagram for the logistic equation Click on Initialconds 2 par range I 2 and fill in the dialog as follows Varyi cx Reset storage N Starti 15 Use old ics Y PEndi 5 Cycle color N Way ey Movies N Crv Array 2 2 Steps2 200 Steps 200 Click on Ok to run the simulation This might take a bit of time you are running 40000 simulations You should see the Mandelbrot set appear on your screen The points in the set itself are empty The reason for this is that in in the Poincare map option of XPPAUT if the condition never occurs in the allotted time no points are captured Thus a point is plotted at cx cy if iterates with that parameter value leave a circle of radius 2 after fewer than 50 iterates Since XPPAUT knows how many iterates it takes to exit we can color code the dots accordingly Look at the first column in the Data Viewer This is the time interpolated at which the iterating left the circle of radius 2 Thus we can code these times by color Click on nU
373. which is about exp 12 As far as XPPAUT is concerned it has found a solution within the tolerances set for the BVP solver Newton s method doesn t really find a zero to a function but only approximates it to within a certain tolerance Click on nUmerics bndVal U V and change the tolerance to 1e 10 Now solve the BVP again using the value k 12 from before as the initial point Notice that it again finds a solution but k 24 This is a significant difference from k 12 which should send up red flags that there is an inability to converge to a true solution Dendrites are branches that occur on neurons and act as the main input regions for the neurons In the classical theory of dendrites they are modeled as passive continuous multi branched cables The steady state voltage distribution on such cables is often of interest in simplified models The distribution of voltages depends on the boundary conditions at the ends of the finite cable There are several boundary conditions that are relevant i fixed in which the potential is 6 1 Boundary value problems 127 held constant at one end ii sealed in which the gradient of the potential is zero and iii leaky in which there are conditions on the voltage and it s derivative Consider a case where the dendrite is held at V Vp at xz 0 and has a leaky boundary condition at x 1 Then the equations are dy V 0 VO aV 1 020 0 The parameter A is the s
374. will show you how to solve a variety of problems with AUTO Keep in mind that AUTO is a tricky program and can fail in a dramatic fashion Nevertheless it is a powerful tool and even though it is 15 years old it is still better than most other similar programs when it comes to tracking periodic solutions and solving BVPs Many physical and biological systems include free parameters One of the goals of applied mathematics is to understand how the behavior of these systems varies as the parameters change This is a numerically challenging task and there is generally no way to systematically explore a system as a function of all it s parameters One of the first things a modeler should do is to reduce the number of parameters by either fixing those which are best known e g gravity and by making the problem dimensionless Dimensional analysis is very useful and the reader should consult any number of books which describe this technique Edelstein Keshet 1989 Murray 1992 Assuming that you can reduce the free parameters to a manageable number there are some useful tools for exploring how a dynamical system changes as these parameters vary The most straightforward technique is called continuation in which a particular solution such as a fixed point or limit cycle is followed as the parameter changes AUTO which is described in this chapter provides some very powerful algorithms for the continuation of fixed points and periodic solutions to differ
375. window Click on Find and type in x for the variable and 100 for the value XPPAUT will try to find the value of x closest to 100 This will be the maximum Click on Get to load this as an initial condition Figure out the unperturbed period One way is to integrate the equations and estimate the period A better way is to let XPPAUT do it for you In the Main Window click on nUmerics Poincare map Max Min and fill in the dialog box as follows Variable x Direction 1 Stop on sect Y and then click on 0k You have told XPPAUT to integrate plotting out only the maxima Direction 1 of x The Stop on section ends the calculation when the section is crossed Click on Transient and choose 4 for the value This is so we don t stop at the initial maximum Transient allows the integrator to proceed for a while before looking at and storing values Now exit the numerics menu by clicking Esc Click on Initialconds Go and the program will integrate until x reaches a maximum In the Data Viewer click on Home and you should see that Time has a value of around 6 6647 This is the unperturbed period Set the parameter t0 to the value in the Data Viewer Compute the PRC Now we are all set to compute the PRC Change the perturbation amplitude from 0 to 3 Click on Initialconds Range and fill in the dialog box as follows Range over tan Steps 10 Starts 0 End 6 6647 Reset storage N and click Ok I
