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1. markt The t value to marker with a small ball 48 maxx The upper bound for the x axis of the plot maxy The upper bound for the y axis of the plot maxz The upper bound for the z axis of the plot minx The lower bound for the x axis of the plot miny The lower bound for the y axis of the plot minz The lower bound for the z axis of the plot odd_tick_ length The length of the odd tick marks period _doubling symbol The symbol to use for period doubling bifurcation points ps _colormode The PostScript output mode color gray or monochrome right margin The margin between the graph and the left edge runner The runner object from which to get data smart_label Whether to use a smart but slower label placement algorithm special_point_colors An array of colors used to mark special points special_point_radius The radius of the spheres used to mark special points solution A parsed solution file to plot solution_column defaults A set of solution columns the user is likely to use solution_coordnames Variable names to use instead of U 1 for solutions solution filename The filename of the solution to plot solution_indepvarname Variable name to use instead of t for solutions solution_x The column to plot along the X axis for solutions solution_y The column to pl
2. cusp symbol The symbol to use for Cusp points d0 d1 d2 d3 d4 Redefine d0 d1 d2 etc setting to use with PyPLAUT pp dashes List of dash no dash lengths for dashed lines decorations Turn on or off the axis tick marks etc default option Default d0 d1 d2 etc setting to use with PyPLAUT Cpp elevation Elevation of the axes in 3D plots error symbol The symbol to use for error points even_tick_ length The length of the even tick marks flip_torus_symbol The symbol to use for flip torus points fold_flip_symbol The symbol to use for fold flip points fold_torus_symbol The symbol to use for fold torus points foreground The background color of the plot generalized_hopf_symbol The symbol to use for Generalized Hopf points grid Turn on or off the grid height Height of the graph hopf_symbol The symbol to use for Hopf bifurcation points index An array of indices to plot label An array of labels to plot or all for all labels labelnames A dictionary mapping names in fort 7 to axis labels label_defaults A set of labels that the user is likely to use left_margin The margin between the graph and the left edge letter_symbols Whether to use letter True or symbols False for special points limit_point_symbol The symbol to use for limit points line_width Width to use for lines and curves
3. Ag 2 3 2 2 yl A y t x As 17 1 Z t 2 2 with objective functional 1 w Ey zi An AAs 2a de 0 where g x y z A1 A2 A3 A4 Az Thus in this application a one parameter extremum of g corresponds to a fold with respect to the problem parameter A3 and multi parameter extrema correspond to generalized folds Note that in general the objective functional is an integral along the periodic orbit so that a variety of optimization problems can be addressed For the case of periodic solutions the extended optimality system can be generated auto matically i e one need only define the vector field and the objective functional as in done in the file ops f90 For reference purpose it is convenient here to write down the full extended system in its general form u t Tf u t A TER period u f ER AER w t T fu u t A w t Kult ygu u t A w ER k y ER u 1 u 0 0 w 1 w 0 0 fy ult uh t dt 0 17 2 Je w g u t A dt 0 feo wt wlt 24 ya dt 0 a ER Sa F ult d w t var u t A m dt 0 ER fo Th lut A wE 792 u t A dt 0 4 ER i 1 e n Above uy is a reference solution namely the previous solution along a solution family 190 In the computations below the two preliminary runs with IPS 1 and IPS 2 respectively locate periodic solutions The subsequent runs are with IPS 15 and
4. Type dl xxx to delete the data files b xxx s xxx d xxx Type cnvc xxx yyy to convert old format constants files c xxx and h xxx to a new format constant file c yyy The command cnvc xxx overwrites the file c xxx with the new style file and deletes h xxx if it exists Type rn to rename within the current directory all old named constants HomCont bifurcation diagram and solution files starting with r s p and q to files with the new prefixes c h b and s Type rc to do a recovery by swapping the backup files fort 7 and fort 8 with the files fort 7 and fort 8 Type rc xxx to do a recovery by swapping the backup files b xxx and s xxx with the files b xxx and s xxx Type gz to compress using gzip all output files in the current directory Type Cuz to decompress using unzip all output files in the current directory Type sr xxx y to copy C xxx to C xxxy Diagnostics Type Clp to list the value of the limit point function in the output file fort 9 This function vanishes at a limit point fold 68 bp hb ho sp it st ss ev efl Ono Type lp xxx to list the value of the limit point function in the data file d xxx This function vanishes at a limit point fold Type bp to list the value of the branch point function in the output file fort 9 This function vanishes at a branch point Type bp xxx to list the value of the
5. 235 Chapter 24 HomCont Demo cir 24 1 Electronic Circuit of Freire et al Consider the following model of a three variable electronic circuit Freire Rodr guez Luis Gamero amp Ponce 1993 6 v x By ast b3 y ay r y Br B47y 2 bs y a 24 1 ZT Yi These autonomous equations are also considered in the AUTO demo tor First we copy the demo into a new directory and compile dm cir The system is contained in the equation file cir f90 and the initial run time constants are stored in c cir 1 We begin by starting from the data from cir dat for a saddle focus homoclinic orbit at v 0 721309 6 0 6 y 0 r 0 6 A3 0 328578 and B3 0 933578 which was obtained by shooting over the time interval 27 PAR 11 36 13 We wish to follow the family in the G v plane but first we perform continuation in 7 1 to obtain a better approximation to a homoclinic orbit ri run cir c cir 1 yields the output BR PT TY LAB PERIOD L2 NORM ore PAR 1 1 21 UZ 2 1 000000E 02 1 286637E 01 7 213093E 01 1 42 UZ 3 2 000000E 02 9 097899E 02 7 213093E 01 1 50 EP 4 2 400000E 02 8 305208E 02 7 213093E 01 Note that y PAR 1 remains constant during the continuation as the parameter values do not change only the length of the interval over which the approximate homoclinic solution is computed Note from the eigenvalues stored in d 1 that this is a homoclinic orbit to a saddle
6. Index 3 2 11 22 22 23 31 Variable A3 A2 T 72 Ae As 7 Table 17 3 Run 5 file c ops 5 In Run 5 the parameter a which has been replaced by A remains fixed and nonzero The variable 7 monitors the value of the optimality functional associated with A The zero of Ta located in this run signals an extremum with respect to Az Index 3 2 1 11 22 23 31 Variable Az Ao Az T Aa As T Table 17 4 Run 6 file c ops 6 In Run 6 72 which has been replaced by A remains zero Note that 7 and 73 are not used as variables in any of the runs in fact their values remain zero throughout Also note that the optimality functionals corresponding to 7 and 73 or equiv alently to T and Az are active in all runs This set up allows the detection of the extremum of the objective functional with T and Az as scalar equation parameters as a bifurcation in the third run The parameter 4 and its corresponding optimality variable 74 are not used in this demo Also A is used in the last run only and its corresponding optimality variable 7 is never used 192 AUTO COMMAND ACTION mkdir ops cd ops demo ops create an empty work directory change directory copy the demo files to the work directory ri run e ops c ops locate a Hopf bifurcation uzr 3 0 92 0 93 r2 run ri HB1
7. NPR 5 NMX 20 IFIXED 11 UZR DS save ri2 12 BR PT TY LAB PAR 7 PAR 8 PAR 6 1 5 9 7 38787E 19 2 91178E 10 3 25000E 01 1 10 10 5 27166E 19 2 23972E 10 8 25000E 01 1 15 11 6 15227E 19 2 91908E 10 1 32500E 00 1 20 EP 12 5 96426E 19 3 20088E 10 1 82500E 00 The orbit flip continues to be defined by a planar homoclinic orbit at PAR 7 PAR 8 0 21 5 Detailed AUTO Commands AUTO COMMAND ACTION mkdir san cd san demo san san load san IPS 9 NDIM 3 ISP 0 ILP 0 ITNW 7 JAC 1 NTST 35 IEQUIB 0 DS 0 05 create an empty work directory change directory copy the demo files to the work directory configure common constants ri run san ICP 1 8 UZR 1 0 25 continuation in PAR 1 r2 run ri ICP 9 8 ITWIST 1 NMX 2 UZR generate adjoint variables r3 run r2 ICP 4 8 10 21 33 IPSI 1 13 NMX 50 NPR 20 UZR 4 1 0 21 0 33 0 save r3 3 continue homoclinic orbit and adjoint save output files as b 3 s 3 d 3 r4 run r1 ICP 4 8 10 21 33 ITWIST 1 IPSI 1 13 NMX 50 UZR 33 0 sv 4 no convergence without dummy step save output files as b 4 s 4 d 4 ro pun ra UZ2 ICPS 1 8 10 21 33 NMX 20 DS sv 5 continue non orientable orbit save output files as b 5 s 5 d 5 Table 21 1 Detailed AUTO Commands for running demo san 217 AUTO COMMAND ACTION r6 r
8. 3 87300E 01 1 35 EP 14 9 33416E 01 6 84317E O0O1 4 32927E 01 The results of both computations are presented in Figure 25 1 from which we see that the orbit shrinks to zero as PAR 1 p gt 0 25 2 Detailed AUTO Commands 241 AUTO COMMAND ACTION mkdir she cd she demo she create an empty work directory change directory copy the demo files to the work directory ri run she c she 1 sv 1 continue heteroclinic orbit start from she dat save output files as b 1 s 1 d 1 r2 run she c she 2 repeat with IEQUIB 1 r3 run r2 2 c she 3 save r2tr3 2 continue in reverse direction restart from label 2 of r2 Save appended results to b 2 s 2 d 2 Table 25 1 Detailed AUTO Commands for running demo she Figure 25 1 Projections into x y z space of the family of heteroclinic orbits 242 Chapter 26 HomCont Demo rev 26 1 A Reversible System The fourth order differential equation an Py y uP Z 0 arises in a number of contexts e g as the travelling wave equation for a nonlinear Schr dinger equation with fourth order dissipation Buryak amp Akhmediev 1995 and as a model of a strut on a symmetric nonlinear elastic foundation Hunt Bolt amp Thompson 1989 It may be expressed as a system ul U2 oo 26 1 ug Us Ua Puz u u Note that 26 1 is invari
9. and by setting ISTART 2 we try to locate a 2 homoclinic orbit r6 run r5 c fhn 6 sv 6 In fact we find many of them exactly as is predicted by the theory BR PT TY LAB PAR 21 as PAR 1 PAR 22 1 174 UZ 45 1 64799E 02 2 74218E 01 3 44422E 11 1 178 UZ 46 1 44799E 02 2 74218E 01 3 29142F 14 1 182 UZ 47 1 24854E 02 2 74218E 01 1 70138E 15 1 187 UZ 48 1 04789E 02 2 74218E 01 8 57896E 14 1 191 UZ 49 8 49517E 01 2 74218E 01 1 93804E 13 1 196 UZ 50 6 45145E 01 2 74218E 01 2 26551E 09 Each of these homoclinic orbits differ by about 20 in the value 7 This is about the time it takes to make one half turn close to and around the equilibrium so that orbits differ by the number of half turns around the equilibrium before a big excursion in phase space Note that the variation of c is so small that it does not appear A plot of T vs e gives insight into how the gap is opened and closed in the continuation process This is depicted in Figure We are now in a position to continue each of these Column 6 4 006 03 3 00e 03 2 00e 03 0 006 00 2 008 02 1 008 02 Column 0 Figure 27 5 A plot of e as a function of 7 during our computation of Shil nikov type two homoclinic orbits Each zero corresponds to a different orbit orbits as a normal homoclinic orbit by setting ISTART 1 and ITWIST 0 We leave this as an exercise to the reader 257 27 3 Branch switching
10. denotes a user chosen data set name This user s guide includes the following information 1 A description of the PLAUTO4 window system 2 A list of PLAUTO4 configuration options 3 An example of using PLAUTO4 8 1 Quick start 8 1 1 Starting and stopping Plaut04 Starting The starting command for PLAUTO4 is plaut04 A short Unix command is also provided as Qpl In the Python CLUI one can start PLAUTO4 by typing plot3 p3 or com mandPlotter3D This command can have no argument one argument or two arguments If no argument is provided then the system uses the AUTO default data files fort 7 fort 8 and fort 9 as inputs If one argument is given it must be the name of the data set which we want to view This data set should be in the current directory When two arguments are given the first is always the path to the data set and the second is the data set name Note that the AUTO data set name does not mean the full name of an AUTO file It refers to the postfix of AUTO data files For example if we have the AUTO data files s H1 b H1 and d H1 the AUTO data file name is H1 79 Stopping One can exit the system by clicking the cross at the top right corner of the window or from the File menu of the system 8 1 2 Changing the Type Often one will frequently change between the solution diagram and the bifurcation diagram The Type men
11. fhn 2 sv 2 BR PT TY LAB PERIOD L2 NORM ae PAR 2 1 189 UZ 11 3 00000E 02 7 37932E 02 1 79286E 01 Next we stop using the homotopy technique and increase the period even further to 1000 r3 run r2 UZ1 c fhn 3 sv 3 BR PT TY LAB PERIOD L2 NORM gs PAR 2 1 80 UZ 14 1 00000E 03 4 04183E 02 1 79286E 01 A continuation in PAR 2 a and PAR 1 c needs to be performed to arrive at the place where we wish to find a 2 homoclinic orbit a 0 At the same time we monitor PAR 22 to locate Belyakov points r4 run r3 UZ1 c fhn 4 sv 4 BR PT TY LAB PAR 2 L2 NORM age PAR 1 PAR 22 1 6 UZ 16 1 31812E 01 3 28710E 02 2 17166E 01 6 31253E 06 1 23 UZ 20 8 55398E 08 1 56158E 02 2 74218E 01 9 88772E 02 Hence there exists a Belyakov point at a c 0 131812 0 21766 At label 19 we have a lower value of a than at the Belyakov point and by inspection of the file d 4 we can observe that the equilibrium has one positive eigenvalue and a complex conjugate pair of eigenvalues with negative real part and conclude that this orbit is of Shil nikov type Before starting the homoclinic branch switching we calculate the adjoint to obtain a Lin vector r5 run r4 UZ5 c fhn 5 sv 5 256 BR PT TY LAB PAR 9 L2 NORM ee PAR 3 1 2 EP 29 1 00000E 00 1 56158E 02 2 50000E 03 Next we continue in the time 7 PAR 21 the gap PAR 22 and c PAR 1
12. sP EP HB3 BPO UZ3 turn on the detection of folds and Hopf bifurcations turn off detection of branch points and stop at the third Hopf bifurcation or third user defined point whichever comes first 10 8 3 ISP This constant controls the detection of Hopf bifurcations branch points period doubling bifur cations and torus bifurcations ISP 0 This setting disables the detection of Hopf bifurcations branch points period doubling bifurcations and torus bifurcations and the computation of Floquet multipliers ISP 1 Branch points and Hopf bifurcations are detected for algebraic equations Branch points period doubling bifurcations and torus bifurcations are not detected for periodic solutions and boundary value problems However Floquet multipliers are computed ISP 2 This setting enables the detection of all special solutions For periodic solutions and rotations the choice ISP 2 should be used with care due to potential inaccuracy in the computation of the linearized Poincar map and possible rapid variation of the Floquet multipliers The linearized Poincar map always has a multiplier z 1 If this multiplier becomes inaccurate then the automatic detection of secondary periodic bifurcations will be discontinued and a warning message will be printed in fort 9 See also Section 11 4 ISP 3 Hopf bifurcations will not be detected Branch points will be detected and AUTO will monitor the Floquet multipliers Pe
13. xxx to list the value of the branch point function in the info file d xxx Aliases br bp commandQueryBranchPoint hopf Print the value of the Hopf function Type hopf x to list the value of the Hopf function in the diagnostics of the bifurcation diagram object x This function vanishes at a Hopf bifurcation point Type hopf to list the value of the Hopf function in the output file fort 9 Type hopf xxx to list the value of the Hopf function in the info file d xxx Aliases hb hp commandQueryHopf secondaryperiod Print value of secondary periodic bif fen Type secondaryperiod x to list the value of the secondary periodic bifurcation func tion in the diagnostics of the bifurcation diagram object x This function vanishes at period doubling and torus bifurcations Type secondaryperiod to list the value of the secondary periodic bifurcation function in the output file fort 9 Type secondaryperiod xxx to list the value of the secondary periodic bifurcation function in the info file d xxx Aliases sc sp commandQuerySecondary Period iterations Print the number of Newton interations Type iterations x to list the number of Newton iterations per continuation step in the diagnostics of the bifurcation diagram object x Type iterations to list the number of Newton iterations per continuation step in fort 9 Type iterations xxx to list the numb
14. 2 6v 15 17 si 0s Furthermore this demo continues canard orbits in parameter space Typically trajectories in slow fast systems such as this one consist of a slow part that follows the attracting slow manifold followed by a fast part when the trajectory hits a fold with respect to the fast direction After this jump the trajectory follows a slow segment again A canard orbit on the other hand does not jump at the fold but follows the repelling slow manifold A central role is played by the two dimensional critical manifold S which is given by the nullcline of the fast variable in the limit for e 0 It consists of attracting and repelling sheets S and S which generically meet at fold curves F with respect to the fast flow direction For details see Desroches Krauskopf amp Osinga 2008 For system 15 17 where we fix y 0 5 and 6 0 565 the critical manifold is given by S v h s R 2h v v 1 vs 0 15 18 which is folded with respect to the fast variable v along the folded node curve F v h s S 1 3v 0 15 19 The computation of the slow manifolds is performed in three steps They start with a constant in time solution at the folded node singularity v h s 0 49 0 6176 0 2797 for e 0 015 on the scaled time interval 0 1 with time lengths T 0 and 7 0 respectively Then for the computation of the attracting manifold v h s via bo
15. 87 Set the radius of the spheres used for labels The normal size is 1 0 For smaller radius use 0 For bigger radius use X XXX Label Sphere Radius 1 0 XXX Disk Rotation Disk Rotation 1 0 0 0 0 0 1 570796 Disk Position Disk Position 0 0 0 0 0 0 Disk Radius Disk Radius 1 0 Disk Height Disk Height 0 001 Disk Transparency 0 1 Disk Transparency 0 7 Read Disk From File Disk From File No Sphere Position Sphere Position 0 0 0 0 0 0 Sphere Radius Sphere Radius 1 0 Sphere Transparency 0 1 Sphere Transparency 0 7 Read Sphere From File Sphere From File No Axes color X Axis Color 1 0 0 0 0 0 Y Axis Color 0 0 1 0 0 0 Z Axis Color 0 0 0 0 1 0 o Color of the satellite large primary and small primary in animation satellite Color 1 0 0 0 0 0 large primary Color 0 0 1 0 0 0 large primary tail Color 0 0 1 0 1 0 small primary Color 0 0 0 0 1 0 small primary tail Color 0 5 0 5 0 0 Surface color Surface Color 0 0 1 0 0 0 Stable solution color Stable Solution Color 0 0 0 0 1 0 88 Stable solution color Unstable Solution Color 1 0 0 0 0 0 Set the radius of the satellite large primary and small primary The normal size is 1 0 For smaller radius use 0 xxx For bigger radius use X XXX Satellite Radius 1 0 Large Primary Radius 1 0 Small Primary Radius 1 0 Libration
16. Another run starting from a longer initial orbit which computes part of the manifold The free parameters are the same as in the preceding run This computation results in the detection of a connecting orbit Osv hetVib save hetV1b hetV1b Save the results in b hetV1b s hetV1b and d hetV1b Table 14 29 Detailed AUTO shell and Python commands for the V1b demo 160 Chapter 15 AUTO Demos BVP 15 1 exp Bratu s Equation This demo illustrates the computation of a solution family to the boundary value problem 15 1 with boundary conditions u 0 0 u 1 0 This equation is also considered in Doedel Keller amp Kern vez 1991a AUTO COMMAND ACTION mkdir exp create an empty work directory cd exp change directory demo exp copy the demo files to the work directory ri run e exp c exp lst run compute solution family containing fold r2 run r1 NTST 20 2nd run restart at the last labeled solution using increased accuracy save r1 r2 exp save output to b exp s exp d exp Table 15 1 Commands for running demo exp 161 15 2 int Boundary and Integral Constraints This demo illustrates the computation of a solution family to the equation ugi U2 2 Uy mie ES with a non separated boundary condition and an integral constraint T 0 24 1 0 ad pa The solution family contains a fold which in the second run is
17. Continuation of traveling wave solutions to a system of parabolic PDEs Starting data can be a Hopf bifurcation point from a previous run with IPS 11 or a traveling wave from a previous run with IPS 12 Run 3 and Run 4 of demo wav IPS 14 Time evolution for a system of parabolic PDEs subject to periodic boundary conditions Starting data may be solutions from a previous run with IPS 12 or 14 Starting data can also be specified in STPNT in which case the wave length must be specified in PAR 11 and the diffusion constants in PAR 15 16 AUTO uses PAR 14 for the time variable DS DSMIN and DSMAX govern the pseudo arclength continuation in the space time variables Note that the time discretization is only first order accurate so that results should be carefully interpreted Indeed this option is mainly intended for the detection of stationary waves Run 5 of demo wav 109 IPS 15 Optimization of periodic solutions The integrand of the objective functional must be specified in the user supplied routine FOPT Only PAR 1 9 should be used for problem parameters PAR 10 is the value of the objective functional PAR 11 the period PAR 12 the norm of the adjoint variables PAR 14 and PAR 15 are internal optimality variables PAR 21 29 and PAR 31 are used to monitor the optimality functionals asso ciated with the problem parameters and the period Computations can be started at a solution computed with IPS 2 or IPS 15 For a de
18. No Draw Background No Initialize the default graph type 0 Solution fort 8 1 Bifurcation fort 7 Graph Type 0 initialize the default graph style 0 LINES 1 TUBES 2 SURFACE 3 nurbs curve graph Style 1 set X Y Z and Label O is Time for X Y Z O is All for Label Solution X Axis 1 Solution Y Axis 2 Solution Z Axis 3 Labels 0 set the parameter indices parameter 1D 1 2 3 10 15 21 22 23 Based on the above settings the solution diagram for the CR3BP family 1 for y 0 01215 appears in Figure 90 Figure 8 11 Example 91 Chapter 9 The Graphical User Interface GUI94 9 1 General Overview The AUTO eraphical user interface GUI is a tool for creating and editing equations files and constants files see Section 3 for a description of these files The GUI can also be used to run AUTO and to manipulate and plot output files and data files see Section 5 for corresponding commands To use the GUI for a new equation change to an empty work directory For an existing equations file change to its directory Do not activate the GUI in the directory auto 07p or in any of its subdirectories Then type Qauto or its abbreviation Qa Here we assume that the AUTO aliases have been activated see Section The GUI includes a window for editing the equations file and four groups of buttons namely the Menu Bar at the top of the GUI the Define Constants buttons at
19. Zlabel_fontsize The font size for the z axis label zticks The number of ticks on the z axis 4 12 The Table 4 8 This table shows the options that can be set for the PyPLAUT plotting window and their meanings Plotting Tool PLAUTO4 The AUTO plotting tool PLAUT04 as described in Chapter 8 can be run from the Python CLUI using the command plot3 or commandPlotter3D in a similar fashion to plot It does not use the options that are used for the plotting window produced by plot 4 13 Quick Reference In this section we have created a table of all of the AUTO CLUI commands their abbreviations and a one line description of what function they perform Each command may be entered using its full name or any of its aliases Command Long name Description Aliases append ap commandAppend Append data files cat commandCat Print the contents of a file cd commandCd Change directories clean cl commandClean Clean the current directory demo dm commandCopyAndLoadDemo Copy a demo into the current direc tory and load it copy cp commandCopyDataFiles Copy data files copydemo commandCopyDemo Copy a demo into the current direc tory save sv commandCopy Fort Files Save data files gui commandCreateGUI Show AUTOs graphical user inter face delete dl commandDeleteDataFiles Delete data files df deletefort commandDeleteFortFiles Clear the current directory o
20. demo pv1 copy the demo files to the work directory ri run e pvl c pvl compute a solution family save r1 pvl save output files as b pvl s pvl d pvl Table 19 1 Commands for running demo pvl 202 19 2 ext Spurious Solutions to BVP This demo illustrates the computation of spurious solutions to the boundary value problem Uy Ug 0 uy Am sin u u u 0 t 0 1 19 2 uy 0 0 uz 1 0 Here the differential equation is discretized using a fixed uniform mesh This results in spuri ous solutions that disappear when an adaptive mesh is used See the AUTO constant IAD in Section 10 3 This example is also considered in Beyn amp Doedel 1981 and Doedel Keller amp Kern vez 19910 AUTO COMMAND ACTION mkdir ext create an empty work directory cd ext change directory demo ext copy the demo files to the work directory ri run e ext c ext detect bifurcations from the trivial solution family r2 run ri BP3 ISW 1 NCOL 3 compute a bifurcating family containing spurious bifurcations save ritr2 ext save all output to b ext s ext d ext Table 19 2 Commands for running demo ext 203 193 tim A Test Problem for Timing AUTO This demo is a boundary value problem with variable dimension NDIM It can be used to time the performance of AUTO for various choices of NDIM which must be even NTST and NCOL The equations
21. kpr 6 sv 6 increase the period save output files as b 6 s 6 d 6 Table 23 1 Detailed AUTO Commands for running demo kpr 234 L 3 5 Te Y dng a Figure 23 8 Projection onto the PAR 3 PAR 2 plane of the non central saddle node homo clinic orbit curves labeled 1 and 2 and the inclination flip curves labeled 3 and 4 AUTO COMMAND ACTION r7 run r6 UZ1 c kpr 7 sv 7 recompute the family of homoclinic orbits sv 7 save output files as b 7 s 7 d 7 r8 run r7 UZ1 c kpr 8 sv 8 continue central saddle node homoclinics save output files as b 8 s 8 d 8 r9 run r8 UZ1 c kpr 9 sv 9 continue homoclinics from codim 2 point save output files as b 9 s 9 d 9 r10 run r3 UZ1 c kpr 10 3 parameter curve of inclination flips sv 10 save output files as b 10 s 10 d 10 rii run r3 UZ2 c kpr 11 another curve of inclination flips sv 11 save output files as b 11 s 11 d 11 ri2 run r7 UZ1 c kpr 12 continue non central saddle node homoclinics sv 12 save output files as b 12 s 12 d 12 r12 r12 run r8 UZ1 c kpr 13 continue non central saddle node homoclinics ap 12 append output files to b 12 s 12 d 12 Table 23 2 Detailed AUTO Commands for running demo kpr
22. ues in the ratio of the masses until a spec ified mass ratio is reached It then com putes a family of periodic orbits for each pair of purely complex eigenvalues n body to matlab xauto A script which takes a set of AUTO data files and creates a set of files formatted for importing into Matlab for either plotting or further calculations Table 4 1 The various demonstration scripts for the AUTO CLUI 27 Unix COMMAND ACTION auto start the AUTO CLUI AUTO CLUI COMMAND ACTION demo ab copy the demo files to the work directory load the filename ab f90 into the variable ab load the contents of the file c ab 1 into the variable ab run AUTO with the current set of files ab load equation ab ab load ab constants ab 1 run ab Table 4 2 Running the demo ab files gt auto Python 2 5 2 r252 60911 Nov 14 2008 19 46 32 GCC 4 3 2 on linux2 Type help copyright credits or license for more information AUTOInteractiveConsole AUTO gt demo ab Copying demo ab Runner configured AUTO gt ab load equation ab Runner configured AUTO gt ab load ab constants ab 1 Runner configured AUTO gt run ab gfortran fopenmp 0 c ab f90 o ab o gfortran fopenmp 0 ab o o ab exe home bart auto 07p lib o done Starting ab BR PT TY LAB PAR 2 L2 NORM U 1 U 2 1 1 EP 1 8 00000E 00 0 00000E 00 0 00000E 00 0
23. y 1 0 6 The value of e is 0 5 in the first run In the second run continuation is used to decrease the value of e to 0 1 or if desired to a smaller value as already mentioned above The norm of the endpoint 2 1 y 1 is fixed in the second run while T is variable In the third run the norm of the endpoint remains fixed However the initial point x 0 y 0 is allowed to move around the small circle of radius r around the orgin The endpoint thereby moves around the large circle of radius 0 6 The integration time T remains variable The orbits computed in this run generate the local manifold When viewing the orbits computed in the third run as written in the file s 3 notice that orbits near the weakly unstable direction which here corresponds to the x direction have been well computed Such orbits are sensitively dependent on the initial condition 0 y 0 when the problem is considered as an initial value problem It is in fact the continuation of the entire orbits using a boundary value approach which enables their determination As already mentioned this feature is even more visible when running this demo with a smaller value of e 15 13 169 15 10 um3 A 2D unstable manifold in 3D This demo uses orbit continuation to compute part of the 2D unstable manifold of the origin of the equations eg e y re y yt 15 14 z 2 e The origin has unstable eigenvalues e and 1 where e gt 0 s
24. 0 0 ui 1 0 As in demo exp a solution curve requires one free parameter here p Note that additional parameters are specified in the user supplied subroutine PVLS in file pvl f90 namely pa the L2 norm of u1 p3 the minimum of uz on the space interval 0 1 pa the boundary value uz 0 ps the same boundary value u2 0 These additional parameters should be considered as solution measures for output purposes they should not be treated as true continuation parameters Note also that four free parameters are specified in the AUTO constants file c pvl namely P1 P2 P3 pa and ps The first one in this list p is the true continuation parameter The parameters po P3 pa and ps are overspecified so that their values will appear in the output However it is essential that the true continuation parameter appear first For example it would be an error to specify the parameters in the following order po P1 P3 Pa Ps In general true continuation parameters must appear first in the parameter specification in the AUTO constants file Overspecified parameters will be printed and can be defined in PVLS but they are not part of the intrinsic continuation procedure As this demo also illustrates see the UZR values in c pvl labeled solutions can also be output at selected values of the overspecified parameters AUTO COMMAND ACTION mkdir pvl create an empty work directory cd pvl change directory