376. with 0 lt b lt 5 and 0 lt ui lt 7 As with many internal parameters most of the AUTO parame ters can be set with options in the ODE file Here is my version of the above equations bruss2 ode two Brusselator equations 7 2 Maps boundary value problems and forced systems 179 f u v a b 1 x u v u 2 g u v b ru v u 2 u1 f ul v1 dux u2 u1 vi g ul v1 dv v2 v1 u2 f u2 v2 du u1 u2 v2 g u2 v2 dv v1 v2 par b 1 5 a 1 du 0 dv 2 init ui 1 u2 1 v1 1 5 v2 1 5 ntst 30 dsmin 1e 5 dsmax 0 1 parmin 0 parmax 5 autoxmin 0 autoxmax 5 autoymin 0 autoymax 7 done Note that a few years ago this computation and some discussion of the resul tant system would have been considered a decent research problem Now it s an exercise in a book 7 2 Maps boundary value problems and forced systems XPPAUT allows you to study continuation of discrete dynamical systems boundary value problems and with some preparation periodically forced systems The use of AUTO with maps is rather restricted since all one can do is follow branch points and two parameter continuations of limit points and Neimark Sacker points Recall the Neimark Sacker bifurcation is the discrete analogue of the Hopf bifurcation 7 2 1 Maps XPPAUT makes some aspects of tracking the behavior of maps pretty easy mainly because it allows you to study the continuation of f x the kt iterate of the map rather than just the map For
377. x2 fires I update similar intervals but also the timing difference phi t t1f Try this file with the parameters as in the file Plot t1 t2 on the same plot Plot phi Increase a to 0 06 Try this again Change a to 0 06 and solve it again Finally set a 0 1 and w2 0 53 and integrate the equations Look at t1 t2 and interpret what this means 9 6 3 Initial data depending on parameters One unfortunate shortcoming in XPPAUT is the inability of the program to set initial data that depend on parameters Parameters can be defined in terms of other 240 Chapter 9 Tricks and advanced methods parameters with the lt name gt lt formula gt declaration However there is no easy way to define initial data in terms of a parameter In spite of this shortcoming there is a little trick that you can use This trick exploits the fact that global flag conditions can set the state variables to anything Let s consider the classic problem of firing a cannonball The parameter that you want to vary is the angle of the cannon 6 Thus the differential equation is mi f a my mg fy with x 0 y 0 0 0 vocos y 0 vosin0 Here is the trick we will flag t 0 and immediately switch initial data The key here is to use the global sign 0 instead of 1 1 The difference with using 0 is that the flag is set only when equality occurs exactly This essentially never happens except at the start of a problem Thus it is a good way to
378. y xlo 2 xhi 2 ylo 4 yhi 4 total 1 001 dt 01 done Most of this file is to define the plotting area and numerics To change the ranges of plotting just vary the parameters xlo xhi and to change the function just click on File Edit Functions and change the function To keep the original graph and subsequent graphs click on Graphics Freeze On freeze and up to 26 curves can be kept Here is an example in which the first 5 terms of the Taylor series for the function sin x are plotted on the interval 7 11 sintayl ode first 5 terms of the Taylor series for sin zi x z2 z1 x 3 6 z3 z2 x 5 120 z4 z3 x 7 5040 z5 z4 x 9 362880 x 2x pi init x 3 1415926 aux y sin x aux y 1 5 z 3 total 1 dt 005 xlo 3 15 xhi 3 15 ylo 1 yhi 1 nplot 6 xp x yp y ypL2 61 y j 11 xpLj x done NOTE By defining a series of internal variables z1 I can just add the re quired new terms sequentially The last line tells XPPAUT that all these quantities should be plotted They will appear as different colored curves You can easily use XPPAUT to make polar coordinate plots of the form r f 8 or 0 f r polarpl ode polar plots f z 1 cos z theta init theta 12 r f theta aux x r cos theta aux y r sin theta total 25 4 1 Plotting functions 47 O xp x yp y x10 2 xhi 2 yl0 2 yhi 2 done There is a parameter a so that one could plot this over a range of values of a Edit t