25. A u1 d dz which changes the above equation into uy 2 A u1 ue un 2 cuz B ur uz C u 18 6 Sharp traveling waves then correspond to heteroclinic connections in this transformed system 199 Finally we map 0 7 0 1 by the transformation z T With this scaling of the independent variable the reduced system becomes u TA u1 uz us E T cuz B ur uz C u1 18 7 For Case 1 this equation has a known exact solution namely 4 Mr OT Ir This solution has wave speed c 1 In the limit as T oo its phase plane trajectory connects the stationary points 1 0 and 0 5 The sharp traveling wave in Case 2 can now be obtained using the following homotopy Let a1 a2 bo b1 b2 1 A 2 0 2 0 0 A 2 1 0 1 0 Then as varies continuously from 0 to 1 the parameters a1 a2 bo b1 b2 vary continously from the values for Case 1 to the values for Case 2 AUTO COMMAND ACTION mkdir stw create an empty work directory cd stw change directory demo stw copy the demo files to the work directory ri run e stw c stw continuation of the sharp traveling wave save r1 stw save output files as b stw s stw d stw Table 18 4 Commands for running demo stw 200 Chapter 19 AUTO Demos Miscellaneous 201 19 1 pvl Use of the Routine PVLS Consider Bratu s equation uy U2 al 19 1 with boundary conditions u
26. LTR 23 83 Flip torus bifurcation maps PTR 77 87 Torus torus bifurcation maps TTR 88 Table 4 6 This table shows the various types of codimension two points as in Table 4 5 The ab solute value of the number divided by 10 gives the type of the branch on which the codimension two bifurcation occurs number These keys are also useful to change bifurcation diagram files as is illustrated in Figure 4 18 Note that you can interactively change branch numbers solution numbers and delete branches and solutions using the 1b command see Chapter 5 4 7 1 Solutions You can also obtain solutions from a bifurcation diagram structure by using the syntax directly on the diagram instead of on individual branches For example in the above example the command s data returns an object which encapsulates all solutions in a easy to use form The object returned in this way is a list of all of the solutions in the appropriate bifurcation solution file and each solution is a Python dictionary with entries for each piece of data for the solution For example the sequence of commands in Figure 4 19 prints out the label of the first solution in a bifurcation solution The query able parts of the object are listed in Table The individual elements of the list may again be accessed in two ways either by the index of the solution using the syntax or by label number or type name using the syntax For example assume that the pa
27. Note that these solutions were obtained by choosing a smaller step DS and more output smaller NPR in c kpr 4 A blow up of the region close to the origin of this figure is shown in Figure 23 4 It illustrates the flip of the solutions of the adjoint equation while moving through the bifurcation point Note that the data in this figure were plotted after first performing an additional continuation of the solutions with respect to PAR 11 Continuing in the other direction r3 r3 run r3 0 c kpr 5 ap 3 we approach a Bogdanov Takens point BR PT TY LAB PAR 1 a PAR 10 os PAR 33 1 50 EP 10 1 93828E 00 7 52334E 00 3 19793E 01 Note that the numerical approximation has ceased to become reliable since PAR 10 has now become large Phase portraits of homoclinic orbits between the BT point and the first inclination flip are depicted in Figure 23 5 Note how the computed homoclinic orbits approaching the BT point have their endpoints well away from the equilibrium To follow the homoclinic orbit to the BT point with more precision we would need to first perform continuation in T PAR 11 to obtain a more accurate homoclinic solution 23 3 More Accuracy and Saddle Node Homoclinic Or bits Continuation in T in order to obtain an approximation of the homoclinic orbit over a longer interval is necessary for parameter values near a non hyperbolic equilibrium either a saddle 230 node or BT where the convergence to the
28. OR flq 2 ext flgq run sext c flq 2 e flq Continue the approximate multiplier If all goes well then the actual multiplier will be detected as a branch point BP with Label 2 Free scalar variables in this run are PAR 1 unfolding parameter PAR 4 multiplier PAR 5 norm of eigenfunction sv flq save flq flq OR flq 3 flq run flq e flq c 1q 3 Switch branches at the BP thereby generating the nonzero Floquet eigenfunction The free scalar variables are PAR 1 unfolding parameter PAR 4 multiplier PAR 5 norm of eigenfunction If all goes well then PAR 5 should become nonzero and the corresponding solution should have Label 4 sv flq save flq flq Table 14 24 Detailed AUTO shell and Python commands for Remark 1 157 OR flq 4 flq run flq e flq c flq 4 Free scalar variables in this run are PAR 1 unfolding parameter PAR 4 multiplier PAR 11 period The norm PAR 5 of the eigenfunction is fixed in this run sv flq save flq flq Table 14 25 Detailed AUTO shell and Python commands for Remark 2 OR man Hia 1 startHla Hla run startHla e man c man Hla 1 Look at c man Hla 1 to see from which label in s startHla this run starts In this run the x coordinate of the end point PAR 21 is kept fixed while the period PAR 11 i e the total integration time is allowed to vary as is the va
29. The number of mesh intervals used to compute the solu NTST a tion See Section 10 3 1 Parameters The value of all of the parameters for the solution parameters p and parameters are aliases p Other syntax s PAR 1 s PAR 1 7 PT The number of the point in the given branch TY A short string which describes the type of the solution see Table 4 5 TY number A number which describes the type of the solution see Table 4 5 Active ICP The values of the one based indices of the free param eters rldot The values of the parameter direction vector This table shows the strings that can be used to query a solution object and their 44 the list and provide new starting solutions Solutions can be deleted using the commands dlb dsp klb and ksp described in the reference Section The command relabel as described before gives each solution a unique label starting at 1 Finally you can change individual solution labels using the relabel method data relabel 9 57 relabels label 9 to label 57 making sure that the label changes both in the bifurcation diagram and in the solution 4 7 2 Summary and reference We have defined the following objects Bifurcation diagram object bd run bd loadbd Branch bd 0 bd 1 bd 2 Branch AUTO constants bd 0 c bd 1 c bd 2 c bd c refers to bd 0 c Branch column bd PAR 1 bd 0 PAR 1 ba 1 L2 NORM Here bd
30. UZSTOP see Section 10 9 12 101 By specifying the maximum number of steps NMX By specifying a negative parameter index in the list associated with the constant UZR see Section 0 9 11 By appropriate choice of the computational window defined by the constants RLO RL1 AO and A1 One should always check that the starting solution lies within this computational window otherwise the computation will stop immediately at the starting point Most demos do not specify these constants and use an unbounded window 10 6 1 STOP This constant adds stopping conditions It is specified as a list of bifurcation type strings fol lowed by a number n greater than zero These strings specify that the contination should stop as soon as the nth bifurcation of the associated type has been reached Example STOP HB3 UZ3 Stop at the third Hopf bifurcation or third user defined point see Sec tion 10 9 11 whichever comes first 10 6 2 NMX The maximum number of steps to be taken along any family 10 6 3 RLO The lower bound on the principal continuation parameter This is the parameter which appears first in the ICP list see Section 10 7 1 10 6 4 RL1 The upper bound on the principal continuation parameter 10 6 5 AO The lower bound on the principal solution measure By default if IPLT 0 the principal solution measure is the L norm of the state vector or state vector function See the AUTO constant IPLT in
31. ab Figure 4 6 The commands from Figure 4 2Jand they would appear in a AUTO CLUI script file The source for this script can be found in AUTO_DIR demos python demol auto 4 5 Second Example In Section we showed a very simple AUTO CLUI script in this Section we will describe a more complex example which introduces several new AUTO CLUI commands as well as a 30 gt cat demol auto demo ab ab load equation ab ab load ab constants ab 1 run ab gt auto demol auto Copying demo ab done Runner configured Runner configured Runner configured gfortran fopenmp 0 c ab f90 o ab o gfortran fopenmp 0 ab o o ab exe home bart auto 07p lib o Starting ab BR PT TY LAB PAR 2 L2 NORM U 1 U 2 1 1 EP 1 8 00000E 00 0 00000E 00 0 00000E 00 0 00000E 00 1 31 UZ 2 1 40000E 01 0 00000E 00 0 00000E 00 0 00000E 00 1 36 UZ 3 1 50000E 01 0 00000E 00 0 00000E 00 0 00000E 00 1 41 UZ 4 1 60000E 01 0 00000E 00 0 00000E 00 0 00000E 00 1 46 UZ 5 1 70000E 01 0 00000E 00 0 00000E 00 0 00000E 00 1 51 EP 6 1 80000E 01 0 00000E 00 0 00000E 00 0 00000E 00 Total Time 0 193E 01 ab done gt Figure 4 7 This Figure starts by listing the contents of the demol auto file using the Unix cat command The file is then run through the AUTO CLUI by typing auto demol auto and the output is shown basic Python construct for looping We will not provide an exhaustive reference for the Python
32. and should generally be used only to locate stationary states AUTO COMMAND ACTION mkdir pd2 create an empty work directory cd pd2 change directory demo pd2 copy the demo files to the work directory ri run e pd2 c pd2 time integration towards stationary state save r1 1 save output files as b 1 s 1 d 1 r2 run r1 IPS 17 ICP 1 ISP 2 NMX 15 continuation of stationary states read NPR 50 DS 0 1 DSMAX 1 0 UZR 1 0 0 restart data from s 1 save r2 2 save output files as b 2 s 2 d 2 Table 16 2 Commands for running demo pd2 183 16 3 wav Periodic Waves This demo illustrates the computation of various periodic wave solutions to a system of coupled parabolic partial differential equations on the spatial interval 0 1 The equations that model an enzyme catalyzed reaction Doedel amp Kern vez 1986b are du 0t Ou 0x p paR u1 u2 pa u1 Ou2 Ot P0 u2 0x p paR us u2 pr p3 ua 16 2 All equation parameters except p3 are fixed throughout AUTO COMMAND ACTION mkdir wav create an empty work directory cd wav change directory demo wav copy the demo files to the work directory ri run e wav c wav 1st run stationary solutions of the system without diffusion save r1 ode save output files as b ode s ode d ode r2 run e wav c wav IPS 11 2nd run dete
33. autorc and autorc as explained in Section can be used to customize Py PLAUT s behaviour and appearance 7 1 Basic PLAUT Commands The principal PLAUT commands are bdo This command is useful for an initial overview of the bifurcation diagram as stored in fort 7 If you have not previously selected one of the default options d0 d1 d2 d3 or d4 described below then you will be asked whether you want solution labels grid lines titles or labeled axes bd This command is the same as the bd0 command except that you will be asked to enter the minimum and the maximum of the horizontal and vertical axes This is useful for blowing up portions of a previously displayed bifurcation diagram ax With the az command you can select any pair of columns of real numbers from fort 7 as horizontal and vertical axis in the bifurcation diagram The default is columns 1 and 2 To determine what these columns represent one can look at the screen ouput of the corresponding AUTO run or one can inspect the column headings in fort 7 2d Upon entering the 2d command the labels of all solutions stored in fort 8 will be listed and you can select one or more of these for display The number of solution components is 76 also listed and you will be prompted to select two of these as horizontal and vertical axis in the display Note that the first component is typically the independent time or space variable scaled to the interval 0 1 sav To
34. branch point function in the data file d xxx This function vanishes at a branch point Type Chb to list the value of the Hopf function in the output file fort 9 This function vanishes at a Hopf bifurcation point Type hb xxx to list the value of the Hopf function in the data file d xxx This function vanishes at a Hopf bifurcation point The command Cho is an alias to hb above Type sp to list the value of the secondary periodic bifurcation function in the output file fort 9 This function vanishes at period doubling and torus bifurcations Type sp xxx to list the value of the secondary periodic bifurcation function in the data file d xxx This function vanishes at period doubling and torus bifurcations Type Cit to list the number of Newton iterations per continuation step in fort 9 Type it xxx to list the number of Newton iterations per continuation step in d xxx Type st to list the number of stable eigenvalues or stable Floquet multipliers per continuation step in fort 9 Type st to list the continuation step size for each continuation step in fort 9 Type st xxx to list the continuation step size for each continuation step in d xxx Type Cev to list the eigenvalues of the Jacobian in fort 9 Algebraic problems Type ev xxx to list the eigenvalues of the Jacobian in d xxx Algebraic problems Type f1 to list the Floquet multipliers in the output file fort 9 Differential e
35. c mth 2 ap 1 one obtains BR PT TY LAB PAR 1 a8 PAR 2 PAR 35 PAR 36 1 34 UZ 10 5 18039E 00 6 38554E 02 8 86654E 10 7 28132E 02 which means another non central saddle node homoclinic bifurcation occurs at D K Z 5 1803 0 063855 Note that these data were obtained using a smaller value of NTST than the original computation compare c mtn 1 with c mtn 2 The high original value of NTST was only necessary for the first few steps because the original solution is specified on a uniform mesh 221 22 3 Switching between Saddle Node and Saddle Homo clinic Orbits Now we can switch to continuation of saddle homoclinic orbits at the located codim 2 points D and Dao ri ri run ri UZ1 c mtn 3 ap 1 starts from D Note that now IEQUIB 1 has been specified in c mtn 3 Also test functions Wo and 419 have been activated in order to monitor for non hyperbolic equilibria along the homoclinic locus We get the following output BR PT TY LAB PAR 1 et PAR 2 PAR 29 PAR 30 1 10 12 7 11454E 00 7 08176E 02 4 64986E 01 3 18355E 03 1 20 13 9 17683E 00 7 67874E 02 4 68491E 01 1 60931E 02 1 30 14 1 21084E 01 8 54348E 02 4 71887E 01 3 06966E 02 1 40 EP 15 1 50379E 01 9 42805E 02 4 74379E 01 4 14457E 02 The fact that PAR 29 and PAR 30 do not change sign indicates that there are no further non hyperbolic equilibria along this family Note that restarting in the o
36. focus with a one dimensional unstable manifold 236 We now restart at LAB 3 corresponding to a time interval 27 200 and change the prin cipal continuation parameters to be v 8 The new constants defining the continuation are given in c cir 2 We also activate the test functions pertinent to codimension two singularities which may be encountered along a family of saddle focus homoclinic orbits viz Ya Ya Us Wg and wo This must be specified in three ways by appropriate IPSI in c cir 2 by adding the corresponding parameter labels to the list of continuation parameters ICP I in c cir 2 recall that these parameter indices are 20 more than the corresponding w indices and finally adding UZR functions defining zeros of these parameters in c cir 2 Running r2 run ri UZ2 c cir 2 sv 2 results in BR PT TY LAB PAR 1 PAR 2 PAR 25 PAR 29 1 17 UZ 5 7 256925E 01 4 535645E 01 1 765251E 05 2 888436E 01 1 75 UZ 6 1 014704E 00 9 998966E 03 1 664509E 00 5 035997E 03 1 78 UZ 7 1 026445E 00 2 330391E 05 1 710804E 00 1 165176E 05 1 81 UZ 8 1 038012E 00 1 000144E 02 1 756690E 00 4 964621E 03 1 100 EP 9 1 164160E 00 1 087732E 01 2 230329E 00 5 042736E 02 with results saved in b 2 s 2 d 2 Upon inspection of the output note that label 5 where PAR 25 0 corresponds to a neutrally divergent saddle focus 5 0 Label 7 where PAR 29 0 corresponds to a local bifurcation
37. language but only the very basics For more extensive documentation we refer the reader to Lutz 1996 or the web page http www python org In this section we will describe each line of the script in detail and the full text of the script is in Figure The script begins with a section extracted into Figure which performs a task identical to that shown in Figure except that the shorthand discussed in Section is used for the run command Up to this point all of the commands presented have had analogs in the command language discussed in Section 5 and the AUTO CLUI has been designed in this way to make it easy for users to migrate from the old command language to the AUTO CLUI The next section of the script extracted into Figure introduces a new command namely branchpoints bvp BP which is the first command which has no analog in the old command language The command bvp BP given the output variable bvp from the first run returns a Python object 31 demo bvp bvp run bvp branchpoints bvp BP for solution in branchpoints bp load solution ISW 1 NTST 50 Compute forwards print Solution label bp LAB forwards fw run bp Compute backwards print Solution label bp LAB backwards bw run bp DS both fw bw merged merge both bvp bvp merged bvp relabel bvp save bvp bvp plot bvp wait Figure 4 8 This Figure shows a more complex AUTO CLU
38. line def myRun demoname begins the function definition and creates a function named myRun which takes one argument demoname The rest of the script is the same except that it has been indented to indicate that it is part of the function definition all occurrences of string bvp have been replaced with the variable demoname and the variable bvp was replaced by the variable r Finally we have added a line myRun bvp which actually calls the function we have created and runs the same computation as the original script from auto import def myRun demoname demo demoname r run demoname branchpoints r BP for solution in branchpoints bp load solution ISW 1 NTST 50 Compute forwards print Solution label bp LAB forwards fw run bp Compute backwards print Solution label bp LAB backwards bw run bp DS both fw bw merged merge both r r merged r relabel r save r demoname plot r wait myRun bvp Figure 4 14 This Figure shows a complex AUTO CLUI script written as a function The source for this script can be found in AUTO_DIR demos python userScript xauto 36 While the script in Figure 4 14 is only slightly different then the one showed in Figure 4 8 it is much more powerful Not only can it be used as a script for running any demo by modifying the last line it can be read back into the interactive mode of the AUTO CLUI and used
39. merge relabel save and plot commands have the shorter forms 1d r mb rl sv and pl respectively All algebraic and functional expressions can be combined in the usual way Combining these techniques a shorter version of the complex AUTO CLUI script is given in Figure 4 12 Even shorter forms are possible to save you typing using features borrowed from ipython both in auto i and in plain auto Those forms use auto parentheses and auto quotes for a command that does not assign to a variable you can skip the parentheses and by using a on the first character of a line you force all parameters to be quoted For example you can just type pl bvp to plot the bifurcation diagram and solutions in the object bvp Beware that these extra short forms are only possible in normal auto scripts and at the AUTO CLUI prompt and only if they start in the first column but not in the expert scripts described in the next section The even shorter version of the complex AUTO CLUI script is given in Figure 4 13 It should be clear that these super short forms save you typing at the command prompt but do not help readability in scripts 4 6 Extending the AUTO CLUI The code in Figure 4 8 performed a very useful and common procedure it started an AUTO calculation and performed additional continuations at every point which AUTO detected as a bifurcation Unfortunately the script as written can only be used for the bvp demo In this 34 dem
40. pylab plot so1 U 1 so01 U 2 gt gt lt matplotlib lines Line2D object at 0x9c3348c gt AUTO gt pylab show Figure 4 19 This figure shows an example of parsing solutions The first command s data extracts a list of all solutions stored in the bifurcation diagram object data from the lrz demo in Figure 4 16Jand puts it into the variable s The command sol s 3 obtains the fourth solution Next print sol displays it The last commands illustrate how to extract components from this fourth solution its label its first point its time array its first coordinate array its first parameter and a plot of the first two coordinates 43 Table 4 7 meanings Query string Meaning An array which contains the AUTO output Each array entry is a Python dictionary with a scalar entry t denot data ing time and sub arrays for the solution vector u and the solution direction vector u dot Other syntax s 0 s t s UC 1 BR The number of the branch to which the solution belongs IPRIV A private field for use by toolboxes IPS The user specified problem type See Section 10 8 12 ISW The ISW value used to start the calculation See Sec tion LAB The label of the solution The number of collocation points used to compute the NCOL solution See Section 10 3 2 NDIM The user specified number of dimensions See Sec tion 10 2 1
41. s2 L2 NORM lt 1le 4 where all solutions are compared with each other and s2 is deleted if the given condition is satisfied which causes pruning of solutions that are close to each other Type information is not kept in the bifurcation diagram Alias commandDeleteSpecialPoints 61 ksp Keep special points Type ksp x list to only keep the special points in list in the Python object x which must be a solution list or a bifurcation diagram Type ksp list xxx to keep them in the data files b xxx and s xxx Type ksp list xxx yyy to save to b yyy and s yyy instead of xxx Type ksp list to keep them in fort 7 and fort 8 list is a label number or type name code or a list of those such as 1 or 2 3 or UZ or BP LP or it can be None or omitted to mean BP LP HB PD TR EP MX deleting UZ and regular points Alternatively a boolean user defined function f that takes a solution can be specified for list such as def f s return s PAR 9 lt 0 where only solutions are kept that satisfy the given condition Type information is not kept in the bifurcation diagram Alias commandKeepSpecialPoints merge Merge branches in data files Type y merge x to return the python object x with its branches merged into continuous curves as a new object y Type merge xxx to merge branches in s xxx b xxx and d xxx Backups of the original files a
42. the equilibrium Then follow N 1 shifted copies of the orbit which travel from the point Xo back to the point zo The last part Uy goes from the point zo back to the equilibrium The derivatives to with respect to time of the point that is furthest from the equilibrium are stored at the parameters PAR NP NDIM 1 NP where NP max NPARX NPAR If ITWIST 1 and this was also the case in the preceding run then a copy of the adjoint vec tor Y at xy is stored at the parameters PAR NP NDIM 2 1 NP NDIM and Lin s method can be used to do homoclinic branch switching To be more precise the individual parts ui and u are at distances away from each other along the Lin vector Psi at the 207 left and right hand end points These gaps e are at parameters PAR 20 2 i Maore over each part except uy 1 ends at at a Poincar section which goes through zo and is perpendicular to Zo The times T that each part u takes are stored as follows T PAR 10 Ty PAR 11 and T PAR 19 2 i fori 1 N 1 Through a continuation in problem parameters gaps e and times T it is possible to switch from a 1 homoclinic to an N homoclinic orbit If ITWIST 0 the adjoint vector is not computed and Lin s method is not used Instead AUTO produces a gap PAR 22 at the right hand end point p of uyy measuring the distance between the stable manifold of the equilibrium and p This technique can also be used to find 2 homoclinic orbits b
43. then you will find seven icons that allow you to use various zoom functions a home button to go back to the original plot back and forward buttons to go back and forwards between zooms a button to select pan zoom mode a button to select rectangular zoom mode a button to which brings up sliders that adjust margins and a floppy disk button that you can use to save the plot to a file In zoom to rect mode the left mouse button may be held down to create a box in the plot When the left button is released the plot will zoom to the selected portion of the diagram Similarly the right mouse button can be used to zoom out In pan zoom mode dragging with the left mouse button pressed pans shifts the graph whereas dragging with the right mouse button zooms in and out If you are not using matplotlib then pressing the right mouse button in the plotting window brings up a menu of buttons which control several aspects of the plotting window The top two toggle buttons control what function the left button performs The print value button causes the left button to print out the numerical value underneath the pointer when it is clicked When zoom button is checked the left mouse button may be held down to create a box in the plot When the left button is released the plot will zoom to the selected portion of the diagram The unzoom button returns the diagram to the default zoom The Postscript button allows the user to save the plot as a Postscri
44. this example it is harmless since the test functions are irrelevant for heteroclinic continuation 240 Alternatively for this problem there exists an analytic expression for the two equilibria This is specified in the subroutine PVLS of she f90 Re running with IEQUIB 1 we obtain the output r2 run she c she 2 1 1 EP 1 5 00000E 01 4 05950E 01 1 63875E 01 1 2 4 43202E 01 3 65772E 01 1 31056E 01 1 10 3 3 72309E 01 3 14244E 01 9 30098E 02 1 15 4 3 00884E 01 2 61156E 01 5 93397E 02 1 20 5 2 28665E 01 2 06219E 01 3 17994E 02 1 25 6 1 55541E 01 1 49165E 01 1 23990E 02 1 30 EP 7 8 10746E 02 9 14311E 02 2 38662E 03 This output is similar to that above but note that it is obtained slightly more efficiently because the extra parameters PAR 12 21 representing the coordinates of the equilibria are no longer part of the continuation problem Also note that AUTO has chosen to take slightly larger steps along the family Finally we can continue in the opposite direction along the family from the original starting point again with IEQUIB 1 r3 run r2 2 c she 3 save r2 r3 2 BR PT TY LAB PAR 3 L2 NORM PAR 1 1 5 8 4 99759E 01 4 06015E 01 1 63732E 01 1 10 9 5 70530E 01 4 55187E 01 2 06526E 01 1 15 10 6 41644E 01 5 03184E 01 2 50783E 01 1 20 11 7 13330E 01 5 50067E O1 2 95934E 01 1 25 12 7 85769E 01 5 95871E 01 3 41549E 01 1 30 13 8 59097E 01 6 40618E 01
45. to deal with problem is that the user need only specify f D and c Another advantage is the compatibility of output data for restart purposes This allows switching back and forth between evolution calculations and wave computations 2 5 Discretization AUTO discretizes ODE boundary value problems which includes periodic solutions by the method of orthogonal collocation using piecewise polynomials with 2 7 collocation points per mesh interval de Boor amp Swartz 1973 The mesh automatically adapts to the solution to equidistribute the local discretization error Russell amp Christiansen 1978 The number of mesh intervals and the number of collocation points remain constant during any given run although they may be changed at restart points The implementation is AUTO specific In particular the choice of local polynomial basis and the algorithm for solving the linearized collocation systems were specifically designed for use in numerical bifurcation analysis 21 Chapter 3 User Supplied Files The user must prepare the two files described below This can be done with the GUI described in Chapter 9 or independently 3 1 The Equations File xxx f90 or xxx f Or xxx c A source file xxx f90 containing the Fortran routines FUNC STPNT BCND ICND FOPT and PVLS Here xxx stands for a user selected name If any of these routines is irrelevant to the problem then its body need not be completed Examples are in auto 07p demos
46. 0 dmg and http ftp coin3d org coin src all SoQt 1 5 0 tar gz Try to make sure that the native Aqua Qt is used by setting QTDIR if you also have fink installed e PLAUT In pre Leopard OS X it appears that you do not see fonts To solve this issue you need to obtain a different version of xterm see http sourceforge net project showfiles php group_id 21781 e GUI94 Perhaps possible using Fink but not attempted e manual BTFX and transfig comes with xfig Notes for 64 bit Snow Leopard to be able to compile PLAUT04 e To compile and install SoQt after installing Qt and Coin3D run configure CFLAGS m32 CXXFLAGS m32 LDFLAGS m32 FFLAGS m32 make sudo make install e Then in the AUTO 07p folder configure and compile AUTO as described above 1 1 3 Installation on Windows A native light weight solution for running AUTO on Windows is to use GFortran MSYS 1 0 11 or higher see http www mingw org combined with a native Win32 version of Python obtained at http www python org To install this setup e Install Python as of this writing preferably version 2 7 not 3 2 from www python org NumPy from numpy scipy org and Matplotlib from matplotlib sf net which all come with installers e Install MinGW make sure to include msys base gcc and fortran using the MinGW Graphical Installer at http www mingw org 14 e Start MSYS using the Start menu Start gt All Programs gt MinGW
47. 00000E 00 1 31 UZ 2 1 40000E 01 0 00000E 00 0 00000E 00 0 00000E 00 1 36 UZ 3 1 50000E 01 0 00000E 00 0 00000E 00 0 00000E 00 1 41 UZ 4 1 60000E 01 0 00000E 00 0 00000E 00 0 00000E 00 1 46 UZ 5 1 70000E 01 0 00000E 00 0 00000E 00 0 00000E 00 1 51 UZ 6 1 80000E 01 0 00000E 00 0 00000E 00 0 00000E 00 Total Time 0 181E 01 ab done lt _ bifDiag instance at 0x0972198c gt AUTO gt Figure 4 2 Typing auto at the Unix shell prompt starts the AUTO CLUI The rest of the commands are interpreted by the AUTO CLUI 28 by AUTO unless it is loaded in again after the changes are made Finally run ab Section in the reference uses the user defined functions loaded by the load equation ab command and the AUTO constants loaded by the load ab constants ab 1 to run AUTO The run command returns a bifurcation diagram structure It can be referenced using the special _ variable in interactive sessions or assigned as result run ab The result can then be referred to in further calculations plotted and saved Figure showed two of the file types that the load command can read into memoty namely the user defined function file and the AUTO constants file Section 3 There are two other files types that can be read in using the load command and they are the restart solution file Section 6 and the HomCont parameter file Section 20 2 The load command can also directly load AUTO constants Note that the name given to th
48. 06 PAR 24 6 37184E 02 3 62449E 02 The output is appended to the Python variable r6 Note that this continuation is very similar to the one where we found a 2 homoclinic orbit In fact we have now found a 2 homoclinic orbit numerically followed by a broken 1 homoclinic orbit only the mesh is not aligned The next step is to close the gap corresponding to ez to obtain a 3 homoclinic orbit We replace the continuation parameter e by Tz because T PAR 23 still has to be decreased from its high value 35 and e needs to stay at 0 r12 run r11 ICP 4 5 23 24 NMX 32 NTST 40 DS 1 DSMAX 1 UZR 24 0 4 0 18 r6 r6 r12 BR PT TY LAB PAR 4 ow PAR 5 PAR 23 PAR 24 3 16 UZ 32 1 98395E 01 6 05536E 03 2 01311E 01 1 82491E 08 3 24 UZ 33 1 80000E 01 6 50293E 03 1 27554E 01 3 14294E 02 3 30 UZ 34 1 66990E 01 6 89269E 03 9 41745E 00 1 03179E 06 3 32 EP 35 1 78172E 01 6 55364E 03 9 50300E 00 7 20367E 02 The output is appended to the Python variable r6 Note that we have found two zeros of PAR 24 at labels 32 and 34 respectively The two zeros correspond to two different 3 homoclinic orbits which when followed from periodic orbits both emanate from from the same saddle node bifurcation These two 3 homoclinic orbits are depicted in Figure 27 3 b We can follow both of these back to the inclination flip point by setting ITWIST back to 0 ri3 run r6 UZ7 ICP 4 5 NMX 30 DS 0 01 DSMAX 0 1 UZR 4 0
49. 11 specified as free parameters Note that one must set ISW 2 for computing such loci of special solutions Also note that in the continuation of folds the principal continuation parameter must be the one with respect to which the fold was located Actually one may only specify the problem parameters as AUTO will automatically add the period For example in Run 3 of demo plp where a locus of folds is computed for periodic orbits we have ICP 4 1 with PAR 4 and PAR 1 specified as free parameters However in this case the period will not appear in the screen output and in the fort 7 output file To continue a locus of folds period doubling or torus bifurcations with fixed period simply specify three problem parameters not including PAR 11 For torus bifurcations it is also possible to specify four problem parameters possibly including PAR 11 In that case the angle of the torus PAR 12 stays fixed 10 7 6 Boundary value problems The simplest case is that of boundary value problems where NDIM NBC and where NINT 0 Then generically one free problem parameter is required for computing a solution family For example in demo exp we have NDIM NBC 2 NINT 0 Thus in this demo one free parameter is designated namely PAR 1 More generally for boundary value problems with integral constraints the generic number of free parameters is NBC NINT NDIM 1 For example in demo lin we have NDIM 2 NBC 2 and NINT 1 Thus ICP 1 3 I
50. 13 ISTART 1 ITWIST 0 r6 r6 r13 BR PT TY LAB PAR 4 L2 NORM ee PAR 5 3 13 UZ 36 1 29999E 01 5 04807E 01 2 33902E 03 3 30 EP 37 9 27258E 02 5 06560E 01 2 76788E 04 r14 run r6 UZ9 ICP 4 5 NMX 30 DS 0 01 DSMAX 0 1 UZR 4 0 145 ISTART 1 ITWIST 0 r6 r6 r14 save r6 6 BR PT TY LAB PAR 4 L2 NORM oes PAR 5 3 7 UZ 38 1 45000E 01 5 47347E 01 4 79400E 03 3 30 EP 39 8 39399E 02 5 52605E 01 7 36611E 05 All the combined appended output is saved to b 6 s 6 and d 6 The bifurcation diagram and the paths we followed when closing the Lin gaps are depicted in Figure It is possible and straightforward to obtain 4 5 6 homoclinic orbits by extending the above strategy 254 Figure 27 4 Parameter space diagram near an inclination flip The curve through label 17 corresponds to a 1 homoclinic orbit The opening of the Lin gaps occurs along the vertical line from label 16 to label 23 The curves through labels 23 and 30 denote the path that is followed when closing the Lin gaps The approximately overlaid curves though labels 25 and 35 correspond to the 2 and one of the 3 homoclinic orbits Finally the curve through label 37 corresponds to the other 3 homoclinic orbit which was obtained for PAR 23 7 gt 12 03201 27 2 Branch switching for a Shil nikov type homoclinic orbit in the FitzHugh Nagumo equations The FitzHugh Nagumo FHN equations FitzHugh 1961 Nagumo Arimoto amp Yosh
51. 1995 X A XY X RX 1 2 DoK l Ai XY As ZY i Y u DY Bi X B3 Y The values of all parameters except K Z are set as follows R 0 5 Ay 0 4 B 0 6 Do 0 01 E 0 6 A 1 0 Bo 0 5 D 0 15 The parametric portrait of the system on the Z K plane is presented in Figure 22 1 It contains fold t12 and Hopf H bifurcation curves as well as a homoclinic bifurcation curve P The fold curves meet at a cusp singular point C while the Hopf and the homoclinic curves originate at a Bogdanov Takens point BT Only the homoclinic curve P will be considered here the other bifurcation curves can be computed using AUTO or for example locbif Khibnik Kuznetsov Levitin amp Nikolaev 1993 22 2 Continuation of Central Saddle Node Homoclinics Local bifurcation analysis shows that at K 6 0 Z 0 06729762 the system has a saddle node equilibrium X Y 5 738626 0 5108401 with one zero and one negative eigenvalue Direct simulations reveal a homoclinic orbit to this saddle node departing and returning along its central direction i e tangent to the null vector Starting from this solution stored in the file mtn dat we continue the saddle node central homoclinic orbit with respect to the parameters K and Z by copying the demo and running it dm mtn ri run mtn c mtn 1 sv 1 220 The file mtn f90 contains approximate parameter values K PAR 1 6 0 Z PAR
52. 2 0 06729762 as well as the coordinates of the saddle node X PAR 12 5 738626 Y PAR 13 0 5108401 and the length of the truncated time interval To PAR 11 1046 178 Since a homoclinic orbit to a saddle node is being followed we have also made the choice IEQUIB 2 in c mtn 1 The two test functions 415 and 46 to detect non central saddle node homoclinic orbits are also activated which must be specified in three ways Firstly in c mtn 1 IPSI is set to 15 16 so the active test functions are chosen as 15 and 16 This sets up the monitoring of these test functions Secondly in c mtn 1 user defined functions UZR are set up to look for zeros of the parameters corresponding to these test functions Recall that the parameters to be zeroed are always the test functions plus 20 Finally these parameters are included in the list of continuation parameters ICP Among the output there is a line BR PT TY LAB PAR 1 ae PAR 2 PAR 35 PAR 36 1 26 UZ 4 6 61046E 00 6 93248E 02 5 23950E 09 6 42344E 02 indicating that a zero of the test function IPSI 1 15 This means that at D K Z 6 6105 0 069325 the homoclinic orbit to the saddle node becomes non central namely it returns to the equilib rium along the stable eigenvector forming a non smooth loop The output is saved in b 1 s 1 and d 1 Repeating computations in the opposite direction along the curve IRS 1 DS 0 01 in c mtn 2 ri ritrun ri 1
53. 2 The other fixed point 1 0 is a saddle point A family of orbits connecting the two fixed points requires one free parameter see Friedman amp Doedel 1991 Here we take this parameter to be the wave speed c In the first run a starting connecting orbit is computed by continuation in the period T This procedure can be used generally for time integration of an ODE with AUTO Starting data in STPNT correspond to a point on the approximate stable manifold of 1 0 with T small In this demo the free end point of the orbit necessary approaches the unstable fixed point 0 0 A computed orbit with sufficiently large T is then chosen as restart orbit in the second run where typically one replaces T by c as continuation parameter However in the second run below we also add a phase condition and both c and T remain free 18 2 AUTO COMMAND ACTION mkdir fsh create an empty work directory cd fsh change directory demo fsh copy the demo files to the work directory ri run e fsh c fsh continuation in the period T with c fixed no phase condition save r1 0 save output files as b 0 s 0 d 0 r2 run r1 EP2 continuation in c and T with active phase ICP 2 11 12 13 14 condition NINT 1 DS UZR 2 1 2 3 5 10 save r2 fsh save output files as b fsh s fsh d fsh Table 18 1 Commands for running demo fsh 197 18 2 nag A Saddle Saddle Connection
54. 2 42 HB 4 2 47368E 01 2 62685E 01 7 95602E 00 7 95602E 00 2 45 EP 5 3 26008E 01 3 41635E 01 9 17980E 00 9 17980E 00 AUTO gt print br 0 TY number 0 PT 1 index 0 section 0 LAB BR 2 data 0 99999994000000003 0 0 0 0 0 0 0 0 TY name No Label AUTO gt print br PAR 1 0 99999994 1 001875 1 00747645 1 01786791 1 03426011 1 05801491 1 08738983 1 12195713 1 16544975 1 21923418 BR 2 o U 3 OOOOOE 00 OOOOOE 00 OOOOOE 00 U 3 37368E 01 16008E 01 U 3 37368E 01 16008E 01 U 3 37368E 01 16008E 01 Figure 4 16 This figure shows an example of parsing a bifurcation diagram First the demo involving the Lorenz equations named Irz is copied and we perform its first run We then print the result its second branch the first point on this branch the column corresponding to PAR 1 and the point with label 4 on this branch 39 AUTO gt import pylab AUTO gt pylab plot br PAR 1 br U 1 lt matplotlib lines Line2D object at 0x9b1356c gt AUTO gt pylab show Figure 4 17 Following the example in Figure we can plot a subset of the bifurcation diagram that is the second branch directly using matplotlib Query string Meaning o TY name The short name for the solution type see Table 4 5 TY number The number of the solution type see Table 4 5 BR The br
55. 3 Loading two files individually 29 AUTO gt ab load e ab c ab 1 Runner configured Figure 4 4 Loading two files at the same time AUTO gt runabi run e ab c ab 1 Runner configured Figure 4 5 Loading two files at the same time and run using them Also since it is common that several files will be loaded that have the same base name load ab performs the same action as load e ab c ab s ab h ab Note for the command load ab it is not required that all of the files exist Information from all existing files is used only if they exist and no error message will be given for non existing files However later run commands may cause AUTO to err with incomplete information 4 4 Scripting Section 4 3 showed commands being interactively entered at the AUTO CLUI prompt but since the AUTO CLUI is based on Python one has the ability to write scripts for performing sequences of commands automatically A Python script is very similar to the interactive mode shown in Section except that the commands are placed in a file and read all at once For example if the commands from Figure where placed into the file demol auto in the format shown in Figure then the commands could be run all at once by typing auto demol auto See Figure 4 7 for the full output demo ab ab load equation ab ab load ab constants ab 1 run
56. 6 9 and the continuation stops at po 9 and also makes sure that 0 lt p lt 5 5 AUTO COMMAND ACTION mkdir bvp create an empty work directory cd bvp change directory dm bvp copy the demo files to the work directory auto bvp auto or Run the script bvp auto auto bvp auto Table 15 3 Commands for running demo bvp 163 15 4 lin A Linear ODE Eigenvalue Problem This demo illustrates the location of eigenvalues of a linear ODE boundary value problem as bifurcations from the trivial solution family By means of branch switching an eigenfunction is computed as is illustrated for the first eigenvalue This eigenvalue is then continued in two parameters by fixing the Lo2 norm of the first solution component The eigenvalue problem is given by the equations tip Ss uy p t un 15 4 with boundary conditions u1 0 pz 0 and u 1 0 We add the integral constraint 1 1 u t dt p3 0 0 Then pz is simply the L2 norm of the first solution component In the first two runs pa is fixed while p and pz3 are free In the third run ps is fixed while p and p are free AUTO COMMAND ACTION mkdir lin create an empty work directory cd lin change directory demo lin copy the demo files to the work directory ri run e lin c lin 1st run compute the trivial solution family and locate eigenvalues r2 run ri BP1 NTST 6 ISW 1 DSMAX 0
57. ADVISED OF THE POSSIBILITY OF SUCH DAMAGE Note that the three dimensional plotting tool PLAUT04 optionally depends on libraries that are covered by the GNU General Public License GPL in particular Coin SoQt and Qt In that case the PLAUT04 binaries are also covered by the GPL 10 Chapter 1 Installing AUTO 1 1 Installation The AUTO file auto07p 0 9 1 tar gz is available via http cmvl cs concordia ca auto Here it is assumed that you are using the Unix e g bash shell and that the file auto07p 0 9 1 tar gz is in your main directory See below for OS specific notes While in your main directory enter the commands gunzip auto07p 0 9 1 tar gz followed by tar xvfo auto07p 0 9 1 tar This will result in a directory auto with one subdirectory auto 07p Type cd auto 07p to change directory to auto 07p Then type configure to check your system for required compilers and libraries Once the configure script has finished you may then type make to compile AUTO and its ancillary software The configure script is designed to detect the details of your system which AUTO requires to compile successfully If either the configure script or the make command should fail you may assist the configure script by giving it various command line options Please type configure help for more details Upon compilation you may type make clean to remove unnecessary files To run AUTO you need to set your environment variables correctly Assuming AUT
58. CR3BP Instructions for computing 2 d unstable manifolds of periodic orbits in the Circular Restricted 3 Body Problem CR3BP using AUTO 07p The instructions below are for the Halo family L1 in AUTO demo Lia Instructions for computing 2 d unstable manifolds of other periodic orbits in the CR3BP are similar Demos Hla H1b Hic Vla V1b and are given after these instructions Select a labeled solution which has exactly one Floquet multiplier with absolute value greater than 1 Floquet multipliers can be found in the file d L1 generated by demo r3b Enter the label of the periodic solution in the file Lla auto at label in Lla auto Also enter the size of the initial step into the direction of the unstable manifold there at step Note that representative values of these three quantities have already been entered there Now run the Python script Lla auto auto Lia auto auto Lia auto This will run r3b auto as above if this was not already done Through various computational steps the execution of the Python script will result in AUTO files b Lla s Lla and d Lla where the orbits in s Lla constitute the manifold which can be viewed with the graphics program plaut04 or r3bplaut04 pl Lia or r3b Lia plot3 Lia r3b True 154 The various steps executed by the Python commands in the script file Lla auto are explained below in Tables and which also show the equivalent Unix shell versions of these AUTO commands The
59. Figure 15 2 172 The above sequence of calculations can be carried out by running the Python script pcl auto without constants files or pclc auto with constants files included in the demo See the script and the Fortran file pcl f90 for details on how all parameters are mapped and which precise AUTO constants are changed at every step eta zgamma Z Z 1 24 0 24 3 24 4 rho 24 1 24 2 w T 10 fF oF 15 5 0 5 xgamma x X 10 Figure 15 1 Closing the Lin gap to obtain the point to cycle connection The left panel is a plot of p versus the gap size 7 and the right panel shows the corresponding orbit segments projected onto the x z plane To obtain these figures run plot closegap or Cpp closegap beta zgamma Z Z 50 60 70 80 rho 0 fi fi 10 20 30 40 90 150 125 f 100 f 75 F 25 F 0 40 10 0 10 20 30 40 xgamma X X 30 Figure 15 2 Parameter space diagram left and corresponding orbit segments in phase space right where the connection is continued in p and 8 To obtain these figures run plot cont or pp cont 173 15 14 snh SNH with Global reinjection Point to cycle connections with Lin s method This demo computes point to cycle or EtoP connection for equilibrium to periodic orbit and homoclinic point to point connections in the model vector field
60. MinGW Shell or by clicking on its desktop icon which puts you in a home directory where you can unpack AUTO using gunzip and tar as described above e Make sure that the gfortran and python binaries are in your PATH and that their direc tories are at the front of it You can do this for instance using the shell command export PATH c Python27 bin c Program Files gfortran bin PATH You can also in spect edit and then source the file auto 07p cmds auto env sh to achieve this e Now you should be able to run configure and make to compile AUTO as shown above You can use AUTO using shell commands from the default MSYS shell environment You can also start the CLUI by double clicking on the file auto py in the python folder of AUTO in Windows Explorer or by creating a shortcut to it Alternatively AUTO runs on Windows as above using the Unix like environment Cygwin see http www cygwin com but the non Cygwin setup is more responsive and is much easier to setup for Matplotlib You can however use its X server and lesstif to compile and run the old PLAUT and GUI94 if you so desire With some effort it is possible to compile PLAUT04 on Windows without an X server using Coin SoQt and Qt You can also find precompiled PLAUT04 binaries at http sourceforge net projects auto 07p files 1 2 Restrictions on Problem Size There are no size restrictions in the file auto 07p include auto h any more This file now contains the d
61. ODEs are almost always structurally correct because the mesh adaption strategy if IAD gt 0 safeguards to some extent against spurious solutions If these do occur possibly near infinite period orbits the unusual appearance of the solution family typically serves as a warning Repeating the computation with increased NTST is then recommended 11 4 Bifurcation Points and Folds It is recommended that the detection of folds and bifurcation points be initially disabled For example if an equation has a vertical solution family then AUTO may try to locate one fold after another Generally degenerate bifurcations cannot be detected Furthermore bifurcations that are close to each other may not be noticed when the pseudo arclength step size is not sufficiently small Hopf bifurcation points may go unnoticed if no clear crossing of the imaginary axis takes 117 place This may happen when there are other real or complex eigenvalues near the imaginary axis and when the pseudo arclength step is large compared to the rate of change of the critical eigenvalue pair A typical case is a Hopf bifurcation close to a fold Similarly Hopf bifurcations may go undetected if switching from real to complex conjugate followed by crossing of the imaginary axis occurs rapidly with respect to the pseudo arclength step size Secondary periodic bifurcations may not be detected for similar reasons In case of doubt carefully inspect the contents of the diagnos
62. PAR 5 3 51 UZ 14 2 00000E 00 4 01890E 01 2 66146E 09 250 The output is saved in the files b 4 s 4 and d 4 Note that PAR 5 p remains zero which is exactly what we expect Next we want to add a solution to the adjoint equation to this solution This is achieved by making the change ITWIST 1 Also we set ISTART to 1 to tell HomCont that it should not try to shift the orbit anymore r5 run r4 ICP 5 8 NMX 2 ITWIST 1 ISTART 1 save r5 5 or alternatively rn c sib 5 s 4 sv 5 The output is stored in b 5 s 5 and d 5 BR PT TY LAB PAR 5 L2 NORM aot a PAR 8 3 2 EP 15 2 66146E 09 4 01890E 01 1 00000E 02 Here PAR 8 is a dummy unused parameter and y just stays where it is Now that we have obtained the solution of the adjoint equation we are able to detect inclination flips This can be achieved by setting IPSI to 13 and monitoring PAR 33 r6 run r5b ICP 4 5 33 NMX 30 DS 0 01 DSMAX 1 0 UZR 33 0 4 0 IPSI 13 BR PT TY LAB PAR 4 L2 NORM waa PARES PAR 33 3 19 UZ 16 7 11774E 02 4 01890E 01 1 24376E 11 2 36702E 07 The output is stored in the Python variable r6 Hence an inclination flip was found at a 0 711774 Now we are ready to perform homoclinic branch switching using the techniques described in Oldeman et al 2003 Our first aim is to find a 2 homoclinic orbit The ingredients we need are a homoclinic orbit where n homoclinic orbits are close by and t
63. Parameter Index Parameter Value or indices with lists of values 4Parameter Indez Parameter Value p to designate the parameters and the parameter values at which output is to be written For examples see demos exp int and fsh If such a parameter index is preceded by a minus sign then the computation will ter minate at such a solution point See also STOP in Section 10 6 1 above and UZSTOP in Section 10 9 12 below for alternative termination methods Demos pen and bru Note that fort 8 output can also be written at selected values of overspecified parameters For an example see demo pvl For details on overspecified parameters see Section 10 7 10 10 9 12 UZSTOP This constant specifies parameter values in the same way as UZR in Section 10 9 11 above but the computation will always terminate if any solution point that is specified is encountered 113 114 10 10 Quick reference e s dat sv Define file names equation prefix f f90 c restart solution suffix s user data prefix dat output suffix b s d unames parnames Dictionary mapping of U and PAR to user defined names NDIM Problem dimension IPS Problem type 0 AE 1 FP ODEs 1 FP maps 2 PO 2 IVP 4 BVP 7 BVP with Floquet multipliers 5 algebraic optimization problem 15 optimization of periodic solutions IRS TY Start solution label start solution type ILP Fold
64. This demo illustrates the computation of traveling wave front solutions to Nagumo s equation Wi Wee f w a o0o lt x lt oo t gt 0 18 3 Fw a w 1 w w a 0O lt a lt l We look for solutions of the form w x t u x ct where c is the wave speed This gives the first order system u 2 u2 2 uy 2 cualz fluil a where z z ct and d dz If a 1 2 and c 0 then there are two analytically known heteroclinic connections one of which is given by 18 4 env u 2 Iam ata u4 2 00 lt Z lt 00 The second heteroclinic connection is obtained by reflecting the phase plane representation of the first with respect to the u axis In fact the two connections together constitute a heteroclinic cycle One of the exact solutions is used below as starting orbit To start from the second exact solution change SIGN 1 in the routine STPNT in nag f90 and repeat the computations below see also Friedman amp Doedel 1991 AUTO COMMAND ACTION mkdir nag create an empty work directory cd nag change directory demo nag copy the demo files to the work directory ri run e nag c nag compute part of first family of heteroclinic orbits r2 run e nag c nag DS compute first family in opposite direction save r1 r2 nag save all output to b nag s nag d nag Table 18 2 Commands for running demo nag 198 18 3 stw Continuat
65. XXX d xxx respectively Existing files by these names will be deleted 66 Cap ell ls lbf 5 2 pp pl r3b p ps 5 3 cp Type Cap xxx to append the output files fort 7 fort 8 fort 9 to existing data files b xxx S XXX d xxx resp Type Cap xxx yyy to append b xxx s xxx d xxx to b yyy s yyy d yyy resp Type 11 to list all solutions in fort 8 Type 11 xxx to list all solutions in s xxx Type 1s to list the abbreviated contents of fort 7 Type ls xxx to list the abbreviated contents of b xxx The contents are shown in a similar format as the screen output of AUTO runs Type 1bf to list the contents of fort 7 Type lbf xxx to list the contents of b xxx The contents are shown with less accuracy 6 instead of 11 significant figures than in the actual file for easier viewing Plotting commands Type Cpp xxx to run the graphics program PyPLAUT See Chapter 7 for the graphical inspection of the data files b xxx and s xxx Type Cpp to run the graphics program PyPLAUT for the graphical inspection of the output files fort 7 and fort 8 The command p1 is equivalent to pp but runs the graphics program PLAUT04 instead See Chapter The command r3b is equivalent to pp but runs the graphics program PLAUTO04 instead in R3B mode See Chapter 8 The command Cp is equivalent to pp but runs the graphics program PLAUT instead See Chapter 7 Type Cps fig x t
66. an orbit with a lt 3 and close the gap corresponding to PAR 22 for decreasing a r5 run r4 c kdv 5 sv 5 BR PT TY LAB PAR 2 PAR 3 PAR 22 PAR 24 1 10 12 2 57977E 00 2 15713E 06 7 65450E 04 3 82670E 04 1 13 UZ 13 2 32044E 00 3 86701E 11 1 13817E 10 1 58675E 08 1 20 EP 14 1 47788E 01 9 46232E 04 7 53666E 01 3 43203E 01 and finally close the gap corresponding to 2 PAR 24 r6 run r5 UZ1 c kdv 6 sv 6 BR PT TY LAB PAR 2 PAR 3 PAR 23 PAR 24 1 23 UZ 15 2 32044E 00 3 30393E 12 1 48758E 01 2 30540E 10 1 35 16 2 31894E 00 2 15192E 08 7 69389E 00 1 07760E 05 1 51 UZ 17 2 33846E 00 2 57829E 07 3 48152E 00 1 29755E 04 1 58 UZ 18 3 08085E 00 2 28299E 12 3 50004E 00 1 62266E 10 259 so that a three homoclinic orbit is found Here the zero at label 17 is the one we are looking for Label 15 is a false positive since T PAR 23 is still too high At label 18 a PAR 1 has changed considerably to the extend that a gt 3 and a second 3 homoclinic orbit is found Note that for all zeros of PAR 24 lt the parameter A PAR 3 is also zero within AUTO accuracy which it has to be to remain within the original Hamiltonian system Setting ISTART 1 a normal trivial continuation with NMX 1 of the orbit corresponding to label 17 lets HomCont produce a proper concatenated 3 homoclinic orbit r7 run r6 UZ2 c kdv 7 sv 7 BR PT TY LAB PAR 2 L2 N
67. ap 1 other saddle homoclinic orbit family restart 3rd UZ append output files to b s 1 d 1 r6 run r1 UZ1 c mtn 6 sv 6 3 parameter non central saddle node homoclinic save output files as b 6 s 6 d 6 Table 22 1 Detailed AUTO Commands for running demo mtn 00 0 07 0 14 0 21 0 28 O55 Z Figure 22 1 Parametric portrait of the predator prey system 224 Figure 22 2 Approximation by a large period cycle d_0 0 012 Figure 22 3 Projection onto the K Do plane of the three parameter curve of non central saddle node homoclinic orbit 225 Chapter 23 HomCont Demo kpr 23 1 Koper s Extended Van der Pol Model The equation file kpr f90 contains the equations x ea ky a 3x A Y 2y z 23 1 z co y D 2 with e 0 1 and es 1 Koper 1995 To copy across the demo kpr and compile we type demo kpr gt 23 2 The Primary Branch of Homoclinics First we locate a homoclinic orbit using the homotopy method The file kpr f90 already con tains approximate parameter values for a homoclinic orbit namely A PAR 1 1 851185 k PAR 2 0 15 The file c kpr 1 specifies the appropriate constants for continuation in 2T PAR 11 also referred to as PERIOD and the dummy parameter w PAR 17 starting from a small solution in the local unstable manifold ri run kpr c kpr 1
68. append the output files fort 7 fort 8 fort 9 to existing data files s xxx b xxx and d xxx Type append xxx yyy to append existing data files s xxx b xxx and d xxx to data files s yyy b yyy and d yyy Aliases ap commandAppend 59 4 14 2 Plotting commands plot Plotting of data Type plot x to run the graphics program PyPLAUT for the graphical inspection of bifurcation diagram or solution data in x Type plot xxx to run the graphics program PyPLAUT for the graphical inspection of the data files b xxx and s xxx Type plot to run the graphics program for the graphical inspection of the output files fort 7 and fort 8 Values also present in the file autorc such as color_list black green red blue orange can be provided as keyword arguments as well as hide True which hides the on screen plot The return value for instance p for p plot x will be the handle for the graphics window It has p config and p savefig methods that allow you to configure and save the plot When plotting see help p config and help p savefig for details Aliases p2 pl commandPlotter plot3 Plotting of data using PLAUT04 Type plot3 x to run the graphics program PLAUT04 for the graphical inspection of bifurcation diagram or solution data in x Type plot3 xxx to run the graphics program PLAUT04 for the graphical inspection of the data files b xxx and s xxx Type plot3 to run the graphics program PL
69. close The equations which model a predator prey system with harvesting are Uy pgui l u wu pi l e 14 3 Up U2 P4uyua Here p quota is the principal continuation parameter while pz p4 3 and p3 5 are fixed The variables wu and u gt denote prey and predator for instance fish and sharks The use of PLAUT is also illustrated The saved plots are shown in Figure and Figure You can obtain similar figures using the Python CLUI s plot command and using PLAUTOA4 COMMAND ACTION mkdir pp2 create an empty work directory cd pp2 change directory dm pp2 copy the demo files to the work directory auto pp2 auto or Run the script pp2 auto auto pp2 auto Table 14 9 Commands for running demo pp2 137 AUTO COMMAND ACTION Cp pp2 or Opp pp2 run PLAUT or PyPLAUT to graph the contents of b pp2 and s pp2 PLAUT PyPLAUT COMMAND ACTION d2 set convenient defaults ax select axes 1 3 select real columns 1 and 3 in b pp2 bdo plot the bifurcation diagram max ui versus p d1 choose other default settings bd get blow up of current bifurcation diagram O 1 0 25 1 enter diagram limits sav save plot see Figure fig 1 or fig1 eps upon prompt enter a new file name e g fig 1 or fig eps cl clear the screen 2d enter 2D mode for plotting labeled solutions 11 15 19 23 select these labeled orbits in s pp2 d default orbit dis
70. continued in two equation parameters AUTO COMMAND ACTION mkdir int create an empty work directory cd int change directory demo int copy the demo files to the work directory ri run e int c int 1st run detection of a fold save ri int save output files as b int s int d int r2 run ri LP1i ICP 1 2 ISW 2 2nd run generate starting data for a curve of folds r3 run r2 3rd run compute a curve of folds restart from the last and only label in r2 save r3 1p save the output files as b lp s lp d lp Table 15 2 Commands for running demo int 162 15 3 bvp A Nonlinear ODE Eigenvalue Problem This demo illustrates the location of eigenvalues of a nonlinear ODE boundary value problem as bifurcations from the trivial solution family The families of solutions that bifurcate at all five computed eigenvalues that is the eigenfunctions are computed in both directions The equations are ti ts 15 3 u pr wh ee with boundary conditions u1 0 0 ui 1 0 We add the integral constraint 1 0 Then p is simply the average of the first solution component The integral constaint gives a measure the exact same continuations could be done without any integral conditions in just the one parameter p however p gives us extra possibilities to plot and stop at desirable solutions The values that bvp auto sets in UZR make sure that solutions are given for pp 3
71. copy above Aliases mv commandMoveFiles df Clear the current directory of fort files Type df to clean the current directory This command will delete all files of the form fort Aliases deletefort commandDeleteFortFiles clean Clean the current directory Type clean to clean the current directory This command will delete all files of the form fort o and exe Aliases cl commandClean delete Delete data files Type delete xxx to delete the data files d xxx b xxx and s xxx if you are using the default filename templates Aliases dl commandDeleteDataFiles 4 14 4 Diagnostics limitpoint Print the value of the limit point function Type limitpoint x to list the value of the limit point function in the diagnostics of the bifurcation diagram object x This function vanishes at a limit point fold Type limitpoint to list the value of the limit point function in the output file fort 9 Type limitpoint xxx to list the value of the limit point function in the info file d xxx Aliases Im lp commandQueryLimitpoint 57 branchpoint Print the branch point function Type branchpoint x to list the value of the branch point function in the diagnostics of the bifurcation diagram object x This function vanishes at a branch point Type branchpoint to list the value of the branch point function in the output file fort 9 Type branchpoint