379. y Click on Kinescope Playback K P and click the mouse button to cycle through them Tap Esc when you are tired of doing this As an exercise you could try this method with some other chaotic system like the Lorenz equations or the Rossler attractor 9 6 4 Poincare maps revisited The use of adaptive integration methods can make it easy to compute Poincare sections for periodically driven systems The idea is to just set the output time stem Dt to the period of the forcing and then just plot the results That is we can ignore the Poincare option completely This only works for systems where the Poincare map is with respect to the time variable Recall the forced Duffing equation from Chapter 3 z v v z fu ccos wt and the ODE file for it duffing ode Run XPPAUT with this file Change c 7 f 3 omega 1 25 In the nUmerics menu change the Method to DorPri 8 3 with a relative and absolute tolerance of le 5 Now click on Dt and type the following string 2 Pi omega XPPAUT evaluates command line numbers that start with the sign and converts them to floats Thus output will only happen at multiples of 27 w the period Now change Total to 2 Pi 4000 omega to get 4000 points Make a 2D window of a v in the plane 2 2 x 2 2 Finally edit the graph type G E and choose 0 Change the Linetype to 1 for large dots Now integrate the equations and you will see the usual attractor 246 Chapter 9 Tricks and advan
380. y to stop the computation when the solution runs into a fixed point Try this with the Fisher equation Click on Sing pts Go Answer no to the Print eigenvalues question At this point a small triangle will appear at the point 1 0 indicating a saddle point A new window will appear that summarizes the information about the fixed point The value of the fixed point is given the number of real and complex eigenvalues with positive and negative real parts and a statement about the stability of the fixed point E g r 1 means that there is one positive real eigenvalue and c 0 means that there are no complex eigenvalues with negative real parts Recall that the eigenvalues of the linearized system inform you of the local stability of that fixed point XPPAUT computes the Jacobi matrix numerically and then uses standard eigenvalue routines to compute the eigenvalues of the resulting matrix Answer yes to the invariant sets question XPPAUT will first compute unstable and then stable manifolds The first trajectory computed yellow goes 6 1 Boundary value problems 133 0 8 F 0 6 F 0 4 F 0 2 F 0 2 F Figure 6 3 Stable and unstable manifolds for the fixed point 1 0 of the Fisher equation off to the right and out of bounds The second trajectory goes to the left and enters the fixed point 0 0 where it is stuck until you click Esc Next two more blue trajectories are computed and these go out of bounds In XPPAUT
381. y glider XC X yc y c cos th s sin th xx 1 14 xg j c yg j s xc lg yy 1 14 xg j s yg j c yc lg 5 total 30 dt 025 bound 1000000 done The structure is similar to the plane glider but the scaling is done after making the glider The picture for the glider is centered at the origin so the coordinates must be rotated by the angle 0 and then translated by the center of mass of the glider Thus I define 14 points for the x and y coordinates of the glider which rotate the little picture and translated it by the center position Finally I scale the whole thing and lift up the y coordinate a bit I don t bother to mod out the coordinates so that it can fly off into the sunset at its leisure The ODE file does all the work and the animation file is but one line fglider ani 8 2 Some first examples 207 ockw and Bullwinkle Figure 8 2 The fancy glider fancy animation for glider line xx 1 131 yy j31 xx 3 1 yy j 1 end I connect the lines for the glider it was designed as a closed curve Try these two files 8 2 5 Coupled oscillators Recall that we looked at the Morris Lecar model in a number of prior chapters Suppose that we take a cluster of such oscillators and couple them together with diffusion through the average voltage of all of them 1 Vj Teot Vj wz dy 5 Vie Vj k 208 Chapter 8 Animation where fio is the local ionic current of each cell We can ask whether they wil
382. yfile ps where I use the latter for color printouts Other ways to get hardcopy Another way to get hardcopy which you can import into documents is to grab the image from the screen In Windows click on Alt PrtSc after making the desired window active For some servers such as the MiX server this will grab the entire X desktop window Paste this into the Paint accessory and then use the tools in Paint to cut out what you want Alternatively you can download a number of programs that let you capture areas of the screen In the UNIX environment you can capture a window using xv an excellent utility that is free and available for most UNIX versions All of the screenshots in this tutorial were captured with xv Finally you can capture the screen or a series of screen images with the Kinescope Capture command and then write these to disk with the Kinescope Save command This produces a gif file that is usable 16 Chapter 2 A very brief tour of XPPAUT Parameters Xx Elose bk Default Eancel Ed Figure 2 6 The parameter window by many software packages including most browsers Getting a good window If you have computed a solution and don t have a clue about the bounds of the graph let XPPAUT do all the work Click on Window zoom F it and the window will be resized to a perfect fit The shortcut is W F and you will likely use it a lot 2 4 Changing Parameters and Initial Data There are many ways to vary the paramete