72. corresponds to a reversible orbit flip for b gt 2 a gt 0 We start from this explicit solution using ISTART 2 for a 3 and b v65 3 4 demo kdv ri run kdv sv 1 BR PT TY LAB PAR 1 L2 NORM ite PAR 3 1 1 EP 1 3 00000E 00 5 56544E 00 0 00000E 00 1 2 EP 2 3 04959E 00 5 49141E 00 4 53380E 18 258 Here PAR 1 a PAR 2 b and PAR 3 A4 We have only done a very small continuation to give AUTO a chance to create a good mesh and avoid convergence problems later Next we set ITWIST 1 and calculate the adjoint r2 run ri c kdv 2 sv 2 BR PT TY LAB PAR 2 L2 NORM PAR 9 1 2 EP 3 2 76557E 00 5 49141E 00 3 12500E 04 We now need to move back to the orbit flip at a 3 r3 run r2 c kdv 3 sv 3 BR PT TY LAB PAR 1 L2 NORM PAR 3 1 14 UZ 5 3 00000E 00 5 47613E 00 1 47725E 09 Now all preparations are done to start homoclinic branch switching This is very similar to the technique used in Sandstede s model in Section 27 1t to find a 3 homoclinic orbit we open 2 Lin gaps until T 3 5 while also varying A PAR 3 r4 run r3 UZ2 c kdv 4 sv 4 BR PT TY LAB PAR 3 PAR 21 PAR 22 PAR 24 1 13 8 5 85315E 10 1 65474E 01 9 20183E 08 6 11537E 07 1 23 UZ 9 1 52986E 09 9 85223E 00 6 68578E 12 2 01956E 07 1 26 10 4 09273E 09 6 87525E 00 2 68679E O7 7 64502E 07 1 33 UZ 11 2 15483E 06 3 49999E 00 7 94022E 04 3 99104E 04 We then look for
73. corresponds to the situation where the eigenvalues of the equilibrium at the origin are a b 1 a b 0 25 and c 0 75 Hence the corresponding bifurcation diagram consists of a complicated structure involving a fan of infinitely many n periodic and n homoclinic orbits for arbitrary n and a region with horseshoe dynamics see also Homburg amp Krauskopf 2000 and the references therein This computation starts from an equilibrium at 2 3 0 0 which exists for a u a 0 Also b is set to 0 625 the value we would like it to be and c is set to 2 5 in stpnt Choosing c 2 at this stage leads to convergence problems This equilibrium is not the one corresponding to the homoclinic orbit but it is an equilibrium with complex eigenvalues that 248 we can follow until it reaches a Hopf bifurcation A periodic orbit emanates from this Hopf bifurcation and can be followed to the homoclinic orbit However first we need to change a from 0 to 0 375 All the following commands except for demo sib are contained within the file sib auto which you can either execute in a batch mode by entering gt auto sib auto or step by step using AUTO gt demofile sib auto We start by copying the demo to the current work directory and running the first step demo sib ri run e sib c sib save r1 1 The equilibrium is followed in a until a or PAR 1 is at our desired value 0 375 BR PT TY LAB P
74. defined in the equations file abc 90 NDIM 3 IPS 1 IRS 0 ILP 1 ICP 1 NTST 15 NCOL 4 IAD 3 ISP 1 ISW 1 IPLT 0 NBC 0 NINT 0 NMX 130 NPR 200 MXBF 10 IID 2 ITMX 8 ITNW 5 NWTN 3 JAC 0 EPSL 1e 07 EPSU 1e 07 EPSS 0 0001 DS 0 02 DSMIN 0 001 DSMAX 0 1 IADS 1 NPAR 5 THL 11 0 0 THU UZR 11 0 4 STOP UZ1 Table 14 4 The constants file c abc 1 for Run 1 stationary solutions of demo abc NDIM 3 IPS 2 TRS 2 ILP 1 ICP 1 11 NTST 25 NCOL 4 IAD 3 ISP 1 ISW 1 IPLT 0 NBC 0 NINT 0 NMX 200 NPR 200 MXBF 10 IID 2 ITMX 8 ITNW 5 NWTN 3 JAC 0 EPSL 1e 07 EPSU 1e 07 EPSS 0 0001 DS 0 02 DSMIN 0 001 DSMAX 0 1 IADS 1 NPAR 5 THL 11 0 0 THU UZR 1 0 25 STOP UZ1 Table 14 5 The constants file c abc 2 for Run 2 periodic orbits of demo abc 135 COMMAND ACTION mkdir abc create an empty work directory cd abc change directory dm abc copy the demo files to the work directory OR abc 1 compute the stationary solution family with four Hopf bifurcations sv abc save output files as b abc s abc d abc R abc 2 compute a family of periodic solutions from the first Hopf point ap abc append the output files to b abc s abc d abc R abc 3 compute a family of periodic solutions from the second Hopf point ap abc append the output files to b ab
75. detection 1 on 0 off ICP Continuation parameters NTST mesh intervals NCOL collocation points IAD Mesh adaption every IAD steps 0 off ISP Bifurcation detection 0 0ff 1 BP FP 3 BP PO BVP 2 all ISW Branch switching 1 normal 1 switch branch BP HB PD 2 switch to two parameter continuation LP BP HB TR 3 switch to three parameter continuation BP IPLT Select principal solution measure NBC boundary conditions NINT integral conditions NMX Maximum number of steps RLO RL1 Parameter interval RLO lt A lt RE AO A1 Interval of principal solution measure AO lt lt Al NPR Print and save restart data every NPR steps MXBF Automatic branch switching for the first MXBF bifurcation points if IPS 0 1 IBR LAB Set initial branch and label number 0 automatic IIS Control solution output of branch direction vector O never 3 always IID Control diagnostic output 0 none 1 little 2 normal 4 extensive ITMX Maximum of iterations for locating special solutions points ITNW Maximum of correction steps NWTN Corrector uses full newton for NWTN steps JAC User defines derivatives 0 no 1 yes EPSL EPSU EPSS Convergence criterion parameters solution components special points DS Start step size DSMIN DSMAX Step size interval DSMIN lt h lt DSMAX IADS Step size adaption every IADS steps 0 off NPAR Maximum number of parameters THL THU list of parameter and solution weights UZR UZSTOP lis
76. entered as individual commands or as prefixes For example one can enter the command sydpbd0 sy Use symbols for special solution points for example open square branch point solid square Hopf bifurcation dp Differential Plot 1 e show stability of the solutions Solid curves represent stable solutions Dashed curves are used for unstable solutions and for solutions of unknown stability For periodic solutions use solid open circles to indicate stability instability or unknown stability st Set up titles and axes labels nu Normal usage reset special options TT 7 3 Other PLAUT Commands The full PLAUT program has several other capabilities for example scr To change the diagram size rss To change the size of special solution point symbols These commands are not available in PyPLAUT 7 4 Printing PLAUT Files Ops Type ps fig 1to convert a saved PLAUT file fig 1 to PostScript format in fig 1 ps Ceps Type eps fig 1 to convert a saved PLAUT file fig 1 to encapsulated PostScript format in fig 1 eps In PyPLAUT you can directly save to a variety of file formats including eps and png 78 Chapter 8 The Graphics Program PLAUTOA4 PLAUTO4 is a graphic tool for AUTO data visualization Here we explain how to view AUTO data sets with PLAUTO4 An AUTO data set contains a solution file s foo a bifurcation file b foo and a diagnostic file d foo Here foo
77. equilibrium is slower First we start from the original homoclinic orbit computed via the homotopy method label 4 which is well away from the non hyperbolic equilibrium Also we shall no longer be interested in in inclination flips so we set ITWIST 0 in c kpr 6 and in order to compute up to PAR 11 1000 we set up a user defined function for this Running AUTO with PAR 11 and PAR 2 as free parameters r r n r2 EP1 c kpr 6 sv 6 we obtain among the output BR PT TY LAB PERIOD L2 NORM ee PAR 2 1 35 UZ 6 1 00000E 03 1 66191E 00 1 50000E 01 We can now repeat the computation of the family of saddle homoclinic orbits in PAR 1 and PAR 2 from this point with the test functions Yg and 4 0 for non central saddle node homoclinic orbits activated r7 run r6 UZi c kpr 7 80 7 The saddle node point is now detected at BR PT TY LAB PAR 1 sit PAR 2 PAR 29 PAR 30 1 29 UZ 8 1 76505E 01 2 40533E 00 1 74004E 06 2 30933E 01 which is stored in s 7 That PAR 29 wg is zeroed shows that this is a non central saddle node connecting the centre manifold to the strong stable manifold Note that all output beyond this point although a well posed solution to the boundary value problem is spurious in that it no longer represents a homoclinic orbit to a saddle equilibrium see Champneys et al 1996 If we had chosen to we could continue in the other direction in order to approach the BT point more accurately by reversing the
78. hence use the automatically generated extended system Run 1 Locate a Hopf bifurcation The free system parameter is Az Run 2 Compute a family of periodic solutions from the Hopf bifurcation Run 3 This run retraces part of the periodic solution family using the full optimality system but with all adjoint variables w x y and hence a equal to zero The optimality parameters 7 and 73 are zero throughout An extremum of the objective functional with respect to A3 is located Such a point corresponds to a branch point of the extended system Given the choice of objective functional in this demo this extremum is also a fold with respect to Az Run 4 Branch switching at the above found branch point yields nonzero values of the adjoint variables Any point on the bifurcating family away from the branch point can serve as starting solution for the next run In fact the branch switching can be viewed as generating a nonzero eigenvector in an eigenvalue eigenvector relation Apart from the adjoint variables all other variables remain unchanged along the bifurcating family Run 5 The above found starting solution is continued in two system parameters here Az and Az i e a two parameter family of extrema with respect to A3 is computed Along this family the value of the optimality parameter 7 is monitored i e the value of the functional that vanishes at an extremum with respect to the system parameter A2 Such a zero of Ta i
79. in the C file but by PAR 11 in the constants file Equation files written in C are used in the homoclinic branch switching demo in Chapter A detailed list of user visible changes can be found in the file AUTO_DIR CHANGELOG 1 4 Parallel Version AUTO contains code which allows it to run in on parallel computers Namely it can use either OpenMP to run most of its code in parallel on shared memory multi processors or the MPI message passing library When the configure script is run it will try to detect if the Fortran compiler supports OpenMP examples are Gfortran 4 2 or later and the Intel Fortran Compiler If it is successful the necessary compiler flags are used to enable OpenMP in AUTO To force the configure script not to use OpenMP one may type configure without openmp and then type make On the other hand unless there is some particular difficulty we recommend that that the configure script be used without arguments since the parallel version of AUTO may easily be controlled and even run in a serial mode through the use of the environment variable OMP_NUM_THREADS For example to run the AUTO executable auto exe in serial mode you just type export OMP_NUM_THREADS 1 To run the same command in parallel on 4 processors you type export OMP_NUM_THREADS 4 Without any OMP_NUM_THREADS set the number of processors that AUTO will use can be equal to the actual number of processors on the system or can be equal to one this is system dep
80. it can be used to verify whether the starting solution is indeed a solution For this purpose the stepsize DS should be small and one should look at the residuals printed in the fort 9 output file Note that the first residual vector printed in fort 9 may be identically zero as it may correspond to the computation of the starting direction Look at the second residual vector in such case This residual vector has dimension NDIM NBC NINT 1 which accounts for the NDIM differential equations the NBC boundary conditions the NINT user defined integral constraints and the pseudo arclength equation For proper inter pretations of these data one may want to refer to the solution algorithm for solving the collocation system as described in Doedel Keller amp Kern vez 19910 TID 5 This setting gives very extensive diagnostic output for differential equations namely debug output from the linear equation solver This setting should not normally be used as it may result in a huge fort 9 file It gives incomplete results when used in combination with MPI parallellization 10 9 10 IPLT This constant allows redefinition of the principal solution measure which is printed as the second real column in the screen output and in the fort 7 output file If IPLT 0 then the L norm is printed Most demos use this setting For algebraic problems the standard definition of L2 norm is used For differential equations the L norm is defined a
81. o 186 16 6 bru Euler Time Integration the Brusselator 187 17 AUTO Demos Optimization 188 17 1 opt A Model Algebraic Optimization Problem 189 17 2 ops Optimization of Periodic Solutions lt i acres lt a eee dee dados 190 17 3 obv Optimization for a BVP lt 4 644 4 26 44644444448 24d 4 5 194 18 AUTO Demos Connecting orbits 18 1 fsh A Saddle Node Connection 18 2 nag A Saddle Saddle Connection 18 3 stw Continuation of Sharp Traveling Waves 19 AUTO Demos Miscellaneous 19 1 pvl Use of the Routine PVLS 19 2 ext Spurious Solutions to BVP 19 3 tim A Test Problem for Timing AUTO ooo e a 20 HomCont Aut e alos ai Aires 20 3 1 NUNSTAB a xe occaeca er 20 3 2 NSTAB epa ratos a 20 3 3 TEQUIB E 0 3 4 ITWIST ori eee 20 3 5 ISTART A 20 3 6 TREV 4 8 hed Gh we oe eg 20 3 7 EI AA DUR TPST AA 20 4 Restrictions on HomCont Constants 0 0020 08 a 20 5 Restrictions on the Use of PAR 20 6 Test Functions 202 2 20 7 Starting Strategies 20 8 Notes on Running HomCont Demos 0 0 002 21 HomCont Demo san 21 1 Sandstede s Modell 2 2 21 2 Inclination Flip 2 24 lt 44 lt 21 3 Non orientable Resonant Eigenvalues 0 0 00 0 00 000002022 ee 21 4 Orbit A ee ee 21 5 Detailed AUTO Commands 22 HomCont Demo mtn 22 1 A Predat
82. parameters are fixed namely p2 0 25 p3 0 5 pa 4 ps 3 and pg 5 However both p and pa are free in the computation of loci of Hopf points The script in pp3 auto first computes the bifurcation diagram involving the stationary solu tions It finds four Hopf bifurcations A periodic orbit family is computed from each of these four Hopf bifurcations Then the second Hopf bifurcation from the first run is continued in two parameters also producing the other locus COMMAND ACTION mkdir pp3 create an empty work directory cd pp3 change directory dm pp3 copy the demo files to the work directory auto pp3 auto or Run the script pp3 auto auto pp3 auto Table 14 14 Commands for running demo pp3 145 AUTO COMMAND ACTION Op pp3 or pp pp3 run PLAUT PyPLAUT to graph the con tents of b pp3 and s pp3 PLAUT PyPLAUT COMMAND ACTION d2 set convenient defaults ax select axes 13 select real columns 1 and 3 in b pp3 bdo plot the bifurcation diagram max u versus p bd get blow up of current bifurcation diagram Oo 0 6 O 1 2 enter diagram limits d1 choose other default settings with labels bd another blow up of the bifurcation diagram 0 0 6 O 0 75 enter diagram limits d2 set defaults 2d enter 2D mode for plotting labeled solutions 13 14 15 select these orbits from s pp3 d default orbit display u versus time 23 select columns 2 and
83. sign of DS in c kpr 7 The file c kpr 8 contains the constants necessary for switching to continuation of the cen tral saddle node homoclinic curve in two parameters starting from the non central saddle node homoclinic orbit stored as label 8 in s 7 r8 run r7 UZ1 c kpr 8 sv 8 In this run we have activated the test functions for saddle to saddle node transition points along curves of saddle homoclinic orbits 415 and 416 Among the output we find BR PT TY LAB PAR 1 Sous PAR 2 PAR 35 PAR 36 1 38 UZ 11 1 76509E 01 2 40533E 00 6 89014E 03 3 09956E 05 which corresponds to the family of homoclinic orbits leaving the locus of saddle nodes in a second non central saddle node homoclinic bifurcation a zero of 436 Note that the parameter values do not vary much between the two codimension two non central saddle node points labels 8 and 11 However Figure 23 6 shows clearly that between the two codimension two points the homoclinic orbit rotates between the two components of the 1D stable manifold i e between the two boundaries of the center stable manifold of the saddle node The overall effect of this process is the transformation of a nearby small saddle 231 100 0 29 S00 OS da Y Hh 1 0 o 9 0 0 2 1 0 1 3 Figure 23 6 Two non central saddle node homoclinic orbits 1 and 3 and 2 a central saddle node homoclinic orbit
84. solution file almost two times bigger than necessary when switching branches is never performed from solutions in this file TIS 0 The direction of the branch is never provided IIS 1 The direction of the branch is only provided at special points from which branch switching can be performed types LP boundary value problems only BP PD TR TIS 2 The direction of the branch is provided at all special points but not at regular points without a type label IIS 3 The direction of the branch is always provided This is the default setting 10 9 9 IID This constant controls the amount of diagnostic output printed in fort 9 the greater IID the more detailed the diagnostic output IID 0 No diagnostic output TID 1 Minimal diagnostic output This setting is not recommended IID 2 Regular diagnostic output This is the recommended value of IID 111 TID 3 This setting gives additional diagnostic output for algebraic equations namely the Jacobian and the residual vector at the starting point This information which is printed at the beginning of fort 9 is useful for verifying whether the starting solution in STPNT is indeed a solution TID 4 This setting gives additional diagnostic output for differential equations namely the reduced system and the associated residual vector This information is printed for ev ery step and for every Newton iteration and should normally be suppressed In particular
85. the center left the Load Constants buttons at the lower left and the Stop and Exit buttons Note Most GUI buttons are activated by point and click action with the left mouse button If a beep sound results then the right mouse button must be used 9 1 1 The Menu bar It contains the main buttons for running AUTO and for manipulating the equations file the constants file the output files and the data files In a typical application these buttons are used from left to right First the Equations are defined and if necessary Edited before being Written Then the AUTO constants are Defined This is followed by the actual Run of AUTO The resulting output files can be Saved as data files or they can be Appended to existing data files Data files can be Plotted with the graphics program PLAUT and various file operations can be done with the Files button Auxiliary functions are provided by the Demos Misc and Help buttons The Menu Bar buttons are described in more detail in Section 9 2 9 1 2 The Define Constants buttons These have the same function as the Define button on the Menu Bar namely to set and change AUTO constants However for the Define button all constants appear in one panel while for 92 the Define Constants buttons they are grouped by function as in Chapter namely Prob lem definition constants Discretization constants convergence Tolerances continuation Step Size diagram Limits designation of free Paramet
86. the item Name to run PLAUT with other data files 9 2 9 Files button This pull down menu contains the item Restart to redefine the restart file Normally when restarting from a previously computed solution the restart data is expected in the file s xxx where xxx is the active equation name Use the Restart button to read the restart data from another data file in the immediately following run The pull down menu also contains the following items Copy to copy b xxx S XXX d xxx c xxx to D yyy s yyy d yyy c yyy resp Append to append data files b xxx s xxx d xxx to b yyy s yyy d yyy resp Move to move b xxx S XXX d xxx c xxx to b yyy s yyy d yyy c yyy resp Delete to delete data files b xxx s xxx d xxx Clean to delete all files of the form fort o and exe 9 2 10 Demos button This pulldown menu contains the items Select to view and run a selected AUTO demo in the demo directory and Reset to restore the demo directory to its original state Note that demo files can be copied to the user work directory with the Equations Demo button 94 9 2 11 Misc button This pulldown menu contains the items Tek Window and VT102 Window for opening windows Emacs and Xedit for editing files and Print for printing the active equations file xxx 90 9 2 12 Help button This pulldown menu contains the items AUTO constants for help on AUTO constants and User Manual for view
87. the third run as written in the file s 3 notice that there appears to be second equilibrium with a 2D stable manifold which intersects the 2D unstable manifold of the origin The intersection curve which corresponds to a heteroclinic orbit is visible in the graphical representation of the manifold 170 15 11 p2c Point to cycle connections In this demo a point to cycle heteroclinic connection is computed via homotopy and then continued in two system parameters in the Lorenz equations f a u psluz u t Uy piu U2 UU Uz UU p2U3 Type auto p2c auto to run the demo and auto clean auto to remove generated files Refer to Doedel Kooi van Voorn amp Kuznetsov 2008a and http www bio vu nl thb research project globif index_main html for background information 15 12 c2c Cycle to cycle connections In this demo a cycle to cycle heteroclinic connection is computed via homotopy and then continued in one system parameter in a food chain model ory t 1 7 ae ory O lyz o gt ee te EA Olyz Type auto c2c auto to run the demo After that it is possible to compute two parameter continuations of folds of the cycle to cycle connection but since this is computationally intensive it is put in a seperate file type auto c2cfolds auto Type auto clean auto to remove all generated files Refer to Doedel Kooi van Voorn amp Kuznetsov 2008b and http www bio vu nl
88. this parameter added to the list of user defined output points UZR in c xxx The default is IPSI 20 4 Restrictions on HomCont Constants Note that certain combinations of these constants are not allowed in the present implementation In particular 208 The computation of orientation ITWIST 1 is not implemented for IEQUIB lt O heteroclinic orbits IEQUIB 2 saddle node homoclinics IREVAX reversible systems ISTART 3 ho motopy method for starting or if the equilibrium contains complex eigenvalues in its linearization The homotopy method ISTART 3 is not fully implemented for heteroclinic connections TEQUIB lt O saddle node homoclinic orbits IEQUIB 2 or reversible systems IREVA Certain test functions are not valid for certain forms of continuation see Section 20 6 below for example PS1 13 and PSI 14 only make sense if ITWIST 1 and PSI 15 and PSI 16 only apply to TEQUIB 2 20 5 Restrictions on the Use of PAR The parameters PAR 1 PAR 9 can be used freely by the user The other parameters are used as follows PAR 11 The value of PAR 11 equals the length of the time interval over which a homoclinic solution is computed Also referred to as period This must be specified in STPNT PAR 10 If ITWIST 1 then PAR 10 is used internally as a dummy parameter so that the adjoint equation is well posed PAR 12 PAR 20 These are used for specifying the equilibria and if ISTART 3
89. u1 u2 1 0 Graphical inspection of the computed periodic orbits for example u versus the scaled time variable t shows how the phase condition has the effect of keeping the peak in the solution in the same location AUTO COMMAND ACTION mkdir phs create an empty work directory cd phs change directory demo phs copy the demo files to the work directory ri run e phs c phs 1 detect Hopf bifurcation r2 run r1i HB1 c phs 2 compute periodic solutions Constants changed IRS IPS NPR save ritr2 phs save output to b phs s phs d phs Table 14 20 Commands for running demo phs 151 14 14 ivp Time Integration with Euler s Method This demo uses Euler s method to locate a stationary solution of the following predator prey system with harvesting p lt Uy pau 1 41 urug pi 1 e Paul Uy Ug P4U U2 14 18 where all problem parameters have a fixed value The equations are the same as those in demo pp2 The continuation parameter is the independent time variable namely PAR 14 Note that Euler time integration is only first order accurate so that the time step must be sufficiently small to ensure correct results Indeed this option has been added only as a convenience and should generally be used only to locate stationary states Note that the AUTO constants DS DSMIN and DSMAX control the step size in the sp
90. user defined function UZR to detect intersections with the plane Dp 0 01 We get among other output BR PT TY LAB PAR 3 L2 NORM PAR 1 PAR 2 1 22 LP 20 1 08120E 02 5 32589E 00 5 67363E 00 6 60818E 02 1 31 UZ 21 1 00000E 02 4 81969E 00 5 18032E 00 6 38551E 02 the first line of which represents the Dp value at which the homoclinic curve P has a tangency with the family t of fold bifurcations Beyond this value of Do P consists entirely of saddle homoclinic orbits The data at label 20 reproduce the coordinates of the point D The results of this computation and a similar one starting from D in the opposite direction with DS 0 01 are displayed in Figure 22 0 Detailed AUTO Commands 223 AUTO COMMAND ACTION mkdir mtn cd mtn demo mtn create an empty work directory change directory copy the demo files to the work directory ri run mtn c mtn 1 sv 1 continue saddle node homoclinic orbit from mnt dat save output files as b 1 s 1 d 1 ri ritrun ri 1 c mtn 2 pape 1 continue in opposite direction restart from label 1 append output files to b 1 s 1 d 1 ri ri run r1 0UZ1 c mtn 3 ap 1 gt switch to saddle homoclinic orbit restart 1st UZ append output files to b 1 s 1 d 1 r4 run ri c mtn 4 sv 4 continue in reverse direction restart from last label save output files as b 4 s 4 d 4 ri ritrun r1 UZ3 c mtn 5
91. with bvp relabel bvp are not part of the loop because they are indented differently This can be confusing first time users of Python but it has the advantage that the code is forced to have a consistent indentation style The next command in the script bp load solution ISW 1 NTST 50 loads a solution with modified AUTO constants All other constants are the same as they were in the first run The ISW value is changed to 1 see Section 10 8 4 so that a branch switch is performed and the NTST value is changed to 50 see Section 10 3 1 Only in memory versions of the AUTO constants are modified the original file c bvp is not modified Some diagnostics are then printed to the screen using a standard Python print command the label number of the branch point that we switch at can be found by using bp LAB In addition as can be seen in Figure 4 10 the character is the Python comment character When the Python interpretor encounters a character it ignores everything from that character to the end of the line We then use a fw run bp command to perform the calculation of the bifurcating branch from solution bp We print additional information and use the command bw run bp DS to change the AUTO initial step size from positive to negative which causes AUTO to compute the bifurcating branch in the other direction see Section 10 5 1 This output is appended to the existing output in the Python variable bvp after some mo
92. work directory ppp run e ppp c ppp compute stationary solutions detect Hopf bifurcations ppp ppptrun ppp HB2 IPS 2 ICP 1 11 compute a family of periodic solutions ILP 0 NMX 15 NPR 50 DS 0 1 DSMAX 0 5 save ppp ppp save the output to b ppp s ppp d ppp hb run ppp HB2 ICP 1 4 ILP 0 compute Hopf bifurcation curves ISW 2 NMX 100 RL1 0 58 DSMAX 0 1 save hb hb save the output files as b hb s hb d hb Table 14 13 Commands for running demo ppp 142 14 7 plp Fold Continuation for Periodic Solutions This demo which corresponds to computations in Doedel Keller amp Kern vez 1991a shows how one can continue folds on a family of periodic solutions in two parameters The calculation of a locus of Hopf bifurcations is also included The equations that model a one compartment activator inhibitor system Kern vez 1980 are given by s so 5 pR s a a a ao a pR s a oe where a en A 0 R s a EE k gt The free parameter is p In the Hopf and fold continuations the parameter s is also free The computed loci of Hopf points and folds suggest the existence of isolas of periodic solutions The computation of one such isola is also included in this demo All calculations can be carried out by running the Python script plp auto included in the demo 143 148 phil Phase Shifting using Continuation This demo whic
93. x is drawn on the X axis 2 is selected for Y which indicates that y is represented on the Y axis 3 is selected for Z which indicates that z is represented on the Z axis Figure 8 9 Displaying multiple components We can also show multiple combinations at the same time For example if we want to show x y z and x y z in the same diagram we can input 1 4 in the X dropdown list to select x and 2 being drawn on the X axis input 2 5 in the Y list to show y and y on the Y axis and input 3 6 in the Z dropdown list to draw z and z on the Z Axis Note that after finishing the input in the dropdown list box we must type ENTER for the input to be accepted by the system Figure 8 9 shows the results of the above choices The combination is flexible For example if X is 1 Y is 3 5 and Z is 4 5 6 the system will automatically reorganize them to 1 3 4 1 5 5 1 3 6 and show the results If X is 1 5 Y is 2 and Z is 3 4 the system reorganizes them to 1 2 3 5 2 4 Different components are drawn with different colors from blue to red The default values can be set in the resource file If no resource file exists then the system will use 1 for X axis 2 for Y axis and 3 for Z axis for both the solution and the bifurcation diagrams 83 8 1 10 Choosing labels From the Label list we can choose the label of the solution to be drawn I
94. 0 30 0 50 0 70 0 90 x T Figure 26 3 Two R2 reversible homoclinic orbits at P 1 6 corresponding to labels 1 smaller amplitude and 5 larger amplitude Ls 0 00 0 20 0 40 0 60 0 80 1 00 0 10 0 30 0 50 0 70 0 90 x T Figure 26 4 An Ro reversible homoclinic orbit at label 8 247 Chapter 27 HomCont Demo Homoclinic branch switching This demo illustrates homoclinic branch switching which is an implementation of Lin s method Lin 1990 Sandstede 1993 Yew 2001 as described in Oldeman et al 2003 We use a direct branch switching method to switch from 1 to 2 and 3 homoclinic orbits near an inclination flip bifurcation in a model due to Sandstede which was introduced in Chapter This also shows how to obtain a homoclinic orbit through continuation of a periodic orbit born at a Hopf bifurcation Thereafter we illustrate homoclinic branch switching for the FitzHugh Nagumo equations and a 5th order Korteweg De Vries model The equation files in these demos are written in C 27 1 Branch switching at an inclination flip in Sand stede s model Consider the system Sandstede 1995a i ar by ax azr 2 3x y bat ay x bx ay az2y 27 1 2 c pur 3rz 0 2 1 2 y as given in the file sib c where for simplicity we have set 7 0 6 1 and y 3 We study an inclination flip that exists for a 0 375 b 0 625 and c 0 75 This
95. 00470E 01 2 70847E 01 2 29 CP 12 2 02768E 12 1 00472E 04 1 00472E 04 3 02837E 08 2 40 13 9 09414E 02 3 56925E 01 3 56925E 01 3 82187E 01 2 60 14 5 73716E 01 6 59512E 01 6 59512E 01 1 30487E 00 2 80 15 1 68023E 00 9 43582E 01 9 43582E 01 2 67104E 00 2 85 UZ 16 1 99995E 00 9 99992E 01 9 99992E 01 2 99995E 00 The CLUI was used to generate the constants file at runtime In the example below the constant file c cusp will be read in and the CLUI will be used to make the appropriate changes to perform the calculation AUTO COMMAND ACTION cusp load cusp load the problem definition cusp mu run cusp execute the run mu mu run cusp DS execute the run backwards and append the results to mu mu rl mu relabel solutions in mu save mu mu save the results in the files b mu s mu and d mu lpi load mu LP1 ISW 2 use the first fold LP in mu as the restart solution and change ISW to 2 cusp run 1p1 execute the third run of demo cusp cusp cusp run 1p1 DS execute the fourth run of demo cusp save cusp cusp save the results in the files b cusp s cusp and d cusp Table 12 4 Selected runs of demo cusp 12 5 Plotting the Results with AUTO The bifurcation diagram computed in the runs above was stored in the files b mu and b cusp while each labeled solution is fully stored in s mu and s cusp To use AUTO to graphically inspect these data files type the AUTO comm
96. 150 14 13 phs Effect of the Phase Condition o o o 02004 151 ate paisa 152 14 15 r3b The Circular Restricted 3 Body Problem CR3BP 153 PONEN 153 se 154 15 AUTO Demos BVP 161 a a e o a ae a 161 15 2 int Boundary and Integral Constraints 162 15 3 bvp A Nonlinear ODE Eigenvalue Problem 163 15 4 lin A Linear ODE Eigenvalue Problem 0 164 hea ey dde ae eres de e eG Stas ee E a 165 15 6 kar The Von Karman Swirling Flows 02 0004 166 15 7 Spo A Singularly Perturbed BVP x asi be oe eee OH ee 167 15 8 ezp Complex Bifurcation in a BVP 2 2 24 4 0 dae ee fae aw ne ee 168 da 169 15 10 um3 A 2D unstable manifold in 3D ges ek osos ero OS 170 15 11 p2c Point to cycle connections o o a a a e a eee eg eds 171 15 12 c2c Cycle to cycle connections o oa 64 a a a 4 hoe RRS YE mE 171 15 13 pcl Lorenz Point to cycle connections with Lin s method Mid epee ee e ey a 174 ipa thts a 3 175 ete hee eee ay 176 15 15 fnc Canards in the FitzHugh Nagumo system 178 16 AUTO Demos Parabolic PDEs 181 16 1 pdi Stationary States 1D Problem o 182 16 2 pd2 Stationary States 2D Problem 183 16 3 wav Periodic IT 184 16 4 bre Chebyshev Collocation in Space o o 185 side oe oe Ae A A A a
97. 2 If you did the optional computation see Remark 2 you may need to change the label of the restart solution autox data py flq n startman data get flq n where n is the different label number The extracted data may be saved in a file called s startman which contains a new starting solution that can be used as a base for the manifold computations The orbit coordinates are at time zero and the Floquet eigenfunction are saved at PAR 25 30 and PAR 31 36 respectively OR man Lia O startman startLia run startman e man c man Lia 0 This step does a time integration using continuation in the period T i e PAR 11 which here is the integration time The labeled solutions from this run all correspond to the same orbit except that the orbit gets longer and longer The starting point of the orbit is the point on the periodic orbit at time zero plus a small distance into the direction of the unstable manifold In AUTO e corresponds to PAR 6 This parameter e is initialized via the script ext py The sign of e is significant The parameters in this run see c man Lla 0 are PAR 3 energy PAR 21 x coordinate at end point PAR 11 integration time PAR 22 y coordinate at end point PAR 12 length of the orbit PAR 23 z coordinate at end point sv startLia save startLia startLila Save the results in b startLla s startLla and d startLla Table 14 22 Detailed AUTO sh
98. 27 3 The 2 homoclinic orbit as a is changed a The two different 3 homoclinic orbits b and is depicted in Figure 27 3 a Next we aim to find a 3 homoclinic orbit To do so we restart at the inclination flip point at label 16 and set ITWIST 3 Moreover we need to continue in one more gap 2 PAR 24 and once again stop when PAR 22 0 Note that the dimension of the boundary value problem we continue is now equal to 12 This is not to be confused with the setting of the AUTO constant NDIM 3 because HomCont handles this internally r10 run r6 UZ1 BR Www w PT TY LAB 10 26 20 27 30 28 38 UZ 29 PAR 21 3 45896E 01 2 73699E 01 1 73719E 01 plot r6 ICP 21 22 24 5 IPSI save ri0 10 1 01451E 01 PAR 22 7 87894E 07 2 91126E 05 4 42289E 03 2 00000E 01 NMX 300 NPR 10 UZR 22 0 2 PAR 24 6 42157E 07 6 51591E 07 1 44090E 04 6 97445E 02 ISTART 3 PAR 5 1 06346E 11 1 63655E 09 3 10188E 05 1 48615E 02 The output is stored in b 10 s 10 and d 10 Now we need to subsequently close the Lin gaps Our strategy is to keep T fixed We first continue in a u e and until e 0 rii run r10 ICP 4 5 22 24 NPR 310 DS 0 01 DSMAX 0 01 UZR 22 0 0 4 0 082 BR 3 3 PT TY LAB 6 UZ 30 32 UZ 31 PAR 4 8 20000E 02 1 98414E 01 r 6 r6 r11 PAR 5 1 29790E 02 6 05495E 03 253 PAR 22 1 76995E 01 2 30717E
99. 3 in s pp3 d display the orbits uz versus u1 2d enter 2D mode for plotting labeled solutions 16 17 18 19 select these orbits d default orbit display uw versus time 23 select columns 2 and 3 in s pp3 d phase plane display uz versus u1 24 select columns 2 and 4 in s pp3 d phase plane display uz versus u1 ex exit from 2D mode end exit from PLAUT Table 14 15 Plotting commands for demo pp3 AUTO COMMAND ACTION p hb or pp hb run PLAUT PyPLAUT to graph the contents of b hb and s hb PLAUT PyPLAUT COMMAND ACTION do set defaults ax select axes 16 select real columns 1 and 6 in b hb bdo plot the bifurcation diagram pa versus p end exit from PLAUT Table 14 16 Plotting the Hopf loci for demo pp3 146 14 10 tor Detection of Torus Bifurcations This demo uses a model in Freire Rodriguez Luis Gamero amp Ponce 1993 to illustrate the detection of a torus bifurcation It also illustrates branch switching at a secondary periodic bifurcation with double Floquet multiplier at z 1 The computational results also include folds homoclinic orbits and period doubling bifurcations Their continuation is not illustrated here see instead the demos plp pp2 and pp3 respectively The equations are x t 8 v x By azz b3 y ay r bx B y y z baly x 14 13 z t y where y 0 6 r 0 6 a3 0 328578 and b3 0 933578 I
100. 5 2nd run compute a few steps along the bifurcating family save rit r2 lin save all output to b lin s lin d lin r3 run r2 UZ1 ICP 1 2 NTST 5 ISW 1 3rd run compute a two parameter curve of eigenvalues save r3 2p save the output files as b 2p s 2p d 2p Table 15 4 Commands for running demo lin 164 15 5 non A Non Autonomous BVP This demo illustrates the continuation of solutions to the non autonomous boundary value prob lem uUi U2 as au Ug Ppie gt 15 5 with boundary conditions u 0 0 ui 1 0 Here x is the independent variable This system is first converted to the following equivalent autonomous system u pe 15 6 ug L with boundary conditions u 0 0 wi 1 0 u3 0 0 For a periodically forced system see demo frc AUTO COMMAND ACTION mkdir non create an empty work directory cd non change directory demo non copy the demo files to the work directory ri run e non c non compute the solution family save ri non save output files as b non s non d non Table 15 5 Commands for running demo non 165 15 6 kar The Von Karman Swirling Flows The steady axi symmetric flow of a viscous incompressible fluid above an infinite rotating disk is modeled by the following ODE boundary value problem Equation 11 in Lentini amp Keller 1980 ui Tug us Tus uz T 2yu
101. 987 up D L uge uv B 1 u A ve Dy L vrs uv Bu with boundary conditions u 0 t u 1 t A and v 0 t v 1 t B A Note that given the non adaptive spatial discretization the computational procedure here is not appropriate for PDEs with solutions that rapidly vary in space and care must be taken to recognize spurious solutions and bifurcations 16 3 AUTO COMMAND ACTION mkdir brc create an empty work directory cd bre change directory demo brc copy the demo files to the work directory ri run e brc c brc compute the stationary solution family with Hopf bifurcations r2 run r1 HB1 IPS 2 ICP 5 11 compute a family of periodic solutions from the first Hopf point r3 run r2 BP1 ISW 1 compute a solution family from a sec ondary periodic bifurcation save r1 r2 r3 brc save all output to b brc s brc d brc Table 16 4 Commands for running demo bre 185 16 5 brf Finite Differences in Space This demo illustrates the computation of stationary solutions and periodic solutions to systems of parabolic PDEs in one space variable A fourth order accurate finite difference approximation is used to approximate the second order space derivatives This reduces the PDE to an autonomous ODE of fixed dimension which AUTO is capable of treating The spatial mesh is uniform the number of mesh intervals as well as the number of equations in the PDE system can be
102. AR 1 ae U 1 U 2 U 3 1 1 EP 1 0 00000E 00 6 66667E 01 0 00000E 00 0 00000E 00 1 5 UZ 2 3 75000E 01 6 66667E 01 1 33333E 01 0 00000E 00 The output is saved in the files b 1 s 1 and d 1 Next we continue in a PAR 4 until a Hopf bifurcation is found r2 run r1 ICP 4 save r2 2 or alternatively rn c sib 2 s 1 sv 2 BR PT TY LAB PAR 4 one U 1 U 2 U 3 1 6 HB 3 3 18429F 01 6 54375E 01 1 34754E 01 7 70102E 02 The output is saved in the files b 2 s 2 and d 2 This Hopf bifurcation can then be continued into a periodic orbit The periodic orbit eventually reaches a homoclinic bifurcation We continue in y PAR 5 and PAR 11 which corresponds to the period and stop when the period is equal to 35 r3 run r2 HB1 IPS 2 ICP 5 11 NMX 200 DS 0 01 DSMAX 0 01 UZR 11 35 save r3 3 BR PT TY LAB PAR 5 L2 NORM ee PERIOD 3 10 5 2 41881E 03 6 70569E 01 1 08975E 01 3 40 8 1 29495E 02 6 14547E 01 1 41297E 01 3 81 UZ 13 1 04657E 04 4 01829E 01 3 50000E 01 249 The output is saved in the files b 3 s 3 and d 3 Note that y first decreases and then increases towards 0 which is precisely what we expect in this model as homoclinic orbits occur on the line u 0 in the a 1 plane It is now instructive to look at a phase space diagram to see what is going on plot r3 Selecting solution for Type 5 6 7 8 9 10 11 12 13 for Label U 1 for X an
103. ART 2 an explicit homoclinic solution can be specified in the routine STPNT in the usual AUTO format that is U T where T is scaled to lie in the interval 0 1 The choice ISTART 3 allows for a homotopy method to be used to approach a homoclinic orbit starting from a small approximation to a solution to the linear problem in the unstable manifold Doedel Friedman Monteiro 1993 For details of implementation the reader is referred to Section 5 1 2 of Champneys amp Kuznetsov 1994 under the simplification that we do not solve for the adjoint u t here The basic idea is to start with a small solution in the unstable manifold and perform continuation in PAR 11 27 and dummy initial condition parameters in order to satisfy the correct right hand boundary conditions which are defined by zeros of other dummy parameters w More precisely the left hand end point is placed in the tangent space to the unstable manifold of the saddle and is characterized by NUNSTAB coordinates satisfying the condition l E care E0 where y is a user defined small number At the right hand end point NUNSTUB values w measure the deviation of this point from the tangent space to the stable manifold of the saddle Suppose that IEQUIB 0 1 and set IP 12 IEQUIB NDIM Then PAR IP amp PAR IP i amp i 1 2 NUNSTAB PAR IP NUNSTAB i wi i 1 2 NUNSTAB 211 Note that to avoid interference with the test functio
104. AUT04 for the graphical inspection of the output files fort 7 and fort 8 Type plot3 r3b True to run PLAUT04 in restricted three body problem mode Aliases p3 commandPlotter3D 4 14 3 File manipulation copy Copy data files Type copy namel name2 or copy namel name2 name3 or copy name1 name2 name3 name4 to copy the data files dir1 c xxx dir1 b xxx dir1 s xxx and dirl d xxx to dir2 c yyy dir2 b yyy dir2 s yyy and dir2 d yyy Type copy xxx yyy to copy the data files c xxx d xxx b xxx and h xxx to C yyy d yyy b yyy and h yyy The values of dir1 xxx and dir2 yyy are as follows depending on whether namel is a directory or name2 is a directory copy name1 name2 no directory names namel and name2 namel is a directory name1 name2 and name2 name2 is a directory namel and name2 namel 56 copy name1 name2 name3 namel is a directory namel name2 and name3 name2 is a directory namel and name2 name3 copy name1 name2 name3 name4 namel name2 and name3 name4 Aliases cp commandCopyDataFiles move Move data files to a new name Type move name1 name2 or move name1 name2 name3 or move name1 name2 name3 name4 to move the data files dirl b xxx dir1 s xxx and dir1 d xxx to dir2 b yyy dir2 s yyy and dir2 d yyy and copy the constants file dirl c xxx to dir2 c yyy The values of dirl xxx and dir2 yyy are determined in the same way as for
105. AUTO O7P CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS Eusebius J Doedel and Bart E Oldeman Concordia University Montreal Canada with major contributions by Alan R Champneys Bristol Fabio Dercole Milano Thomas Fairgrieve Toronto Yuri Kuznetsov Utrecht Randy Paffenroth Pasadena Bjorn Sandstede Brown Xianjun Wang and Chenghai Zhang January 2012 Contents 1 Installing AUTO MA lt 5 ee oS ee ee 4 eh we hk ES ee 2 SE Oe 1 1 1 Installation on Linux Unix 24642445 era ERS 1 1 2 Installation on Mac OS X sara CA ee ee Se ee a 1 1 3 Installation on Windows o lt 6 45544 864 2 es 1 2 Restrictions on Problem SE 290604 0 a o a 1 3 Compatibility with Earlier Versions eee eee 1 4 Parallel Version ce oe ia ee pS a oe toed WE He RA OE OR SRO SOUR Boe a Red 2 2 Algebraic Systems fc eee eed REE EDRE REDE AA E ech a wey Sine es cee ale e Re ZA Poarabone PODES lt lt 644484 4 K 4 45 Re ROR MER do cdo 25 Dis cretiZati n x a Be ROR we SOR oe He GO ee SG we Be ER a eee rer oi 3 2 The Constants File Cs 5 3 2 ean eS Mee oe ee e A SS 3 3 User supplied Routines sois cer proa AAA 3 4 User Supplied Derivatives xa att a a a 4 1 Typographical Conventions 22d a EOS pi AA AA AAA eS 4 3 First Example vicio Bend e E A AR eee de ik ES EIA A oe y da DA A ee E A 4 6 Extending the AUTO CLUI J suse sii a a BS 4 7 Bifurcation Di
106. BT 221 Bogdanov Takens bifurcation on fold curve algebraic systems BT 31 Bogdanov Takens bifurcation on Hopf curve CP 222 Cusp bifurcation on fold curve algebraic systems GH 32 Generalized Hopf bifurcation on Hopf curve ZH 13 Zero Hopf on BP curve algebraic systems ZH 23 Zero Hopf on Fold curve algebraic systems ZH 33 Zero Hopf on Hopf curve R1 25 1 1 resonance on Fold maps R1 55 1 1 resonance on Fold periodic solutions R1 85 1 1 resonance on Torus periodic solutions maps R2 76 1 2 resonance on PD periodic solutions maps R2 86 1 2 resonance on Torus periodic solutions maps R3 87 1 3 resonance on Torus periodic solutions maps R4 88 1 4 resonance on Torus periodic solutions maps LPD 28 Fold flip bifurcation on Fold maps LPD 78 Fold flip bifurcation on PD maps LTR 23 Fold torus bifurcation on Fold maps LTR 83 Fold torus bifurcation on Torus maps PTR 77 Flip torus bifurcation on PD maps PTR 87 Flip torus bifurcation on Torus maps TTR 88 Torus torus bifurcation on Torus maps Table 6 2 Codimension two solution types Note that the absolute value of the numerical code divided by 10 gives the type of the curve on which the special point occurs 74 spected The amount of diagnostic data can be controlled via the AUTO constant IID see Section 10 9 9 The user ha
107. CP 2 8 UZR 2 3 0 get the AUTO constants file and run AUTO HomCont restart from the last la bel in r8 r6 r6 r9 append output to the Python variable r6 save r6 6 save output to the files b 6 s 6 d 6 R san 9 6 use the constants file c san 9 start from the file s 6 ap 6 append output to the files b 6 s 6 d 6 Table 20 2 These two sets of AUTO Commands behave similarly 213 Chapter 21 HomCont Demo san 21 1 Sandstede s Model Consider the system Sandstede 1995a t ax by ax f az x 2 32 y be ay ibs fary f az 2y 21 1 2 ce pr yoy ap z 1 2 y as given in the file san f90 Choosing the constants appearing in appropriately allows for computing inclination and orbit flips as well as non orientable resonant bifurcations see Sandstede 1995a for details and proofs The starting point for all calculations is a 0 b 1 where there exists an explicit solution given by x t y t 2 t 5 4e o This solution is specified in the routine STPNT 21 2 Inclination Flip We start by copying the demo to the current work directory and running the first step demo san san load san IPS 9 NDIM 3 ISP 0 ILP 0 ITNW 7 JAC 1 NTST 35 IEQUIB 0 DS 0 05 ri run san ICP 1 8 UZR 1 0 25 This computation starts from the analytic solution above with a 0 b 1 c 2 a 0 B 1l and y u 0 The homoclinic s
108. E 102 TOGA A AAA 102 iaa a a a e a ee a ae de A 102 MA AA 102 de aes De oe te Be Os es AT Se ge et rege ee ds Bw He 103 MA API 103 WEA Fixed points e a hte RR A oe SG 103 Ge dh ne Mee ee ee oe ee ed 103 ee o as ee 103 10 7 5 Folds and period doublings 2244668 64444 94 e 404 104 bb SS A WS OE O ee ee a h 104 ARA Ge REE RE ERE H ES 104 fb teh RD HE ia BE MOS e Sod 104 bs prt ok BNR te ah hd Ke Rw 105 10 7 10 Parameter overspecitication 64 24 40 es6b Uae ea eases 105 10 8 Computation Constants 2 2424 4 424h6424 044 4 24 a G 105 OSL IER a aa Se he ee oy ee ee ne es Dd ee ee oh we a 105 ee ee ee ee ee ee a ere ee S 106 GW xe ecu ae Gk WP oe es Gaeta em Gee ee a eee 106 We 6er aa o ee eR A ee RA 106 MAA AA poe ee TA 107 AT EEE NS 107 Mer asa a e a a A 107 A A a a e ia Ea e A 107 TOSO PAR AA 108 Mo IA 108 TEST aa aa ore A AR 108 MERA AE AA 108 10 9 Q tp t ARE 110 A ado pa a eee ee oo eee A dd daa EE ee e SA a A be edn ean ont Robes eat estes ts eo Ia oa ot homer aneebapee dane ad oa euve ee E A ee ee ee eee a ee Ge ha Ao bdo Rhee he eh a Pao eee Oe e oe Chay ooo ceuseceeyecea sees e eT A TOD T2UZSTOR coo artos idos Sede a Gen Ge we Bene Ste a A g 10 10 Quick reference sciences dea a a HG 11 Notes on Using AUTO 11 1 Restrictions on the Use of PAR 0 002020 2 0000002 eee eee 11 2 CTO I es eea piana e e Oe ee Hk SB wo ee ee oe ee a A Sse sas Be ee eT ES a in ee se E a boy ea eres ee C
109. I script The source for this script can be found in AUTO_DIR demos python demo2 auto demo bvp bvp run bvp Figure 4 9 The first part of the complex AUTO CLUI script 32 which represents a list of all branchpoint solutions Accordingly this list is stored in the Python variable branchpoints Note variables in Python are different from those in languages such as C in that their type does not have to be declared before they are created branchpoints bvp BP for solution in branchpoints bp load solution ISW 1 NTST 50 Compute forwards print Solution label bp LAB forwards fw run bp Compute backwards print Solution label bp LAB backwards bw run bp DS both fw bw merged merge both bvp bvp merged Figure 4 10 The second part of the complex AUTO CLUI script The next command for solution in branchpoints is the Python syntax for loops The branchpoints object is a list of the branch point solutions from the first run The command for solution in branchpoints is used to loop over all solutions in the branchpoints variable by setting the variable solution to be one of the solutions in branchpoints and then calling the rest of the code in the block Python differs from most other computer languages in that blocks of code are not defined by some delimiter such as END DO in Fortran but by indentation In Figure 4 8 the commands starting
110. IPS 2 ICP 3 11 NMX 150 RLO 0 9 UZR uzr save r1 r2 0 set variable for UZR compute a family of periodic solutions restart from r1 save output to b 0 s 0 d 0 icp 3 11 12 22 22 23 31 r3 run r2 UZ1 IPS 15 ILP 0 ICP icp ISP 2 NMX 25 ITNW 7 DS 0 05 set variable for ICP locate a 1 parameter extremum as a bifur cation restart from r2 r4 run r3 BP1i ISW 1 ISP 0 NMX 5 save r3 r4 1 switch branches to generate optimality starting data restart from r3 save output to b 1 s 1 d 1 icp 1 3 2 11 set variable for ICP uzr 22 0 0 set variable for UZR r5 run r4 ICP icp ISW 1 NMX 150 compute 2 parameter family of 1 RLO 0 8 RL1 1 9 DS UZR uzr parameter extrema restart from r4 save r5 2 save the output files as b 2 s 2 d 2 icp 2 4 1 11 set variable for ICP r6 run r5 UZ4 IRS 15 ICP icp NTST 50 compute 3 parameter family of 2 UZR 1 0 1 0 05 0 01 0 005 0 001 parameter extrema restart from r5 save r6 3 save the output files as b 3 s 3 d 3 Table 17 5 Commands for running demo ops 193 17 3 obv Optimization for a BVP This demo illustrates use of the method of successive continuation for a boundary value opti mization problem A detailed description of the basic method as well as a discussion of the specific application considered here is given in Doedel Keller amp Kern vez 1991b The required e
111. IST has just been increased from zero then AUTO will automatically generate the initial solution to the adjoint In this case a dummy Newton step should be performed see Section 20 7 for more details 20 3 5 ISTART ISTART 1 No special action is taken ISTART 2 If IRS 0 an explicit solution must be specified in the subroutine STPNT in the usual format ISTART 3 The homotopy approach is used for starting see Section for more details Note that this is not available with the choice IEQUIB 2 ISTART 4 A phase shift is performed for homoclinic orbits to let the equilibrium either fixed or non fixed depending on IEQUIB correspond to t 0 and t 1 This is necessary if a periodic orbit that is close to a homoclinic orbit is continued into a homoclinic orbit ISTART 5 default If IRS 0 the restart solution comes from a data file or the restart solution is a homoclinic orbit with problem type IPS 9 no special action is taken as for ISTART 1 For other problem types use a phase shift as for ISTART 4 ISTART N N 1 2 3 Homoclinic branch switching this description is for refer ence only We refer to the demo in Chapter to see how this can be used in actual practice and to Oldeman Champneys amp Krauskopf 2003 for theory and background The orbit is split into N 1 parts and AUTO sees it as an N 1 xNDIM dimensional object The first part uy goes from the equilibrium to the point xy that is furthest from
112. O is installed in your home directory the following commands set your environment variables so that you will be able to run the AUTO commands and may be placed into your login profile or cshrc file as appropriate If you are using a sh compatible shell such as sh bash ksh or ash enter the command source HOME auto 07p cmds auto env sh On the other hand if you are using a csh compatible shell such as csh or tcsh enter the command source HOME auto 07p cmds auto env The Graphical User Interface GUI94 requires the X Window system and Motif or LessTif Note that the GUI is not strictly necessary since AUTO can be run very effectively using the Unix Command Language User Interface CLUI Moreover long or complicated sequences of AUTO calculations can be programmed using the alternative Python CLUI The GUI is not compiled by default To compile AUTO with the GUI type configure enable gui and then make in the directory auto 07p To use the Python CLUI and the App PyPLAUT plotter it is strongly recommended to install NumPy eee TkInter and Matplotlib sourceforge net Note that Matplotlib 0 99 or higher is recommended because it supports 3D plotting in addition to 2D plotting Python itself needs to be at least version 2 3 or higher 11 but 2 4 or higher is strongly recommended and required for NumPy and Matplotlib As of this writing Python 3 x is not yet supported by Matplotlib and therefore not recommended For enhanc
113. ORM ws PAR 3 1 1 EP 20 2 33846E 00 7 50835E 00 2 57829E 07 This 3 homoclinic orbit is depicted in Figure 2 006 01 6 006 01 1 006 00 Columns t Figure 27 6 A 3 homoclinic orbit in a 5th order Hamiltonian Korteweg De Vries model 260 Bibliography Alexander J C Doedel E J amp Othmer H G 1990 On the resonance structure in a forced excitable system SIAM J Appl Math 50 No 5 1373 1418 Anderson E Bai Z Bischof C Blackford S Demmel J Dongarra J Du Croz J Greenbaum A Hammarling S McKenney A amp Sorensen D 1999 LAPACK Users Guide third edn Society for Industrial and Applied Mathematics Philadelphia PA Aronson D G 1980 Density dependent reaction diffusion systems in Dynamics and Mod elling of Reactive Systems Academic Press pp 161 176 Bai F amp Champneys A 1996 Numerical detection and continuation of saddle node homo clinic orbits of codimension one and codimension two J Dyn Stab Sys 11 327 348 Beyn W J 1990 The numerical computation of connecting orbits in dynamical systems IMA J Num Anal 9 379 405 Beyn W J amp Doedel E J 1981 Stability and multiplicity of solutions to discretizations of nonlinear ordinary differential equations SIAM J Sci Stat Comput 2 1 107 120 Buryak A amp Akhmediev N 1995 Stability criterion for stationary bound states of solitons wit
114. PAR 1 is a shortcut for bd 0 PAR 1 Point ba 0 0 ba 1 5 ba 1 UZ1 bd 1 b bd UZ1 b Point list bd 0 UZ bd 0 1 2 balo UZ HB1 7 Solution bd 5 bd UZ1 bd 0 Solution list s bd s UZ bd UZ bd 1 2 bd UZ HB1 7 Solution column bd UZ1 t bd UZ1 U 1 7 Solution measures bd UZ1 L2 NORM This example is a shortcut for bd UZ1 b L2 NORM here you can use any column name from the bifurcation diagram Solution point bd UZ1 0 bd UZ1 0 Here the bd UZ1 t notation gives the point at time t Solution AUTO constants bd 5 c bd UZ1 c bd 0 c These constants are copied from the corresponding branch constants removing the con stants IRS PAR U sv s and dat because those constants need to change between runs The AUTO constant IRS is automatically set to the solution label 45 4 8 Importing data from Python or external tools A solution can be created from a Python list or Numerical Python array using the load com mand The syntax is slightly different depending on whether you create a point or an orbit Use s load u PAR p where u is a Python list or a numpy array representing a solution For a point such an array is just the point itself for example u x y z oru 0 0 O For an orbit such an array must be giv
115. PD1 c pen 2 a family of period doubled and out of phase rotations Constants changed IPS NTST ISW NMX append output to bifurcation diagram ob ject pen pen pentrun pen BP1 c pen 3 a secondary bifurcating family without bi furcation detection Constants changed IRS ISP append output to bifurcation diagram ob ject pen pen pentrun pen BP2 c pen 4 save pen pen another secondary bifurcating family without bifurcation detection Con stants changed IRS append output to bifurcation diagram object pen save pen to output files b pen s pen d pen t run pen PD1 c pen 5 generate starting data for period doubling continuation Constants changed IRS IcP ICP ISW NMX pd run t sv pd compute a locus of period doubling bifur cations restart from t Constants changed IRS save output files as b pd s pd d pd Table 14 18 Commands for running demo pen 149 14 12 chu A Non Smooth System Chua s Circuit Chua s circuit is one of the simplest electronic devices to exhibit complex behavior For related calculations see Khibnik Roose amp Chua 1993 The equations modeling the circuit are ul a uz h u1 Uy u U2 Uz 14 16 uz P u where 1 h a ax 5 ao 01 2 1 2 1 and where we take 8 14 3 ay 1 7 a 2 7 Note that h x is not a smooth functio
116. PS 9 The length in time of the truncated homoclinic or heteroclinic orbit is stored in PAR 11 For the adjoint variational equations PAR 10 is used The equilibria are stored in PAR 11 to PAR 11 NDIM 1 PAR 11 2 NDIM 1 homoclinic heteroclinic Test function values may be stored in PAR 21 to PAR 36 Homoclinic branch switch ing uses PAR 20 and higher to store time intervals and gap sizes IPS 11 12 The wave speed is in PAR 10 and the diffusion constants in PAR 15 16 The period for periodic solutions and at Hopf bifurcations is stored in PAR 11 IPS 14 16 AUTO uses PAR 14 for the time variable and the diffusion constants are in PAR 15 16 The period is stored in PAR 11 The previous time for each step is stored in PAR 12 IPS 17 The diffusion constants are in PAR 15 16 The period is stored in PAR 11 116 IPS 15 Only PAR 1 9 should be used for problem parameters PAR 10 is the value of the objective functional PAR 11 the period PAR 12 the norm of the adjoint variables PAR 14 and PAR 15 are internal optimality variables PAR 21 29 and PAR 31 are used to monitor the optimality functionals associated with the problem parameters and the period 11 2 Efficiency In AUTO efficiency has at times been sacrificed for generality of programming This applies in particular to computations in which AUTO generates an extended system for example com putations with ISW 2 However the user has signifi
117. Point Size 1 0 Set the initial maximum and minimum satellite animation speed Sat Animation Speed 100 Sat Max Animation Speed Sat Min Animation Speed 100 0 Set the initial maximum and minimum orbit change animation speed Orbit Animation Speed 50 Orbit Max Animation Speed Orbit Min Animation Speed 100 0 Set the active AUTO parameter indices parameter 1D 10 Choose 3D or 2D graph 3D Yes Choose 3D or 2D graph for the bifurcation diagram 3DBif Yes Choose 3D or 2D graph for the solution diagram 3DSo1 Yes 8 3 Example In this example we want to view a CR3BP data set We want the diagram to show the x E component on the X axis y component on the Y axis and z component on the Z axis for the solution diagram In the CR3BP we use the parameters 1 2 3 10 21 22 23 in the AUTO calculations and we also want to be able to use these to color the diagram so we set the parameter indices Other preferences include The diagram is drawn using Tubes Coordinate axes are not drawn No animation Reference plane libration points and primaries are drawn 89 e All labels are shown e Data is not normalized The settings are the settings in the resource file are then as follows Initialize the default options Draw Reference Plane Yes Orbit Animation No Satellite Animation No Draw Primaries Yes Draw Libration Points Yes Normalize Data
118. Python script LlaX auto does the same as Lla auto but with additional calculations that generate additional AUTO data files e g to detect heteroclinic connections Some of these additional runs take quite a bit of CPU time and generate big data files Use auto clean auto auto clean auto to remove all generated files REMARK 1 If the run to compute the Floquet eigenfunction is not successful i e if PAR 5 does not become nonzero then try to compute the Floquet eigenfunction in more stages as in Table REMARK 2 One can also follow the orbit its multiplier and eigenfunction as in Table The instructions below are for the Halo family H1 in AUTO demo Hla Follow the instructions for Lla above where you replace L by H throughout for instance you can run everything in one go using auto Hla auto auto Hla auto or with the extra calculations auto HlaX auto auto H1aX auto The Floquet eigenfunction is now computed from label 7 with step size 10 The detailed commands are likewise except for the manifold calculations in Table 14 23 and those are given in Table 14 26 The instructions below are for the Halo family H1 in AUTO demo H1b Follow the instructions for Lla above where you replace L by H and a by b throughout for instance you can run everything in one go using auto Hib auto auto H1b auto or with the extra calculations auto H1bX auto auto H1bX auto The Floquet eigenfunc
119. R the 2nd 4th 6th etc are deleted from the files The commands CRD and RD for case insensitive file systems are equivalent to rd above but are faster though not reliable when interrupting by using moves instead of copies Type mb to merge branches into continuous curves in fort 7 fort 8 and fort 9 Backups of the original files are saved Type mb xxx to merge branches in s xxx b xxx and d xxx Backups of the original files are saved Type mb xxx yyy to merge branches in s xxx b xxx and d xxx and save them to s yyy b yyy and d yyy 71 sb Type sb xxx yyy ref to subtract using interpolation the first branch in b yyy from all branches in b xxx and save the result in b xxx Use ref e g PAR 1 as the reference column in b yyy only the first monotonically increasing or decreasing part is used A Backup of the original file is saved Use optional fourth and fifth arguments m and n to denote the branch m and first point n on that branch within b yyy where m n are in 1 2 3 zr Type zr xxx to zero all small numbers with absolute value less than 10716 in s xxx A backup file is made 5 7 HomCont commands Note that the h and H are obsolete with new style constants files where HomCont constants can be included in the main constant file with a c prefix Ch Use Ch instead of Or when using HomCont i e when IPS 9 see Chapter 20 Type Ch xxx to run AUTO HomCont Restart data if ne