383. you to manip ulate data in the columns much like you would in a spread sheet For example sometimes you might want to numerically differentiate or integrate data in a col umn or look at a logarithmic plot This is all possible by manipulating columns in the Data Viewer So let s look at some of the functionality of the data browser Run the Morris Lecar equation model m1 ode In the Data Viewer click on Add 9 5 Oscillators phase models and averaging 227 col For the name type in il an abbreviation for the leak current Iz For the formula type in gl1 v el and then click on Add it A new column will appear in the data browser You can plot this new quantity and when you integrate the equations this new quantity changes appropriately You can add more quantities as well The Del Col does not work properly so that it has been inactivated If you make a mistake in the definition of the formula for the new quantity you can always edit it using File Edit RHS s in the Main Window If you don t want to add new column but instead just want to look at say the log of a quantity for that particular run you can always click the Replace button in the Data Viewer This prompts you for the name of the column and then the formula The Unreplace button undoes the most recent replacement You can numerically integrate or differentiate a column of data using the sym bols amp and respectively Thus in the Morris Lecar file suppose you w
384. yp cy xlo 1 5 xhi 5 ylo 1 yhi 1 1t 1 maxstor 40000 meth disc total 50 poimap section poivar amp poipln 0 poisgn 1 poistop 1 done Notes This is a rather simple ODE file with lots of controls added We could easily set these controls in XPPAUT but for ease of use set them here The first two lines are the map to iterate written in the real variables z x iy The next lines are so that XPPAUT can use the parameters as coordinate axes XPPAUT only allows auxiliary variables and dynamic variables as axes The auxiliary variable amp changes from negative to positive when the magnitude of z exceeds 2 It can be shown that once this happens Zn diverges to infinity The controls in the first line set up the plot axes and bounds The next line tells XPPAUT to allocate room for 40000 points the default is 4000 the method of integration and the total iterations per run We use only 50 since we are only crudely estimating the Mandelbrot set The last option line starting with poimap section tells XPPAUT that we are going to compute Poincare maps That is we will plot only certain points when certain events happen There are several different types of Poincare maps that are possible in XPPAUT If some quantity say Q crosses a value say v with a positive sign then we plot all the variables at the time at which this happens In the present case 4 2 Discrete dynamics in one dimension 61 we want the quantity amp to vanish as that is
385. ze graphical and geometric methods for solving differential equations There are several ways to gain a qualitative view of the behavior of a differential equation One way is to draw the phaseline Consider the differential equation dx T f x To draw the phaseline for this we plot f a versus x Then we use the sign of f x at any point x to determine the direction along the x axis Fixed points are precisely zeros of f x Thus to draw a phaseline plot we draw f x and then representative trajectories on the x axis XPPAUT can animate this whole process We will construct an animation file which plots the function f x and a moving ball showing the behavior of x t The point of this is to see how the value of f x determines the speed of the particle First we make an ODE file for the scalar system Then we add the animation file Here is the ODE file phase line ode one dimensional autonomous system f x x x 25 1 x x f x init x 26 transformation information for the animator par x0 5 x1 1 5 y0 1 y1 1 tr x x0 x 100 x1 x0 ff x f tr x y0 y1 y0 ylo 25 yhi 1 25 total 40 xhi 40 done The file is pretty straightforward I have defined 4 parameters x0 x1 y0 y1 to scale the x axes and the y axes for the animator The depicted phase space in the animation will be the interval xo 11 The parameters y0 y1 give the maximum and minimum vertical values in which f x is plotted Since the x axis is

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