120. S 1 NPAR 36 THL THU RLO 1 7976e 308 RL1 1 7976e 308 A0 1 7976e 308 A1 1 7976e 308 UZR 4 UZSTOP SP STOP IIS 3 IBR 0 LAB 0 TY NUNSTAB 1 NSTAB 1 IEQUIB 1 ITWIST 0 ISTART 5 IREV IFIXED IPSI The significance of the AUTO constants grouped by function is described in the sections below The HomCont constants NUNSTAB NSTAB IEQUIB ITWIST ISTART IREV IFIXED and IPSI are explained in Chapter Representative demos that illustrate use of the AUTO constants are also mentioned 97 10 2 Problem Constants 10 2 1 NDIM Dimension of the system of equations as specified in the user supplied routine FUNC 10 2 2 NBC The number of boundary conditions as specified in the user supplied routine BCND Demos exp kar 10 2 3 NINT The number of integral conditions as specified in the user supplied routine ICND Demos int lin obv 10 2 4 NPAR Maximum parameter number that can be used in all user supplied routines 10 2 5 JAC Used to indicate whether derivatives are supplied by the user or to be obtained by differencing JAC 0 No derivatives are given by the user Most demos use JAC 0 JAC 1 Derivatives with respect to state and problem parameters are given in the user supplied routines FUNC BCND ICND and FOPT where applicable This may be necessary for sensitive problems It is also recommended for computations in which AUTO gen
121. Section 10 9 10 for choosing another principal solution measure 10 6 6 Al The upper bound on the principal solution measure 102 10 7 Free Parameters 10 7 1 ICP For each equation type and for each continuation calculation there is a typical generic number of problem parameters that must be allowed to vary in order for the calculations to be properly posed The array ICP designates these free parameters The parameter that appears first in the ICP list is called the principal continuation parameter Specific examples and special cases are described below 10 7 2 Fixed points The simplest case is the continuation of a solution family to the system f u p 0 where f u R ef Equation 2 1 Such a system arises in the continuation of ODE stationary solutions and in the continuation of fixed points of discrete dynamical systems There is only one free parameter here As aconcrete example consider Run 1 of demo ab where ICP 1 Thus in this run PAR 1 is designated as the free parameter 10 7 3 Periodic solutions and rotations The continuation of periodic solutions and rotations generically requires two parameters namely one problem parameter and the period For example in Run 2 of demo ab we have ICP 1 11 Thus in this run the free parameters are PAR 1 and PAR 11 Note that AUTO reserves PAR 11 for the period Actually for periodic solutions it is sufficient to only specify the inde
122. TO constant IRS or if IRS is not specified the last solution in s e A string AUTO uses the solution in the file s s together with the constants in the files c s and h s Not all of these files need to be present e A Python list array or a numpy array representing a solution returns a solution with the given contents Such an array must be given column wise as t0 tn x0 xn y0 yn or for a point solution as x y z There are many possible options Long name Short name Description equation e The equations file constants C The AUTO constants file homcont h The Homcont parameter file solution s The restart solution file NDIM IPS etc AUTO constants BR PT TY LAB Solution constants If data is not specified or data is a string then options which are not explicitly set retain their previous value For example one may type s load e ab c ab 1 to use ab f90 ab f or ab c as the equations file and c ab 1 as the constants file Type s load name to load all files with base name This does the same thing as running s load e name c name h name s name You can also specify AUTO Constants e g DS 0 05 or IRS 2 Special values for DS are forwards and backwards Example s load s DS changes s c DS to s c DS Aliases ld commandRunnerLoadName 54 loadbd Load bifurcation diagram files
123. Via Save the results in b Vla s Vla and d Vla Table 14 28 Detailed AUTO shell and Python commands for the Vla demo The instructions below are for the Halo family V1 in AUTO demo V1b Follow the instructions for Lla above where you replace L by V and a by b throughout for instance you can run everything in one go using auto Vib auto auto Vib auto or with the extra calculations auto V1bX auto auto V1ibX auto The Floquet eigenfunction is now computed from label 12 with step size 1075 The detailed commands are likewise except for the manifold calculations in Table 14 23 and those are given in Table 14 29 159 CR man Vib 1 startVib Vib run startVib e man c man Vib 1 Look at c man V1b 1 to see from which label in s startVlb this run starts In this run the z coordinate of the end point PAR 21 is kept fixed while the period PAR 11 i e the total integration time is allowed to vary as is the value of e i e PAR 6 Note that if PAR 6 be comes large then the manifold may no longer be accurate The free parameters in this run are PAR 3 energy PAR 12 length of the orbit PAR 6 starting distance PAR 22 y coordinate at end point PAR 11 integration time PAR 23 z coordinate at end point sv Vib save Vib V1b Save the results in b V1b s V1b and d V1b R man Vib 2 startVib hetVib run startVib e man c man V1b 2
124. Ya 0 which we note from the eigenvalues stored in d 2 corresponds to a Shil nikov Hopf bifurcation Note that PAR 2 is also approxi mately zero at label 7 which accords with the analytical observation that the origin of undergoes a Hopf bifurcation when 6 0 Labels 6 and 8 are the user defined output points the solutions at which are plotted in Fig 24 1 Note that solutions beyond label 7 e g the plotted solution at label 8 do not correspond to homoclinic orbits but to point to cycle heteroclinic orbits c f Section 2 2 1 of Champneys et al 1996 We now continue in the other direction along the family It turns out that starting from the initial point in the other direction results in missing a codim 2 point which is close to the starting point Instead we start from the first saved point from the previous computation label 5 in s 2 r3 run r2 UZ1 c cir 3 ap 2 The output BR PT TY LAB PAR 1 PAR 2 PAR 22 PAR 24 1 9 UZ 10 7 204001E 01 5 912315E 01 1 725669E 00 3 295862E 05 1 18 UZ 11 7 590583E 01 1 26 UZ 12 7 746686E 01 1 28 EP 13 7 746628E 01 7 428734E 01 7 746147E 01 7 746453E 01 3 432139E 05 2 822988E 01 5 833163E 01 1 637611E 07 5 908902E 01 1 426214E 04 contains a neutral saddle focus a Belyakov transition at LAB 10 44 0 a double real leading eigenvalue saddle focus to saddle transition at LAB 11 Y2 0 and a neutral saddle at LAB 12 44 0 Data at s
125. a u3 2uus uz 15 7 j Tus u T 27u 2u9U4 2u105 with left boundary conditions and asymptotic right boundary conditions f F al f y u2 1 za us 1 y tty 0 a foo 1 ELE ua 1 foo al foo 1 ual us 1 0 15 8 uy 1 fos where alfa Felt H EJ ia b foos 7 E tP TA Note that there are five differential equations and six boundary conditions Correspondingly there are two free parameters in the computation of a solution family namely y and f The period T is fixed T 500 The starting solution is u 0 i 1 5 at y 1 fo 0 AUTO COMMAND ACTION mkdir kar create an empty work directory cd kar change directory demo kar copy the demo files to the work directory ri run e kar c kar computation of the solution family save r1 kar save output files as b kar s kar d kar Table 15 6 Commands for running demo kar 166 15 7 spb A Singularly Perturbed BVP This demo illustrates the use of continuation to compute solutions to the singularly perturbed boundary value problem Us u A ujue ut 1 u 15 10 with boundary conditions u1 0 3 2 ui 1 y The parameter A has been introduced into the equations in order to allow a homotopy from a simple equation with known exact solution to the actual equation This is done in the first run In the second run e is decreased by continuation I
126. ace consisting of time here PAR 14 and the state vector here u1 us AUTO COMMAND ACTION mkdir ivp create an empty work directory cd ivp change directory demo ivp copy the demo files to the work directory ri run e ivp c ivp time integration save r1 ivp save output files as b ivp s ivp d ivp Table 14 21 Commands for running demo ivp 152 1415 r3b The Circular Restricted 3 Body Problem CR3BP This demo computes periodic solutions and two dimensional unstable manifolds of those periodic solutions in the restricted three body problem T Tp Y Yp Z 2p THH x l pu Lp 2Yp 1 p dp TH dar lap jy ye E sa p pTY eS a F Up p 1 p Zu de du where dg y 2 u y 22 and dy y 1 1 u y2 2 Here l 4 0 breaks the conservativeness of the system In general continuations involve l as a parameter and will then approximately stay at zero 14 15 1 Computation of Periodic Solutions of the CR3BP Running the Python script r3b auto will generate the families of periodic solutions L1 H1 and V1 for the case of the mass ratio u 0 063 auto r3b auto auto r3b auto where as in the following examples the left hand side command can be used at the shell prompt and the right hand side command at the Python CLUI prompt Note that the commands starting with work in both interfaces but cannot be used in the expert scrip
127. agram Op jects ec a a A e a ATi Sol tions s sso a en o e BE GRE de e Ede BE AP AE ee Se ee ee 4 8 Importing data from Python or external tools 4 9 Exporting output data for use by Python or external visualization tools AR a rn ee ee 11 El 12 13 14 15 15 16 18 18 18 19 20 21 LU Plotting Tollos rios E Boe SH a A Be Sw a BO 4 12 The Plotting Tool PLAUTO4 Ss ba ot A oe a Ew se is se A ok Ge A A A NET cad doe da ee ea PEER AAA Aa a E Le ob eed oe eee oe ew eS Ye raras rra A AA E ad a IN ea e RA EE oe a a de aaa E 4 14 7 Python data structure manipulation functions 2 ES A a a Wee o ye ee a 5 1 Basic commands nee ek kk seras re E A 5 2 Plotting commands ic daa cea cei Rs ba eed bg oe Bae ee Boo File manipulation o a 46 44 46 oo ed a A 54A Diagnostic sa we Be A A se ee ow ee A REE 5 6 Filem intena n e o A ed da 5 7 HomCont commands 2 4 44 de 244 weed owe Oa eae ne es 5 8 Copy ee a den s sos adi se ee Hse ewe Se BS SE mS wR Se a 5 9 Viewing the manual ceo 2404464864646 444554 44 obvawd 6 Output Files 7 The Graphics Programs PLAUT and PyPLAUT 7 1 Basic PLAU T Commands 2 2066 5600464 24 eR ee Eee eR ee 7 2 Default Options ako can oe ee ee ee a Re eee we eh Ree RA 7 3 Other PLAU T Commands oaa 7 4 Printing PLAUT Files 2 lt x oc 6 oe a eS ee a A 8 The Graphics Program PLAUTOA 6 1 Quick start AAN ATA 8 1 1 Star
128. al param eter variables Demos exp int Determine folds and branch points along solution families to the above boundary value problem Branch switching is possible at branch points Curves of folds and branch points can be computed Demos bvp int sspg 2 4 Parabolic PDEs For 2 3 the program can Trace out families of spatially homogeneous solutions This amounts to a bifurcation analysis of the algebraic system 2 1 However AUTO uses a related system instead in order to enable the detection of bifurcations to wave train solutions of given wave speed More precisely bifurcations to wave trains are detected as Hopf bifurcations along fixed point families of the related ODE u z v z u z D le vle f u z p where z x ct with the wave speed c specified by the user Demo wav Run 2 2 4 Trace out families of periodic wave solutions to that emanate from a Hopf bifurcation point of Equation 2 4 The wave speed c is fixed along such a family but the wave length L i e the period of periodic solutions to 2 4 will normally vary If the wave length L becomes large i e if a homoclinic orbit of Equation is approached then the wave tends to a solitary wave solution of 2 3 Demo wav Run 3 20 Trace out families of waves of fixed wave length L in two parameters The wave speed c may be chosen as one of these parameters If L is large then such a continuation gives a family of ap
129. amily with a fixed mesh IAD 0 Be sure to set IAD back to IAD 3 for computing eventual non trivial bifurcating solution families 10 4 Tolerances 10 4 1 EPSL Relative convergence criterion for equation parameters in the Newton Chord method Most demos use EPSL 10 or EPSL 10 which is the recommended value range 10 4 2 EPSU Relative convergence criterion for solution components in the Newton Chord method Most demos use EPSU 10 or EPSU 10 which is the recommended value range 10 4 3 EPSS Relative arclength convergence criterion for the detection of special solutions Most demos use EPSS 107 or EPSS 10 which is the recommended value range Generally EPSS should be approximately 100 to 1000 times the value of EPSL EPSU 10 4 4 ITMX The maximum number of iterations allowed in the accurate location of special solutions such as bifurcations folds and user output points by M ller s method with bracketing The recom mended value is ITMX 8 used in most demos 99 10 4 5 NWIN After NWIN Newton iterations the Jacobian is frozen i e AUTO uses full Newton for the first NWTN iterations and the Chord method for iterations NWIN 1 to ITNW The choice NWTN 3 is strongly recommended and used in most demos Note that this constant is only effective for ODEs i e for solving the piecewise polynomial collocation equations For algebraic systems AUTO always uses full Newton 10 4 6 ITNW The maximum number of comb
130. ampneys amp Spence 1993 Continuing from the initial solution in the other parameter direction r3 r3 run r3 UZ1 c rev 4 ap 37 we obtain the output BR PT TY LAB PAR 1 L2 NORM MAX U 1 1 7 UZ 9 1 60000E 00 3 70171E 01 3 84045E 01 1 33 UZ 10 9 99998E 01 3 61440E 01 1 77504E 01 1 94 UZ 11 5 14775E 08 3 71301E 01 4 69706E 02 1 153 MX 12 2 54464E 01 3 75071E 01 3 00627E 02 which again ends at a no convergence error for similar reasons 26 4 Detailed AUTO Commands AUTO COMMAND ACTION mkdir rev cd rev demo rev create an empty work directory change directory copy the demo files to the work directory ri run rev c rev 1 sv 1 use the starting data in rev dat 1 increase PAR 1 save output files as b 1 s 1 d 1 ri r1 run r1 UZ1 c rev 2 ap 1 continue in reverse direction restart 1st UZ append output files to b 1 s 1 d 1 r3 run rev c rev 3 sv 3 use the starting data in rev dat 3 restart with different reversibility save output files as b 3 s 3 d 3 r3 r3 run r3 UZ1 c rev 4 ap 3 continue in reverse direction restart 1st UZ append output files to b 3 s 3 d 3 Table 26 1 Detailed AUTO Commands for running demo rev 246 0 001 i E AR 0 25 0 50 TOG HS 1 eG 0 00 0 20 0 40 0 60 0 80 1 00 0 10
131. anch number PT The point number LAB The solution label if any section A unique identifier for each branch in a file with multiple branches data An array which contains the AUTO output Table 4 4 This table shows the strings that can be used to query a bifurcation diagram object and their meanings Type Short Name Number No Label No Label Branch point algebraic problem BP 1 Fold algebraic problem LP 2 Hopf bifurcation algebraic problem HB 3 Regular point every NPR steps RG 4 User requested point UZ 4 Fold ODE LP 5 Bifurcation point ODE BP 6 Period doubling bifurcation ODE PD Y Bifurcation to invariant torus ODE TR 8 Normal begin or end EP 9 Abnormal termination MX 9 Table 4 5 This table shows the various types of points that can be in solution and bifurcation diagram files with their short names and numbers 40 Type Short Name Number Bogdanov Takens bifurcation algebraic problem BT 21 31 Cusp algebraic problem CP 22 Generalized Hopf bifurcation algebraic problem GH 32 Zero Hopf bifurcation algebraic problem ZH 13 23 33 1 1 Resonance bifurcation ODE maps Rl 25 55 85 1 2 Resonance bifurcation ODE maps R2 76 86 1 3 Resonance bifurcation ODE maps R3 87 1 4 Resonance bifurcation ODE maps R4 88 Fold flip bifurcation maps LPD 28 78 Fold torus bifurcation maps
132. and given in Table The saved plots are shown in Figure and in Figure 12 2 Figure 12 1 shows the bifurcation diagrams for the first run and Figure 12 2 for the second run The plotting window consists of a menubar at the top a plotting area and a control panel with four control widgets at the bottom By default the first two columns in the bifurcation diagram output are plotted against each other To obtain a u versus x bifurcation diagram you need to plot column mu versus column x You can do that by changing the Y box to say x either by typing it there by using the menu obtained by clicking the downwards 123 facing triangle or by using a scripted command as used in cusp auto You can also change the mode of the plotting tool from bifurcation to solution This is accomplished by clicking on the widget marked Type on the bottom control panel and setting it from bifurcation to solution In the plotting window will appear a plot of the first labeled solution in this case just a point You can plot all points by changing the Label to 1 2 3 4 5 6 7 8 9 10 The plotting tool can also be used to create Postscript files from plots by selecting the File on the menubar and then selecting the Save Postscript from the drop down menu This will bring up a dialog into which the user can enter the filename of the postscript file to save the plot in When using matplotlib
133. ant under two separate reversibilities Ry u U2 UZ Ud t u1 U2 U3 Ua t 26 2 and Ra u1 U2 Ug Ua t gt 41 U2 U3 U4 t 26 3 First we copy the demo into a new directory demo rev For this example we shall make two separate starts from data stored in equation and data files rev c 1 rev dat 1 and rev c 3 rev dat 3 respectively The first of these contains initial data for a solution that is reversible under R and the second for data that is reversible under R 26 2 An R Reversible Homoclinic Solution The first run ri run rev c rev 1 sv 1 243 starts by using the file rev dat 1 via the dat AUTO constant in c rev 1 The orbit contained in the data file is a primary homoclinic solution for P 1 6 with truncation half interval PAR 11 39 0448429 which is reversible under R Note that this reversibility is specified in c rev 1 via IREV 0 1 0 1 Note also from c rev 1 that we only have one free parameter PAR 1 because symmetric homoclinic orbits in reversible systems are generic rather than of codimension one The first run results in the output BR PT TY LAB PAR 1 L2 NORM MAX U 1 1 1 EP 1 1 60000E 00 2 85704E 01 3 62232E 01 1 8 UZ 2 1 70000E 00 2 90288E 01 4 18225E 01 1 11 UZ 3 1 80000E 00 2 95723E 01 4 80604E 01 1 14 UZ 4 1 90000E 00 2 74864E 01 4 43069E 01 1 20 EP 5 1 99678E 00 1 13379E 01 9 59430E 02 which is consistent with the theo
134. are u v i 1 NDIM 2 with boundary conditions u 0 0 u 1 0 Here e u T y k gt k 0 with n 25 The computation requires 10 full LU decompositions of the linearized system that arises from Newton s method for solving the collocation equations The commands for running the timing problem for a particular choice of NDIM NTST and NCOL are given below Note that if NDIM is changed then NBC must be changed accordingly 19 3 i i AUTO COMMAND ACTION mkdir tim create an empty work directory cd tim change directory demo tim copy the demo files to the work directory ri run e tim c tim Timing run save ri tim save output files as b tim s tim d tim Table 19 3 Commands for running demo tim 204 Chapter 20 HomCont 20 1 Introduction HomCont is a collection of routines for the continuation of homoclinic solutions to ODEs in two or more parameters The accurate detection and multi parameter continuation of certain codimension two singularities is allowed for including all known cases that involve a unique homoclinic orbit at the singular point Homoclinic connections to hyperbolic and non hyperbolic equilibria are allowed as are certain heteroclinic orbits Homoclinic orbits in reversible systems can also be computed The theory behind the methods used is explained in Champneys amp Kuznetsov 1994 Bai amp Champneys 1996 Sandstede 1995 1995
135. auto the results can be plotted using plot hen or pp hen AUTO COMMAND ACTION mkdir hen cd hen demo hen ri run hen save beta create an empty work directory change directory copy the demo files to the work directory fixed point solution branch for PB a 1 detects a period doubling PD and a Naimark Sacker TR bifurcation save output files as b beta s beta d beta run ri TR1 ICP alpha beta ISW 2 ILP 0 STOP R11 R21 save hen run DS append hen continue the TR bifurcation in two param eters until a 1 1 or 1 2 resonance is found save output files as b hen s hen d hen compute last continuation the opposite way append output files to b hen s hen d hen run ri PD1 ICP alpha beta ISW 2 ILP 0 append hen run DS append hen continue the PD bifurcation in two param eters append output files as b hen s hen d hen compute last continuation the opposite way append output files to b hen s hen d hen r4 run c hen ICP alpha DS STOP LP1 save alpha fixed point solution branch for a 3 1 detects and stops at a fold LP save output files as b alpha s alpha d alpha run r4 LP1 ICP alpha beta ISW 2 ILP 0 append hen run DS append hen continue the LP bifurcatio
136. behind 3 at left and ahead 4 always at origin Coordinate Type 3 Draw Scale on the Aexs Draw Scale Yes Initialize the default graph type 0 Solution fort 8 86 1 Bifurcation fort 7 Graph Type 0 Initialize the default graph style O LINES 1 TUBES 2 SURFACE Graph Style 0 Set the window width and height Window Width 1000 Window Height 1000 Set X Y Z axes for the solution diagram 0 is Time for X Y Z X Axis Solution 1 Y Axis Solution 2 Z Axis Solution 3 Set X Y Z axes for the bifurcation diagram X Axis Bifurcation 4 Y Axis Bifurcation 5 Z Axis Bifurcation 6 Labeled solutions Labels 0 Set coloring method 6 STABILITY 5 POINT 4 BRANCH 3 TYPE 2 LABEL 1 COMPONENT Otherwise according to the data in the ith column of the solution file It can only be set to an integer value Coloring Method 3 For the solution diagram Coloring Method Solution 3 For the bifurcation diagram Coloring Method Bifurcation 3 Number of Period Animated 1 HH H HH HH OF Line Width Scaler adjusts the thickness of curves Line Width Scaler 1 0 The AniLine Thickness Scaler sets the thickness of animated solution curves AniLine Thickness Scaler 3 0 Background color Background Color 0 0 0 0 0 0 Background transparency Background Transparency 0 0
137. between these two points 1 00 US 0 50 0 70 20 La 10 O 0 0 0 0 5 1 0 La 2410 Figure 23 7 The big homoclinic orbit approaching a figure of eight 232 homoclinic orbit to a big saddle homoclinic orbit i e with two extra turning points in phase space Finally we can switch to continuation of the big saddle homoclinic orbit from the new codim 2 point at label 11 r9 run r8 UZ1 c kpr 9 sv 9 Note that AUTO takes a large number of steps near the line PAR 1 0 while PAR 2 approaches 2 189 which is why we chose such a large value NMX 500 in c kpr 9 This particular computation ends at BR PT TY LAB PAR 1 L2 NORM rare PAR 2 1 500 EP 24 2 04263E 05 2 18126E 01 2 18951E 00 By plotting phase portraits of orbits approaching this end point see Figure we see a canard like like transformation of the big homoclinic orbit to a pair of homoclinic orbits in a figure of eight configuration That we get a figure of eight is not a surprise because PAR 1 0 corresponds to a symmetry in the differential equations Koper 1994 note also that the equi librium stored as PAR 12 PAR 13 PAR 14 in d 9 approaches the origin as we approach the figure of eight homoclinic 23 4 Three Parameter Continuation We now consider curves in three parameters of each of the
138. by appending Yy 0 to the continuation of a saddle homoclinic orbit with IEQUIB 1 The first approach is used in the example mtn for contrast we shall adopt the second approach here r12 run r7 UZ1 c kpr 12 sv 12 ri2 r12 run r8 UZ1 c kpr 13 ap 12 The projection onto the e k plane of all four of these codimension two curves is given in Figure The intersection of the inclination flip lines with one of the non central saddle node homoclinic lines is apparent Note that the two non central saddle node homoclinic orbit curves are almost overlaid but that as in Figure the orbits look quite distinct in phase space 23 5 Detailed AUTO Commands AUTO COMMAND ACTION mkdir kpr create an empty work directory cd kpr change directory demo kpr copy the demo files to the work directory ri run kpr c kpr 1 sv 1 continuation in the time length parameter PAR 11 save output files as b 1 s 1 d 1 r2 run ri UZ1 c kpr 2 8v 2 locate the homoclinic orbit save output files as b 2 s 2 d 2 r3 run r2 UZ1 c kpr 3 sv 3 generate adjoint variables save output files as b 3 s 3 d 3 r3 r3 run r3 c kpr 4 ap 3 continue the homoclinic orbit append output files to b 3 s 3 d 3 r3 r3 run r3 0 c kpr 5 ap 3 continue in reverse direction append output files to b 3 s 3 d 3 r6 run r2 EP1 c
139. c Champneys Kuznetsov amp Sandstede 1996 and references therein The final cited paper contains a concise description of the present version The current implementation of HomCont must be considered as experimental and updates are anticipated The HomCont routines are in the file auto 07p src autlib5 f Expert users wishing to modify the routines may look there Note also that at present HomCont can be run only in AUTO Command Mode and not with the GUI 20 2 HomCont Files and Routines In order to run HomCont one must prepare an equations file xxx f90 where xxx is the name of the example and constants file c xxx These files are in the standard AUTO format but the c xxx file almost always also must contain constants that are specific to homoclinic continuation The choice IPS 9 in c xxx specifies the problem as being homoclinic continuation The equation file kpr f90 serves as a sample for new equation files It contains the Fortran routines FUNC STPNT PVLS BCND ICND and FOPT The final three are dummy routines which are never needed for homoclinic continuation Note a minor difference in STPNT and PVLS with other AUTO equation files in that the common block BLHOM is required The constants file c xxx is identical in format to other AUTO constants files except for the added constants Note that the values of the constants NBC and NINT are irrelevant as these are set automatically by the choice IPS 9 Also the choice JAC 1 is str
140. c s abc d abc R abc 4 compute a family of periodic solutions from the third Hopf point ap abc append the output files to b abc s abc d abc R abc 5 compute a family of periodic solutions from the fourth Hopf point ap abc append the output files to b abc s abc d abc Table 14 6 Unix Commands for running demo abc abc run e abc c abc 1 abc abc run abc HB1 c abc 2 abc abc run abc HB2 c abc 3 abc abctrun abc HB3 c abc 4 abc abc trun abc HB4 c abc 5 save abc abc Table 14 7 Python Commands for running demo abc abc run e abc c abc 1 for solution in abc HB abc abctrun solution c abc 2 abc r1 abc save abc abc Table 14 8 Python Program for running demo abc 136 143 pp2 A 2D Predator Prey Model This demo illustrates the computation of families of stationary solutions including bifurcating stationary families as well as the detection of a Hopf bifurcation The first run computes the families of stationary solutions bounded by 0 lt p lt 1 and u gt 0 25 Then the script pp2 auto scans the first run for Hopf bifurcations finds one and computes the family of periodic solutions that emanates from the Hopf bifurcation This family terminates in a heteroclinic orbit The continuation is configured to stop if the period PAR 11 36 when the heteroclinic orbit is very
141. cant control over computational efficiency in particular through judicious choice of the AUTO constants DS DSMIN and DSMAX and for ODEs NTST and NCOL Initial experimentation normally suggests appropriate values Slowly varying solutions to ODEs can often be computed with remarkably small values of NTST and NCOL for example NTST 5 NCOL 2 Generally however it is recommended to set NCOL 4 and then to use the smallest value of NTST that maintains convergence The choice of the pseudo arclength stepsize parameters DS DSMIN and DSMAX is highly prob lem dependent Generally DSMIN should not be taken too small in order to prevent excessive step refinement in case of non convergence It should also not be too large in order to avoid instant non convergence DSMAX should be sufficiently large in order to reduce computation time and amount of output data On the other hand it should be sufficiently small in order to prevent stepping over bifurcations without detecting them For a given equation appropriate values of these constants can normally be found after some initial experimentation The constants ITNW NWTN THL EPSU EPSL EPSS also affect efficiency Understanding their significance is therefore useful see Section and Section 10 5 Finally it is recommended that initial computations be done with ILP 0 no fold detection and ISP 1 no bifurcation detection for ODEs 11 3 Correctness of Results AUTO computed solutions to
142. ch fork bifurcation in the complex equation This bifurcation is located in the first computation below In the second and third run both legs of the bifurcating solution family are computed On it both solution components u and uz have nontrivial imaginary part AUTO COMMAND ACTION mkdir ezp create an empty work directory cd ezp change directory demo ezp copy the demo files to the work directory ezp run e ezp c ezp 1st run compute solution family containing fold ezp ezp run ezp BP1 ISW 1 2nd run compute bifurcating complex so lution family ap ezp append output files to p ezp s ezp d ezp ezp ezptrun ezp BP1 ISW 1 DS 3rd run compute 2nd leg of bifurcating family save ezp ezp save combined output to b ezp s ezp d ezp Table 15 8 Commands for running demo ezp 168 15 9 um2 Basic computation of a 2D unstable mani fold This demo shows how one can compute a 2D unstable manifold of an equilibrium using orbit continuation The model equations are given by ad ex y ia 15 12 The origin has eigenvalues e and 1 where e gt 0 so that its unstable manifold is indeed 2 dimensional Since the phase space itself is 2 dimensional one can also consider this demo as showing how to generate part of a 2 dimensional phase portrait However the basic steps in this demo also apply to the computation of 2 dimensional unstable manifolds of e
143. clinic point to point connection The starting point for this investigation is a homoclinic orbit connecting the point b to itself where the phase of y is shifted by 27 in other words the homoclinic orbit reinjects once and it is a heteroclinic orbit in the covering space We can most easily compute the homoclinic orbit using a homotopy method see the HomCont section 20 7 for details 1 We locate the homoclinic orbit or here the heteroclinic orbit in the covering space by continuing the one dimensional stable manifold in negative time This way HomCont views the stable manifold as a one dimension unstable manifold to which its standard homotopy method can be applied and which makes the method much more straightfoward than starting with a two dimensional unstable manifold We reach the unstable eigenspace of E b as soon as the artificial dummy parameter w measuring a distance to E Db vanishes 2 We can now improve this connection by continuing in decreasing negative time keeping w fixed and freeing up the system parameter 1 3 The resulting orbit can be continued forwards and backwards in the system parameter v and va using standard HomCont settings Note the setting of TEQUIB 1 HomCont 174 auto detects the phase shift and only continues one equilibrium instead of treating the orbit as a general heteroclinic orbit The resulting homoclinic orbit snakes in parameter space between the two tangencies of the codime
144. codimension two points encountered in this model by freeing the parameter e PAR 3 First we continue the first inclination flip stored at label 7 in s 3 r10 run r3 UZ1 c kpr 10 sv 10 Note that ITWIST 1 in c kpr 10 so that the adjoint is also continued and there is one fixed condition IFIXED 1 13 so that test function 413 has been frozen Among the output there is a codimension three point zero of g where the neutrally twisted homoclinic orbit collides with the saddle node curve BR PT TY LAB PAR 1 eae PAR 2 PAR 3 PAR 29 Su 1 18 UZ 11 1 28270E 01 2 51932E 00 5 74477E O01 2 59151E 06 The other detected inclination flip at label 8 in s 3 is continued similarly rii run r3 UZ2 c kpr 11 sv 11 giving among its output another codim 3 saddle node inclination flip point BR PT TY LAB PAR 1 Lis PAR 2 PAR 3 PAR 29 TE 1 27 UZ 11 1 53542E 01 2 45810E 00 1 17171E 00 1 13312E 06 233 Output beyond both of these codim 3 points is spurious and both computations end in an MX point no convergence To continue the non central saddle node homoclinic orbits it is necessary to work on the data without the solution t We restart from the data saved at LAB 8 and LAB 11 in s 7 and s 8 respectively We could continue these codim 2 points in two ways either by appending the defining condition 416 0 to the continuation of saddle node homoclinic orbits with IEQUIB 2 etc or
145. ct bifurcations to wave train solutions r3 run r2 HB1 IPS 12 ICP 3 11 ILP 0 3rd run wave train solutions of fixed wave ISP 0 RL1 700 DS 0 1 DSMAX 1 0 speed UZR 3 610 0 638 0 11 500 0 save r2 r3 wav save output to b wav s wav d wav uz3 load r3 UZ3 RL1 1000 load restart label DS 0 5 DSMAX 2 0 UZR r4 run uz3 ICP 3 10 NPR 50 4th run wave train solutions of fixed wave length save r4 rng save output files as b rng s rng d rng r5 run uz3 IPS 14 ICP 14 NMX 230 NPR 5 5th run time evolution computation save r5 tim save output files as b tim s tim d tim Table 16 3 Commands for running demo wav 184 16 4 bre Chebyshev Collocation in Space This demo illustrates the computation of stationary solutions and periodic solutions to systems of parabolic PDEs in one space variable using Chebyshev collocation in space More precisely the approximate solution is assumed of the form u x t S77 ug t 4 1 Here uz t corresponds to u xz t at the Chebyshev points her with respect to the interval 0 1 The polynomials TAON are the Lagrange interpolating coefficients with respect to points an be ys where xo 0 and amp 4 1 The number of Chebyshev points in 0 1 as well as the number of equations in the PDE system can be set by the user in the file brc f90 As an illustrative application we consider the Brusselator Holodniok Knedlik amp Kub ek 1
146. d U 2 for Y we obtain the diagram depicted in Figure a where the periodic orbit grows from the Hopf equilibrium to a homoclinic orbit Columns 1 Columns 0 4 00e 01 1 008 00 2 00e 01 4 006 01 4 006 01 0 i i 0 008 00 i 008 00 4 006 01 8 006 01 0 008 00 4 006 01 8 006 01 2 008 01 6 006 01 1 006 00 2 008 01 6 008 01 1 008 00 Columns 0 Columns t Figure 27 1 Periodic orbit growing from a Hopf bifurcation to a homoclinic orbit a The unshifted homoclinic orbit b Note however that the homoclinic orbit has the wrong left hand and right hand end points This can be seen by plotting the solution corresponding to Label 13 using t vs x coordinate U 1 as depicted in Figure 27 1 b Hence in order to continue this as a real homoclinic we have to give HomCont special instructions to do a phase shift in time This can be done by setting ISTART 4 Moreover since we have not specified the value of the equilibrium at the origin in sib c we need to set IEQUIB 1 this is the default value to let HomCont detect the equilibrium Note that in this case this is not strictly necessary however we do this for instructional purposes Now we use HomCont to continue the homoclinic orbit in c and y PAR 3 PAR 5 to get the desired value c 2 0 r4 run r3 IPS 9 ICP 3 5 NPR 60 JAC 1 UZR 3 2 0 ISTART 4 save r4 4 BR PT TY LAB PAR 3 L2 NORM Poe
147. d alias Aliases and internal names may be used interchangably but the intention is that the aliases will be more commonly used A default set of aliases is provided and these aliases will be used in the examples in the rest of this Chapter The default aliases are listed in the reference in Section 4 14 NOTE Defaults for the plotting tool may be included in the autorc file as well 4 11 Plotting Tool The plotting tool can be run by using the command plot bd to plot a bifurcation diagram object bd after a calculation has been run or using the command plot to plot the files fort 7 and fort 8 or using the command plot foo to plote the data in the files s foo and b foo The menu bar provides two buttons The File button brings up a menu which allows the user to save the current plot as a Postscript file or to quit the plotting tool The Options button allows the plotter configuration options to be modified The available options are decribed in Table 4 8 In addition the options can be set and figures can be saved from within the CLUI For example the set of commands in Figure 4 20 shows how to create a plot change its background color to black and save it The demo script auto 07p demo python plotter py contains several examples of changing options in plotters The special argument hide True to plot does not produce an on screen plot which is useful for quick automatic generation of saved figure files If you are using matplotlib
148. d put them in the same directory as the AUTO data files PLAUTO4 first looks for the resource file in the current directory If it cannot find a resource file there then it will try to use the one installed in the AUTO root directory If both these searches fail then the internal default values will be used In order to write a usable resource file one should follow the following rules 1 Comment lines start with 4 Comments may take as many lines as desired 2 Between the variable name and the default value we must use to tell the system that the left side is the variable name and the right side is its corresponding default value must be used between two values 3 If a variable has aggregate values a comma 4 The line type is set using 4 digit hexadecimals starting with Ox Its values can range from 0 invisible to Oxf solid The system default is Oxffff for stable solutions and 0x3333 for unstable ones The line pattern is determined by the number of 1s and Os when the hexadecimal is converted to a 16 bit binary A 1 indicates that the drawing occurs and 0 that it does not on a pixel by pixel basis For example the pattern OxAAAA in binary is 0000100010001000 and PLAUTO4 interprets this as drawing 3 bits off 1 bit on 3 bits off 1 bit on 3 bits off 1 bit on and finally 4 bits off The pattern is read backward because the low order bits are u
149. dated subject to a modifiable maximum length On SGI machines this is 10240 bytes which can be increased for example to 20480 bytes by replacing the line CC cc Wf XNI10240 O in auto 07p gui Makefile by CC cc Wf XNI20480 O On other machines the maximum message length is the system defined maximum string literal length 96 Chapter 10 Description of AUTO Constants 10 1 The AUTO Constants File As described in Section 3 if the equations file is xxx f90 then the constants that define the computation are normally expected in the file c xxx The format of this file is free with constant value pairs separated by commas or spaces Comments start with one of the characters and and run to the end of a line An example with default values is listed below In real constant files you only need to specify those values that are different from these but listing all of them allows for easier editing Note that this file is not strictly necessary when using the Python interface you can define all constants inside the scripts if you so wish Default AUTO Constants file e s dat sv unames parnames U PAR NDIM 2 IPS 1 IRS 0 ILP 1 ICP 1 NTST 20 NCOL 4 IAD 3 ISP 2 ISW 1 IPLT 0 NBC 0 NINT 0 NMX 0 NPR O MXBF 10 IID 2 ITMX 9 ITNW 5 NWIN 3 JAC 0 EPSL 1e 07 EPSU 1e 07 EPSS 1e 05 DS 0 01 DSMIN 0 005 DSMAX 0 1 IAD
150. ds that are described in Chapter 5 and start with an Q sign can be entered directly without the ep 52 AUTO gt p plotQ Created plot AUTO gt p config bg black AUTO gt p savefig black eps AUTO gt p plot hide True Created plot AUTO gt p savefig white eps Figure 4 20 This example shows how a plotter is created how the background color may be changed to black how a figure is saved and how an invisible plot is saved to a file All other configuration options are set similarily Note the above commands assume that the files fort 7 and fort 8 exist in the current directory 4 14 Reference 4 14 1 Basic commands run Run AUTO Type r run data loptions to run AUTO from solution data with the given AUTO constants or file keyword options The results are stored in the bifurcation diagram r which you can later print with print r obtain branches from via r 0 r 1 and obtain solutions from via r 3 r 5 rCLP2 where 3 and 5 are label numbers and LP2 refers to the second LP label run data runs AUTO in the following way for different types of data e A solution AUTO starts from solution data with AUTO constants data c e A bifurcation diagram AUTO start from the solution specified by the AUTO con stant IRS or if IRS is not specified the last solution in data data 1 with AUTO constants data 1 c e A string AUTO uses the solution in the file s data tog
151. e which gives a list of the solutions themselves load xxx typename getLabels returns the list of labels 64 Or use splabs s typename where s is a parsed solution from s1 This is equivalent to the command s typename getLabels Aliases commandSpecialPointLabels wait Wait for the user to enter a key Type wait to have the AUTO interface wait until the user hits any key mainly used in scripts Aliases commandWait quit Quits the AUTO CLUI Aliases q commandQuit gui Show AUTO s graphical user interface Type gui to start AUTO s graphical user interface NOTE This command is not implemented yet Aliases commandCreateGUI 4 14 8 Shell Commands cat Print the contents of a file Type cat xxx to list the contents of the file xxx Aliases commandCat cd Change directories Type cd xxx to change to the directory xxx This command understands both shell vari ables and home directory expansion Aliases commandCd ls List the current directory Type 1s to run the system ls command in the current directory This command will accept whatever arguments are accepted by the Unix command 1s Aliases commandLs shell Run a shell command Type shell xxx to run the command xxx in the Unix shell and display the results in the AUTO command line user interface Aliases commandShell 65 Chapter 5 Running AUTO using Unix Commands Apart from the Python commands described
152. e 14 5 we note that IPS 2 a family of periodic solutions is computed IRS 2 the starting point is the solution with label 2 a Hopf bifurcation point to be read from the solutions file here s abc IcP 1 11 there are two continuation parameters namely PAR 1 and the period PAR 11 UZR 1 0 25 there is one user output point now at PAR 1 0 25 where the calcula tion is to terminate since the index 1 is negative 133 SUBROUTINE FUNC NDIM U ICP PAR IJAC F DFDU DFDP IMPLICIT NONE INTEGER INTENT IN NDIM ICP IJAC DOUBLE PRECISION INTENT IN U NDIM PAR DOUBLE PRECISION INTENT OUT F NDIM DOUBLE PRECISION INTENT INOUT DFDU NDIM NDIM DFDP NDIM DOUBLE PRECISION X1 X2 X3 D ALPHA BETA B S E X1C X1 U 1 X2 U 2 X3 U 3 D PAR 1 ALPHA PAR 2 BETA PAR 3 B PAR 4 S PAR 5 E DEXP X3 X1C 1 X1 F 1 X1 D X1C E F 2 X2 D E X1C S X2 F 3 X3 BETA X3 D B E X1C ALPHA S X2 END SUBROUTINE FUNC Table 14 2 The equations for demo abc as defined in the equations file abc 90 134 SUBROUTINE STPNT NDIM U PAR T IMPLICIT NONE INTEGER INTENT IN NDIM DOUBLE PRECISION INTENT INOUT U NDIM PAR x DOUBLE PRECISION INTENT IN T PAR 1 0 0 PAR 2 1 0 PAR 3 1 55 PAR 4 8 PAR 5 0 04 U 1 0 U 2 0 U 3 0 END SUBROUTINE STPNT Table 14 3 The starting solution for demo abc as
153. e where you replace L by H and a by c throughout for instance you can run everything in one go using auto Hic auto auto Hic auto The Floquet eigenfunction is now computed from label 68 with step size 10 The detailed commands follow the ones for Hla above except that the last run is left out and so the H1cX auto script is not necessary The instructions below are for the Halo family V1 in AUTO demo Via Follow the instructions for Lla above where you replace L by V throughout for instance you can run everything in one go using auto Via auto auto Via auto The Floquet eigenfunction is now computed from label 8 with step size 107 The detailed commands are likewise except for the manifold calculations in Table 14 23 and those are given in Table 14 28 CR man Via 1 startVia Via run startVia e man c man Via 1 Look at c man Vla 1 to see from which label in s startVla this run starts In this run the 2 coordinate of the end point PAR 23 is kept fixed while the period PAR 11 i e the total integration time is allowed to vary as is the value of e i e PAR 6 Note that if PAR 6 be comes large then the manifold may no longer be accurate The free parameters in this run are PAR 3 energy PAR 12 length of the orbit PAR 6 starting distance PAR 21 x coordinate at end point PAR 11 integration time PAR 22 y coordinate at end point sv Via save Via
154. e RB ee E Ee eee ES 94 be oo hee ome Eo A e E 95 tee eee Oe oe oe a eee ee Be ee N 95 EC 2 fs 4 4000 Sha he ew SS SENS SERS RE 95 9 4 Customizing the GU 3 20 4 amp oe o Glee ee oe ee oie E 95 94 1 Printzb ttoml 1424 24454 a ee we A 95 oe de E AA a te Soe ee ee ee oe Ae 95 Be sy ry ae ae eee es he ee ee ee a 96 97 ey oe elke e ee ee ee wee ee 97 aeaa dan da aos a 98 AA A eas Mckee a oe ee ees ea ee a en ee Ge es UE Ss al eo Ey 98 TO22 NBG 5 es tw eu ete A a a ere 98 MIA seit See gs cde Goede Mesa ee wee a Se eek he Ge a ete ee 98 TO2A NBAR oe ee a oe ee eee ee Ge Se ee we GE ee a ere E 98 T020 SACs ar ek hae vA Eo la OB a MR EAS SAE EOS ee 98 ob ae oe ae Oe Oe i OSE 98 10 3 0 NIST ice cw bade wee ae da Swe dak we ha eke ee we a 98 103 2 NCOM homicida ea he ea Oe e 99 aras epi 2 basis ps ee Bet 99 rica Gb RE ee ae OH Se GIR EMS Gee HS A A 99 MER AA 99 MEA ARE NETA 99 e Asa he Ae o ee TR e a 99 O E E E 99 TIAS 1 dr HK Ae A A A EM A A 100 TACTIL ea COR 100 ARA hte ee BEALE AS HESS 100 CREA ee SARE REED A CS ERS OES 100 0 5 2 DSMIN ssa bea 4 Bae a a a ae a 100 MSDS 4 a ag ye RE ae ew A Ge ee ee Ste ee a 100 TOSA TADS goede oh dk eh PR BE HAR HD ERE SH DOS HB 101 IM Tas aa Sey gene ew Hee Ge ee oe ee ee S 101 AAA s yee ob eee Be Se BER hk eee hee oh oe ee SSS 101 10 0 Diagram Limited c e a3 2 s feo ne Gee a ee Cee wR AAA wae G 101 MA gas we eS os be A a ee ee 102 10 6 2 a oe one A PS ee ee 102 10 6 3 A
155. e We a ee See ee ee A ee O a e daa od ge aed He ees Wie aoa oe 12 AUTO Demos Tutorial e A me GOR hem a SR oe e de 12 2 cusp A Tutorial Demo ep oo amp eRe A ae ee Whee 2 Ga IA ay o Dio Ge ke oe os A 12 5 Plotting the Results with AUTO 0 o o ooo Seer eee ee ee eee ee 12 7 Exporting the Results for different plotters ooa aa aa 12 8 ab A Programmed Demo 13 AUTO Demos Fixed points 13 1 enz Stationary Solutions of an Enzyme Model AR E Hee hie Se E a ee PRA AAA AA eo a 14 AUTO Demos Periodic solutions 14 1 lrz The Lorenz Equations xicos e oe be eee a A 14 2 abc The A gt B gt C Reaction o ee 14 3 pp2 A 2D Predator Prey Model 22 0 o ie e e oes 14 4 lor Starting an Orbit from Numerical Data 14 5 fre A Periodically Forced SysteM o 14 6 ppp Continuation of Hopf Bifurcations 2 04 116 116 117 117 117 118 118 119 120 120 120 121 123 124 124 126 127 127 128 129 130 14 7 plp Fold Continuation for Periodic Solutions o o 143 14 8 phl Phase Shifting using Continuation 2 2 45 ee ee eee ees 144 OS 145 14 10 tor Detection of Torus Bifurcations 4 55445 854 eee eee es 147 14 11 pen Rotations of Coupled Pendula 42 62 e228 4 2 ee 5 148 14 12 chu A Non Smooth System Chua s Circuit
156. e load command is not the same as the filename which is read in for example load constants ab 1 reads in the file c ab 1 This difference is a result of the automatic transformation of the filenames by the AUTO CLUI into the standard names used by AUTO The standard filename transformations are shown in Table Long name Short name Name entered Transformed file name equation e foo foo f90 foo f foo c constants c foo c foo solution S foo s foo bifurcationDiagram b foo b foo diagnostics d foo d foo homcont h foo h foo Table 4 3 This table shows the standard AUTO CLUI filename translations In load and run commands either the long name or the short name may be used for loading the appropriate files Since the load command is so common there are various shorthand versions of it First there are short versions of the various arguments as shown in Table For example the command load constants ab 1 can be shortened to load c ab 1 Next several different files may be loaded at once using the same load command For example the two commands in Figure 4 3 have the same effect as the single command in Figure Last you can bypass the load command unless the intermediate result is needed and use the run command directly on the load arguments as in Figure 4 5 AUTO gt ab load e ab Runner configured AUTO gt ab load ab c ab 1 Runner configured Figure 4
157. e of Technology and Concordia University All rights reserved Redistribution and use in source and binary forms with or without modification are permitted provided that the following conditions are met e Redistributions of source code must retain the above copyright notice this list of conditions and the following disclaimer e Redistributions in binary form must reproduce the above copyright notice this list of conditions and the following disclaimer listed in this license in the documentation and or other materials provided with the distribution e Neither the name of the copyright holders nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS AS IS AND ANY EXPRESS OR IMPLIED WARRANTIES INCLUDING BUT NOT LIMITED TO THE IMPLIED WAR RANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT INDIRECT INCIDENTAL SPECIAL EXEMPLARY OR CONSEQUENTIAL DAMAGES INCLUDING BUT NOT LIMITED TO PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES LOSS OF USE DATA OR PROFITS OR BUSINESS INTERRUPTION HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY WHETHER IN CONTRACT STRICT LIABILITY OR TORT INCLUDING NEGLIGENCE OR OTHERWISE ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE EVEN IF
158. e of the limit point function note nt commandQueryNote Print notes in info file secondaryperiod commandQuerySecondaryPeriod Print value of secondary periodic sc sp bif fen stepsize ss st commandQueryStepsize Print continuation step sizes quit q commandQuit Quit the AUTO CLUI relabel rl commandRelabel Relabel data files run r rn commandRun Run AUTO ch changecon commandRunnerConfigFort2 Modify continuation constants stant cc hch commandRunnerConfigFort12 Modify HomCont continuation con stants load ld commandRunnerLoadName Load files into the AUTO runner printconstant commandRunnerPrintFort2 Print continuation parameters pe pr hpr commandRunnerPrintFort12 Print HomCont continuation pa rameters shell commandShell Run a shell command splabs command5SpecialPointLabels Return special labels subtract sb commandSubtractBranches Subtract branches in data files 51 triple tr command Triple Triple a solution us userdata commandUserData Covert user supplied data files wait command Wait Wait for the user to enter a key auto execfile ex Execute an AUTO CLUI script demofile dmf Execute an AUTO CLUI script line by line demo mode 1 For convenience you can use to run a shell command Moreover the common shell com mands clear less mkdir rmdir cp mv rm 1s cd and cat and all AUTO Unix comman
159. ed at http gcc gnu org wiki Gfortran The following packages and their de 12 pendencies are recommended for Fedora e Python python matplotlib tk and ipython e PLAUTO0O4 SoQt devel For SoQt versions older than 1 5 0 to see pictures of stars the earth and the moon instead of white blobs compile simage from source see needs libjpeg devel e PLAUT xterm e GUI94 lesstif devel or openmotif devel e manual TFX tetex or texlive and transfig and the following for Ubuntu and Debian Python python matplotlib and ipython e PLAUTO4 SoQt libsoqt dev or libsoqt4 dev and libsimage dev e PLAUT xterm e GUI94 lesstif2 dev or libmotif dev e manual WIX tetex or texlive and transfig Other distributions may have packages with similar names If you need to compile and install one of the above PLAUT04 libraries from the source code available at the above web site and you find that after that PLAUT04 still does not work then you might need to adjust the environment variable LD_LIBRARY_PATH to include the location of these libraries for instance usr local lib 1 1 2 Installation on Mac OS X AUTO runs on Mac OS X using the above instructions provided that you have the development tools installed see the Mac OS X Install DVD Optional Installs Xcode You do not need to start an X server to run AUTO Furthermore the following packages are recommended e Gfortran See http gcc gnu org wiki GFortranBinaries Alte
160. ed interactive use of the Python CLUI it is also worth installing Python http 7 xpython scipy org The graphic tool for 3D AUTO data visualization PLAUT04 is compiled by default but depends on a few libraries that may not be in a standard installation of a typical Unix like system These libraries may be available as optional packages though In order of preference these are 1 Coin3D version 2 2 or higher SoQt 1 1 0 or higher and simage 1 6 or higher With SoQt 1 5 0 or higher simage is no longer required 2 Coin3D with the SoXt library which interfaces with Open Motif or LessTif version 2 0 or higher instead of Qt The user interface has a few problems with LessTif though in particular it is likely to crash on 64 bit machines so the Qt version or Open Motif is recommended 3 One can download SGI s implementation of the Open Inventor libraries from ftp oss sgi com projects inventor download Because SGI s implementation for Linux can not show text correctly we recommend that Coin is used instead of SGI s implementation The configure script checks for these libraries and outputs a warning if any of these libraries cannot be found It first checks for SoQt and then for SoXt unless you pass disable plaut04 qt as an option to configure If the libraries are not available you can still compile all other components of AUTO using make The Fortran code uses several routines that were not standardardized pri
161. eded are expected in s xxx AUTO constants in c xxx and HomCont constants in h xxx Type Ch xxx yyy to run AUTO HomCont with equations file xxx f90 and restart data file s yyy AUTO constants must be in c xxx and HomCont constants in h xxx Type Ch xxx yyy zzz to run AUTO HomCont with equations file xxx 90 restart data file s yyy and constants files c zzz and h zzz H The command CH xxx is equivalent to the command Ch xxx above Type CH xxx i to run AUTO HomCont with equations file xxx f90 and constants files c xxx i and h xxx i and if needed restart data file s xxx Type CH xxx i yyy to run AUTO HomCont with equations file xxx 90 constants files c xxx i and h xxx i and restart data file s yyy Use H on case insensitive file systems 5 8 Copying a demo dm Type dm xxx to copy all files from auto 07p demos xxx to the current user directory Here xxx denotes a demo name e g abc Note that the dm command also copies auto files to the current user directory To avoid the overwriting of existing files always run demos in a clean work directory 5 9 Viewing the manual mn Use gv or evince to view the PDF version of this manual 72 Chapter 6 Output Files AUTO writes to standard output and three output files standard output A summary of the computation is written to standard output which usually corresponds to the window in which AUTO is run Only special labeled solution poi
162. ee Alexander Doedel amp Othmer 1990 AUTO COMMAND ACTION mkdir frc create an empty work directory cd frc change directory demo frc copy the demo files to the work directory ri run e fre c frc homotopy to r 0 2 save r1 0 save output files as b 0 s 0 d 0 r2 run r1 UZ1 ICP 5 11 compute solution family restart from r1 NMX 20 DS 0 5 DSMAX 5 0 save r2 frc save output files as b frc s frc d frc Table 14 12 Commands for running demo frc 141 14 6 ppp Continuation of Hopf Bifurcations This demo illustrates the continuation of Hopf bifurcations in a 3 dimensional predator prey model Doedel 1984 This curve contain branch points where one locus of Hopf points bifurcates from another locus of Hopf points and generalized Hopf Bautin bifurcations GH where the Hopf bifurcation changes from sub to supercritical The diagnostics file d hb can be inspected to see where the Hopf bifurcation is subcritical and where it is supercritical The equations are uy uj l u1 pauu Uy p2U2 pau uz psuzuz pi 1 en Pou 14 7 Us P3U3z P5UgU3z Here pz 1 4 pg 1 2 pa 3 ps 3 pe 5 and p is the free parameter In the continuation of Hopf points the parameter pa is also free AUTO COMMAND ACTION mkdir ppp create an empty work directory cd ppp change directory demo ppp copy the demo files to the
163. efault effective number of equation parameters NPAR set to 36 upon installation It can be overridden in constant files See also Section 11 1 The default can be changed by editing auto h This must be followed by recompilation by typing make in the directory auto 07p src Note that in certain cases the effective dimension may be greater than the user dimension For example for the continuation of folds the effective dimension is 2NDIM 1 for algebraic equa tions and 2NDIM for ordinary differential equations respectively Similarly for the continuation of Hopf bifurcations the effective dimension is 3NDIM 2 1 3 Compatibility with Earlier Versions Unlike earlier versions AUTO can no longer be compiled using a pure Fortran 77 compiler but you need at least a Fortran 90 compiler A free Fortran 95 compiler GFortran is shipped with most recent Linux distributions or can be obtained at which contains binaries for Linux Mac OS X and Windows AUTO was also tested with the free compiler g95 and there exist various commercial Fortran 9x compilers as well The AUTO input files are now called c xxx the constants file and h xxx the HomCont constants file only used with HomCont the output files are called b xxx the bifurcation diagram file s xxx the solution file and d xxx the diagnostics file The command rn can be used to rename all these files from their old names There are also minor changes in the formatting of these files co
164. ell and Python commands for the Lla demo part 1 156 OR man Lia 1 startLia Lia run startLia c man Lia 1 Look at c man Lla 1 to see from which label in s startLla this run starts In this run the y coordinate of the end point PAR 22 is kept fixed while the period PAR 11 i e the total integration time is allowed to vary as is the value of epsilon i e PAR 6 Note that if PAR 6 becomes large then the manifold may no longer be accurate The free parameters in this run are PAR 3 energy PAR 12 length of the orbit PAR 6 starting distance PAR 21 x coordinate at end point PAR 11 integration time PAR 23 z coordinate at end point sv Lia save Lia Lia Save the results in b Lla s Lla and d Lla CR man Lia 2 startLia L1a2 run startLia c man Lia 2 Another run starting from a longer initial orbit which computes part of the manifold The free parameters are the same as in the preceding run This computation results in the orbit winding around the selected periodic L1 orbit sv Lia2 save Lia2 L1a2 Save the results in b Lla2 s Lla2 and d Lla2 Table 14 23 Detailed AUTO shell and Python commands for the Lla demo part 2 Give the label of the selected solution and a value that is smaller than the associated Floquet multiplier magnitude greater than 1 autox ext py L1 3 1le 5 2000 import ext sext ext get L1 3 1e 5 2000
165. en column wise as u t0 tn x0 xn y0 yn The PAR p keyword argument takes a dictionary p as a in Section 10 8 9 for instance PAR 1 5 0 2 0 0 to set PAR 1 5 0 and PAR 2 0 0 You can check your new solution using the command print s An external orbit from an ASCII file can be directly imported by AUTO via the dat AUTO constant see Section 4 9 Exporting output data for use by Python or external visualization tools The bifurcation and solution file classes have three methods that are particularily useful for creating data which can be used in other programs First there is a method called toArray which takes a bifurcation diagram or solution and returns a Python array a list of lists Second the method toarray returns a Numerical Python numpy array which works for branches points and solutions but not for lists of branches Third there is a method called writeRawFilename which will create a standard ASCII file which contains the bifurcation diagram or the solution In the solution ASCII file the first element of each row will be the t value and the following elements will be the values of the components at that t value Such ASCII files can be readily parsed and plotted by external tools such as Gnuplot and MATLAB For example we assume that a bifurcation diagram object is contained in a variable bd for instance using bd loadbd ab If one wanted to have the bifurcation diagram
166. endent The MPI message passing library is not used by default You can enable it by typing configure with mpi If OpenMP and MPI are both used then AUTO uses mixed mode with MPI parallelisation occurring at the top level Running the MPI version is somewhat more complex because of the fact that MPI normally uses some external program for starting the computational processes The exact name and command line options of this external program depends on your MPI installation A common name for this MPI external program is mpirun and a common command line option which 16 defines the number of computational processes is np Accordingly if you wanted to run the MPI version of AUTO on four processors with the above external program you would type mpirun np 4 file exe Please see your local MPI documentation for more detail Both the Python CLUI and the commands in the auto 07p cmds directory described in Chapter 5 may be used with the MPI version as well by setting the AUTO COMMAND PREFIX environment variable For example to run AUTO in parallel using the MPI library on 4 pro cessors just type export AUTO_COMMAND_PREFIX mpirun np 4 before you run the Python CLUI auto or the commands in auto 07p cmds normally The previous example assumed you are using the sh shell or the bash shell for other shells you should modify the commands ap propriately for example setenv AUTO_COMMAND_PREFIX mpirun np 4 for the csh and tcsh shells A
167. ent most demos use a value that was found appropriate after some experimentation See also the discussion in Section 11 2 10 5 3 DSMAX The maximum allowable absolute value of the pseudo arclength stepsize DSMAX must be positive It is only effective if the pseudo arclength step is adaptive i e if IADS gt 0 The choice of DSMAX is highly problem dependent most demos use a value that was found appropriate after some experimentation See also the discussion in Section 11 2 100 10 5 4 IADS This constant controls the frequency of adaption of the pseudo arclength stepsize IADS 0 Use fixed pseudo arclength stepsize i e the stepsize will be equal to the specified value of DS for every step The computation of a family will be discontinued as soon as the maximum number of iterations ITNW is reached This choice is not recommended Demo tim TADS gt 0 Adapt the pseudo arclength stepsize after every IADS steps If the New ton Chord iteration converges rapidly then DS will be increased but never beyond DSMAX If a step fails then it will be retried with half the stepsize This will be done repeatedly until the step is successful or until DS reaches DSMIN In the latter case nonconvergence will be signalled The strongly recommended value is IADS 1 which is used in almost all demos 10 5 5 THL By default the pseudo arclength stepsize includes all state variables or state functions and all free parameters Under c
168. entered inertial system Figure displays the layout of the Center menu 80 F Draw Reference Plane F Draw Primaries F Draw Libration Pts Orbit Animation 2 NONE F Satellite Animation Draw Background F Add Legend Normalize Data Coord Origin Left and Back Left and Ahead Draw Scale PREFERENCES Figure 8 3 The Draw Coordinate Axes Menu Figure 8 4 The Options Menu 2 Rotating Frame Bary Centered Big Primary Centered About Small Primary Centered HELP Figure 8 5 The Center Menu Figure 8 6 The Help Menu 8 1 7 Help The Help menu provides an on line help on how to use PLAUTO4 8 1 8 Picking a point in the diagram The picking operation is useful when we want to know data corresponding to a certain point in the diagram In order to execute a picking operation we should follow these steps e Click the arrow icon to change the mouse to picking state e Move the mouse to the point of interest e Click the left button of the mouse to pick the point Once a point has been picked a new window is popped up In this new window the Floquet multipliers of the point are shown in an x y plane Black crosses in the diagram indicate the Floquet Multipliers The solution and the values of the corresponding Floquet Multipliers are given in the lower part of the window A unit circle is drawn in the diagram Figure 8 7 is an example of the picking operation From this diagram we can see that two Floquet Multipliers are out
169. er Anal 28 No 5 1446 1462 FitzHugh R 1961 Impulses and physiological states in theoretical models of nerve mem brane Biophys J 1 445 446 Freire E Rodriguez Luis A Gamero E amp Ponce E 1993 A case study for homo clinic chaos in an autonomous electronic circuit A trip from Takens Bogdanov to Hopf Shilnikov Physica D 62 230 253 Friedman M Doedel E J amp Monteiro A C 1994 On locating connecting orbits Applied Math And Comp 65 1 3 231 239 Friedman M J amp Doedel E J 1991 Numerical computation and continuation of invariant manifolds connecting fixed points SIAM J Numer Anal 28 789 808 Henderson M E amp Keller H B 1990 Complex bifurcation from real paths STAM J Appl Math 50 No 2 460 482 Hodgkin A L amp Huxley A F 1952 A quantitative description of membrane current and its applications to conduction and excitation in nerve J Physiol 117 500 544 Holodniok M Knedlik P amp Kub cek M 1987 Continuation of periodic solutions in parabolic differential equations in T K pper R Seydel amp H Troger eds Bifurcation Analysis Algorithms Applications Vol INSM 79 Birkhauser Basel pp 122 130 Homburg A amp Krauskopf B 2000 Resonant homoclinic flip bifurcations J Dyn Diff Eqns 12 4 807 850 Hunt G W Bolt H M amp Thompson J M T 1989 Structural
170. er of Newton iterations per continuation step in the info file d xxx Aliases it commandQuerylterations note Print notes in info file Type note x to show any notes in the diagnostics of the bifurcation diagram object x Type note to show any notes in the output file fort 9 Type note xxx to show any notes in the info file d xxx Aliases nt commandQueryNote 58 stepsize Print continuation step sizes Type stepsize x to list the continuation step size for each continuation step in the diagnostics of the bifurcation diagram object x Type stepsize to list the continuation step size for each continuation step in fort 9 Type stepsize xxx to list the continuation step size for each continuation step in the info file d xxx Aliases ss st commandQueryStepsize eigenvalue Print eigenvalues of Jacobian algebraic case Type eigenvalue x to list the eigenvalues of the Jacobian in the diagnostics of the bifurcation diagram object x Algebraic problems Type eigenvalue to list the eigenvalues of the Jacobian in fort 9 Type eigenvalue xxx to list the eigenvalues of the Jacobian in the info file d xxx Aliases ev eg commandQuery Eigenvalue floquet Print the Floquet multipliers Type floquet x to list the Floquet multipliers in the diagnostics of the bifurcation diagram object x Differential equations Type floquet to list the in the output file fort 9 Type floquet xxx to list the F
171. erates an extended system for example when ISW 2 For ISW see Section L0 8 4 Demos int dd2 obt plp ops JAC 1 Asfor JAC 1 but derivatives with respect to problem parameters may be omitted in FUNC Demo san 10 3 Discretization Constants 10 3 1 NTST The number of mesh intervals used for discretization NTST remains fixed during any particular run but can be changed when restarting For mesh adaption see IAD in Section 10 3 3 Recommended value of NTST As small as possible to maintain convergence Demos exp ab spb 98 10 3 2 NCOL The number of Gauss collocation points per mesh interval 2 lt NCOL lt 7 NCOL remains fixed during any given run but can be changed when restarting at a previously computed solution The choice NCOL 4 used in most demos is recommended If NDIM is large and the solutions very smooth then NCOL 2 may be appropriate 10 3 3 IAD This constant controls the mesh adaption TAD 0 Fixed mesh Normally this choice should never be used as it may result in spurious solutions Demo ext IAD gt 0 Adapt the mesh every IAD steps along the family Most demos use TAD 3 which is the strongly recommended value When computing trivial solutions to a boundary value problem for example when all solution components are constant then the mesh adaption may fail under certain circumstances and overflow may occur In such case try recomputing the solution f
172. ers constants defining the Computation and constants that specify Output options 9 1 3 The Load Constants buttons The Previous button can be used to load an existing AUTO constants file Such a file is also loaded if it exists by the Equations button on the Menu Bar The Default button can be used to load default values of all AUTO constants Custom editing is normally necessary 9 1 4 The Stop and Exit buttons The Stop button can be used to abort execution of an AUTO run This should be done only in exceptional circumstances Output files if any will normally be incomplete and should be deleted Use the Exit button to end a session 9 2 The Menu Bar 9 2 1 Equations button This pull down menu contains the items Old to load an existing equations file New to load a model equations file and Demo to load a selected demo equations file Equations file names are of the form xxx f90 The corresponding constants file c xxx is also loaded if it exists The equation name xxx remains active until redefined 9 2 2 Edit button This pull down menu contains the items Cut and Copy to be performed on text in the GUI window highlighted by click and drag action of the mouse and the item Paste which places editor buffer text at the location of the cursor 9 2 3 Write button This pull down menu contains the item Write to write the loaded files xxx f90 and c xxx by the active equation name and the item Write As to write these fi
173. ers may wish to add their own test functions by editing the function PSIHO in autlib5 f It is important to remember that in order to specify activated test functions it is required to also add the corresponding label 20 to the list of continuation parameters and a zero of this parameter to the list of user defined output points Having done this the corresponding parameters are output to the screen and zeros are accurately located 210 20 7 Starting Strategies There are four possible starting procedures for continuation i ii iii iv Data can be read from a previously obtained output point from AUTO e g from contin uation of a periodic orbit up to large period note that if the end point of the data stored is not close to the equilibrium a phase shift must be performed by setting ISTART 4 These data can be read from fort 8 saved to s xxx by making IRS correspond to the label of the data point in question Data from numerical integration e g computation of a stable periodic orbit or an ap proximate homoclinic obtained by shooting can be read in from a data file using the AUTO constant dat see Section 10 8 7 The numerical data should be stored in a file xxx dat in multi column format according to the read statement READ T J UCI J I 1 NDIM where T runs in the interval 0 1 When starting from this solution IRS should be set to O and the value of ISTART is irrelevant By setting IST
174. ertain circumstances one may want to modify the weight accorded to individual parameters in the definition of stepsize For this purpose THL defines the parameters whose weight is to be modified If THL then all weights will have default value 1 0 else one must enter pairs Parameter Index Weight For example for the computation of periodic solutions it is recommended that the period not be included in the pseudo arclength continuation stepsize in order to avoid period induced limitations on the stepsize near orbits of infinite period This exclusion can be accomplished by setting THL 11 0 0 If THL is not specified this is the default for computing periodic solutions IPS 2 Most demos that compute periodic solutions use this option see for example demo ab 10 5 6 THU Under certain circumstances one may want to modify the weight accorded to individual state variables or state functions in the definition of stepsize For this purpose THU defines the number of states whose weight is to be modified If THU then all weights will have default value 1 0 else one must enter pairs State Index Weight At present none of the demos use this option 10 6 Diagram Limits There are five ways to limit the computation of a family By specifying a stopping condition in the list associated with the constant STOP see Section 10 6 1 By specifying parameters and parameter values in the list associated with the constant
175. ese two folders the manifolds can be computed by running the commands auto attr auto and auto rep auto respectively The alternative script files attrc auto and repc auto use constants files for every step instead of specifying the constants in the script The main folder contains AUTO files to continue in parameter space six of the secondary canards of this systems The script fnc auto first computes the attracting and repelling slow manifolds in their respective folders and next concatenates matching manifolds in Python Six of these canard orbits are then continued in e in both decreasing and increasing direction If T is the integration time for the canard segment 5 on the attracting slow manifold and T is the integration time for the orbit segment on the repelling slow manifold corresponding to the same canard solution then the concatenated orbit segment has Tf T T as integration time The orbit is obtained from and by concatenating f and and rescaling the time so that it runs monotonically from 0 to 1 Parameters are then also copied where only those relating to the start of the attracting manifold and the end of the repelling manifold are kept for these parameters are used for the boundary conditions of the canard orbits The new integration time 7 is stored in PAR 11 Two constant files are provided They correspond to the continuation in epsilon in both decreasing c fnc and and increasing c fnc ep
176. estart data from the last label of r1 save r2 2 save output files as b 2 s 2 d 2 Table 16 1 Commands for running demo pdl 182 16 2 pd2 Stationary States 2D Problem This demo uses Euler s method to locate a stationary solution of a nonlinear parabolic PDE followed by continuation of this stationary state in a free problem parameter The equations are Ou Ot D 07u Ox p u 1 u urus uz t Dz Pus 0x uz uo 16 1 on the space interval 0 L where L PAR 11 1 is fixed throughout as are the diffusion constants D PAR 15 1 and Da PAR 16 1 The boundary conditions are u 0 u1 L 0 and u2 0 ua L 1 for all time In the first run the continuation parameter is the independent time variable namely PAR 14 while p 12 is fixed The AUTO constants DS DSMIN and DSMAX then control the step size in space time here consisting of PAR 14 and w x u2 x Initial data at time zero are u x sin rx L and us x 1 Note that in the subroutine STPNT the initial data must be scaled to the unit interval and that the scaled derivatives must also be provided see the equations file pv2 f90 In the second run the continuation parameter is p A branch point is located during this run Euler time integration is only first order accurate so that the time step must be sufficiently small to ensure correct results Indeed this option has been added only as a convenience
177. ether with the constants in the files c data and h data Not all of these files need to be present If no solution data is specified then the global values from the load command below are used instead where options which are not explicitly set retain their previous value Keyword argument options can be AUTO constants such as DS 0 05 or ISW 1 or specify a constant or solution file These override the constants in s c where applicable See load run s options is equivalent to run load s options Example given a bifurcation diagram bd with a branch point solution switch branches and stop at the first Hopf bifurcation hb run bd BP1 ISW 1 STOP HB1 53 Special keyword arguments are sv and ap sv is also an AUTO constant run bd BP1 ISW 1 STOP HB1 sv hb ap all saves to the files b hb s hb and d hb and appends to b all s all and d all Aliases r rn commandRun load Load files into the AUTO runner or return modified solution data Type result load options to modify the AUTO runner Type result load data loptions to return possibly modified solution data The type of the result is a solution object load data options returns a solution in the following way for different types of data e A solution load returns the solution data with AUTO constants modified by options e A bifurcation diagram or a solution list returns the solution specified by the AU
178. everal points on the complete family are plotted in Fig If we had continued further by increasing NMX the computation would end at a no convergence 237 0 00 0 20 0 40 0 60 0 80 1 00 0 10 0 30 0 50 0 70 0 90 Time Figure 24 1 Solutions of the boundary value problem at labels 6 and 8 either side of the Shil nikov Hopf bifurcation Figure 24 2 Phase portraits of three homoclinic orbits on the family showing the saddle focus to saddle transition 238 error TY MX owing to the homoclinic family approaching a Bogdanov Takens singularity at small amplitude To compute further towards the BT point we would first need to continue to a higher value of PAR 11 24 2 Detailed AUTO Commands AUTO COMMAND ACTION mkdir cir create an empty work directory cd cir change directory demo cir copy the demo files to the work directory ri run c cir 1 sv 1 increase the truncation interval restart from cir dat save output files as b 1 s 1 d 1 r2 run ri UZ2 c cir 2 sv 2 continue saddle focus homoclinic orbit restart from r1 save output files as b 2 s 2 d 2 r3 run r2 UZ1 c cir 3 ap 2 generate adjoint variables restart from r2 append output files as b 2 s 2 d 2 Table 24 1 Detailed AUTO Commands for running demo cir 239 Chapter 25 Ho
179. eys et al 1996 209 i 1 Resonant eigenvalues neutral saddle y A1 1 2 Double real leading stable eigenvalues saddle to saddle focus transition 41 ua 1 3 Double real leading unstable eigenvalues saddle to saddle focus transition i Ao 1 4 Neutral saddle saddle focus or bi focus includes i 1 Re 111 Re A1 i 5 Neutrally divergent saddle focus stable eigenvalues complex Re A1 Re u1 Re u2 1 6 Neutrally divergent saddle focus unstable eigenvalues complex Re pi1 Re A1 Re A2 1 7 Three leading eigenvalues stable Re A1 Re u1 Re u2 1 8 Three leading eigenvalues unstable Re u Re A1 Re A i 9 Local bifurcation zero eigenvalue or Hopf number of stable eigenvalues decreases Re p11 0 1 10 Local bifurcation zero eigenvalue or Hopf number of unstable eigenvalues decreases Re A 0 i 11 Orbit flip with respect to leading stable direction e g 1D unstable manifold i 12 Orbit flip with respect to leading unstable direction e g 1D stable manifold i 13 Inclination flip with respect to stable manifold e g 1D unstable manifold i 14 Inclination flip with respect to unstable manifold e g 1D stable manifold i 15 Non central homoclinic to saddle node in stable manifold i 16 Non central homoclinic to saddle node in unstable manifold Expert us
180. f ALL is chosen all solutions are shown in the diagram If NONE is chosen none of the solutions is shown HALF shows the solutions with odd labels and special solutions only SPEC lets the system show the special solutions only We can also show selected solutions by inputting their labels in the list box separated by commas For example typing 1 10 15 20 will lead the system to show only the solutions with label 1 10 15 and 20 We can set the default value for this list in the PLAUTO4 resource file 8 1 11 Coloring Many coloring methods are provided They can be classified into three groups The first group is coloring by variables This group provides as many choices as the number of variables of a problem plus 1 for the time The second group is coloring by parameters These parameters are defined by the AUTO user in the AUTO constants file There are as many choices as the number of parameters defined in the AUTO constants file The third group includes TYPE type of solution PONT point number BRAN the branch to which the solution belongs and LABL label of the solution Different coloring methods cannot be used at the same time Figure 8 10 shows the difference between coloring by type and coloring by label From Figure 8 10 a we can see that there is only one branching orbit in this family which is shown in cyan In Figure b the start solution is colored in blue and the last sol
181. f fort files dlb commandDeleteLabels Delete special labels dsp commandDeleteSpecialPoints Delete special points double commandDouble Double a solution man commandInteractiveHelp Get help on the AUTO commands 50 klb commandKeepLabels Keep special labels ksp commandKeepSpecialPoints Keep special points ls commandLs List the current directory merge mb commandMergeBranches Merge branches in data files move mv command MoveFiles Move data files to a new name cn constantsget commandParseConstantsFile Get the current continuation con stants bt diagramand solutionget commandParseDiagramAndSolutionFile Parse both bifurcation diagram and solution dg diagramget commandParseDiagramFile Parse a bifurcation diagram sl solutionget commandParseSolutionFile Parse solution file plot p2 pl commandPlotter plotting of data plot3 p3 commandPlotter3D PLAUT04 plotting of data branchpoint br bp commandQueryBranchPoint Print the branch point function eigenvalue ev commandQueryEigenvalue Print eigenvalues of Jacobian alge eg braic case floquet fl commandQueryF loquet Print the Floquet multipliers hopf hb hp commandQueryHopf Print the value of the Hopf func tion iterations it commandQuerylterations Print the number of Newton inter ations limitpoint lm lp commandQueryLimitpoint Print the valu
182. f the right hand equilibrium the equilibrium approached by the orbit as t gt 00 The same default as for NUNSTAB applies here 20 3 3 IEQUIB TEQUIB 0 Homoclinic orbits to hyperbolic equilibria the equilibrium is specified explic itly in PVLS and stored in PAR 11 1 I 1 NDIM TEQUIB 1 default Homoclinic orbits to hyperbolic equilibria the equilibrium is solved for during continuation Initial values for the equilibrium are stored in PAR 11 1 I 1 NDIM in STPNT TEQUIB 2 Homoclinic orbits to a saddle node initial values for the equilibrium are stored in PAR 11 I I 1 NDIMin STPNT TEQUIB 1 Heteroclinic orbits to hyperbolic equilibria the equilibria are specified explic itly in PVLS and stored in PAR 11 1 I 1 NDIM left hand equilibrium and PAR 11 1 I NDIM 1 2 NDIM right hand equilibrium 206 TEQUIB 2 Heteroclinic orbits to hyperbolic equilibria the equilibria are solved for during continuation Initial values are specified in STPNT and stored in PAR 11 1 I 1 NDIM left hand equilibrium PAR 11 1 I NDIM 1 2 NDIM right hand equilibrium 20 3 4 ITWIST ITWIST 0 default the orientation of the homoclinic orbit is not computed ITWIST 1 the orientation of the homoclinic orbit is computed For this purpose the adjoint variational equation is solved for the unique bounded solution If IRS 0 an initial solution to the adjoint equation must be specified as well However if IRS gt 0 and ITW
183. f the variable indicating the orientation is small compared to its value at the other regular points Data for the adjoint equation at LAB 4 6 and 8 at and on either side of the inclination flip are presented in Fig The switching of the solution between components of the leading unstable left eigenvector is apparent Finally we remark that the Newton step in the dummy parameter PAR 20 performed above is crucial to obtain convergence Indeed if instead we try to continue the homoclinic orbit and the solution of the adjoint equation directly by setting 215 ITWIST 1 IRS 2 NMX 50 ICP 1 4 and running r4 run r1 ICP 4 8 10 21 33 ITWIST 1 IPSI 1 13 NMX 50 UZR 33 0 we obtain a no convergence error 21 3 Non orientable Resonant Eigenvalues Inspecting the output saved in the third run we observe the existence of a non orientable homoclinic orbit at the second UZ label 6 We restart at this label with the first continuation parameter being once again a PAR 1 by changing constants according to DS 0 05 NMX 20 IcP 1 1 Running r5 run r3 UZ2 ICP 1 8 10 21 33 NMX 20 DS sv 5 the output at label 9 BR PT TY LAB PAR 1 PAR 8 PAR 10 PAR 21 1 8 UZ 9 1 30447E 07 3 41490E 12 1 63406E 09 2 60894E 07 indicates that AUTO has detected a zero of PAR 21 implying that a non orientable resonant bifurcation occurred at that point 21 4 Orbit Flip In this section we compute an orbit
184. flip To this end we restart from the original explicit solution without computing the orientation We begin by separately performing continuation in a 1 6 1 a fe b 1 and u A in order to reach the parameter values a b a 0 u 0 5 3 1 0 0 25 The sequence of continuations up to the desired parameter values are run via r6 run san ICP 4 8 UZR 4 1 r7 run r6 ICP 5 8 UZR 5 0 DS r8 run r7 ICP 1 8 UZR 1 0 5 DS r9 run r8 ICP 2 8 UZR 2 3 0 ri0 run r9 ICP 7 8 UZR 7 0 25 with appropriate continuation parameters and user output values set The desired output is stored in r10 The final saved point LAB 6 contains a homoclinic solution at the desired parameter values From here we perform continuation in the negative direction of u A PAR 7 PAR 8 with the test function 4 11 for orbit flips with respect to the stable manifold activated 216 rii run ri0 1CP 7 8 31 1PSI 11 UZR 431 0 0 7 0 5 D8 save rii 11 The output detects an inclination flip by a zero of PAR 31 at PAR 7 0 BR PT TY LAB 1 5 UZ PAR 7 7 6 33545E 06 PAR 8 1 70968E 06 PAR 31 8 70508E 05 at which parameter value the homoclinic orbit is contained in the x y plane see Fig 21 2 Finally we demonstrate that the orbit flip can be continued as three parameters PAR 6 PAR 7 PAR 8 are varied of r1i UZ1 r1i2 run of ICP 7 8 6 IPSI
185. ft and corresponding orbit segments in phase space right where the connection is continued in v and v2 To obtain these figures run plot cb or pp cb and plot3 cb or pl cb Label 13 denotes the largest connection at a saddle node bifurcation of limit cycles and label 16 the smallest one where the periodic orbit disappears in a Hopf bifurcation 15 14 3 The codimension zero point to cycle connection Next we compute the codimension zero connection back to the cycle which must also exist near the accumulation of the parameter space curve h This computation starts in the same way as 176 the computation of the codimension one connection steps 1 to 4 are the same except that in step 3 we now find the negative Floquet exponent y instead of the positive exponent Steps 5 to 8 proceed as follows nu2 5 1 45F 1 46 F 1 47 F 1 48 1 49 1 50 Similarly to step 5 before we compute an orbit in the stable manifold of the periodic orbit However Lin s method is not used because it is easier to use a homotopy method to connect directly to the unstable eigenspace E b which is given by the section where p arccos 12 2 Because we compute an approximation to the stable rather than the unstable manifold using almost the same boundary value problem we let T be negative Also the distance from the periodic orbit to the connection is now given by 6 107 flipping its sign Improve t
186. h radiationless oscillating tails Physical Review E 51 3572 3578 Champneys A amp Kuznetsov Y 1994 Numerical detection and continuation of codimension two homoclinic bifurcations Int J Bifurcation amp Chaos 4 795 822 Champneys A amp Spence A 1993 Hunting for homoclinic orbits in reversible systems a shooting technique Adv Comp Math 1 81 108 Champneys A Kuznetsov Y amp Sandstede B 1996 A numerical toolbox for homoclinic bifurcation analysis Champneys A R amp Groves M D 1997 A global investigation of a solitary wave solutions to a fifth order two parameter model equation for water waves J Fluid Mechanics 342 199 229 de Boor C amp Swartz B 1973 Collocation at gaussian points SIAM J Numer Anal 10 582 606 Dercole F 2008 BPcont An AUTO driver for the continuation of branch points of algebraic and boundary value problems SIAM J Sci Comput 30 2405 2426 to appear 261 Desroches M Krauskopf B amp Osinga H M 2008 Mixed mode oscillations and slow man ifolds in the self coupled fitzhugh nagumo system CHAOS 18 1 015107 Doedel E J 1981 AUTO a program for the automatic bifurcation analysis of autonomous systems Cong Numer 30 265 384 Doedel E J 1984 The computer aided bifurcation analysis of predator prey models J Math Biol 20 1 14 Doedel E J amp Heine
187. h uses the activator inhibitor model in Doedel Keller amp Kern vez 1991a shows how one can phase shift a periodic solution This can be useful in applications for example when one wants a component of a periodic solution to have a specific value at time 0 The equations are given by s so s pR s a a a a a pR s a S where si R s a Eo Ra The first two runs compute a family of stationary solutions and a bifurcating family of periodic solutions The free problem parameter in these runs is p The results are saved in the files b sa s sa and d sa The third run starts at a specified periodic solution in s sa namely the solution with label 6 and phase shifts this solution in time until s 0 30 The above sequence of calculations can be carried out by running the Python script phl auto included in the demo The basic idea for doing the phase shift in the third run is to drop the integral phase condition which is automatically added when the AUTO constant IPS has value 2 For this purpose the third run uses the value 4 for IPS as specified in c ph1 in which case the periodicity conditions must be specified explicitly in the subroutine BCND in the equations file ph1 f90 Also the interval of periodicity must be scaled explicitly to the interval 0 1 which introduces the period T as an explicit parameter in the differential equations Note that no integral phase condition is specified in ICND The problem form
188. he connection computed in step 5 by decreasing the negative value of T7 fixing the starting point in b and freeing Follow the codimension zero connection in the system parameter v4 together with u T 6 and T7 also adding an integral condition for the connection AUTO detects two fold points LP corresponding to tangencies of W b and W D Continue the two folds forwards and backwards in two parameters by adding the system parameter va and setting the AUTO constant ISW 2 The folds terminate where I dis appears in a Hopf bifurcations small v and disappears in a saddle node bifurcation of limit cycles large v In both cases the continued connection and orbit stop converging so AUTO reports MX phigamma 0 50 po 7 0 25 000 yas o SS Gamma EE O ye 0 75 0 70 0 71 0 72 0 73 0 74 nui Figure 15 5 Parameter space diagram left and the corresponding homoclinic orbit on the snaking curve ht for label 20 right The snaking curve is in between the two tangencies for the codimension zero EtoP connection t and terminates at a segment of the codimension one EtoP connection c To obtain these figures run plot all or pp all and plot3 all or pl all 177 15 15 fnc Canards in the FitzHugh Nagumo system This demo computes attracting and repelling slow manifolds in the self coupled FitzHugh Nagumo system v h v v 1 2 ys h e 2h
189. he current user directory Here xxx denotes a demo name e g abc To avoid the overwriting of existing files always run demos in a clean work directory Aliases copydemo commandCopyDemo 4 14 7 Python data structure manipulation functions All commands here except for man gui and wait are only provided for backwards compat ibility Alternatives are given man Get help on the AUTO commands Type man to list all commands with a online help Type man xxx or help xxx to get help for command xxx Aliases commandInteractiveHelp cn Get the current continuation constants Type cn xxx to get a parsed version of the constants file c xxx This is equivalent to the command loadbd xxx c Aliases constantsget commandParseConstantsFile dg Parse a bifurcation diagram Type dg xxx to get a parsed version of the diagram file b xxx This is equivalent to the command loadbd xxx but without the solutions in s xxx and without the diagnostics in d xxx Aliases diagramget commandParseDiagramPFile sl Parse solution file Type s1 xxx to get a parsed version of the solution file s xxx This is equivalent to the command loadbd xxx 0 Aliases solutionget commandParseSolutionFile 63 bt Parse both bifurcation diagram and solution Type bt xxx to get a parsed version of the diagram file b xxx and solution file s xxx This is equivalent to the command loadbd xx
190. he solution to the adjoint equation to obtain the Lin vector Since both ingredients are there we can now continue in p e and T to obtain the initial Lin gap Recall from Chapter 20 that the Lin gaps e correspond to PAR 20 i 2 and the time intervals T correspond to PAR 21 i 2 We stop when e 0 2 We need to specify ITWIST 2 to tell HomCont we aim to find a 2 homoclinic orbit so that it will split it up in three parts with two potential Lin gaps We effectively have a 9 dimensional system at this point r7 run r6 UZ1 ICP 21 22 5 NMX 300 NPR 10 UZR 22 0 2 ISTART 2 IPSI save r7 7 BR PT TY LAB PAR 21 L2 NORM e PAR 22 PAR 5 3 10 18 3 45897E 01 4 46818E O1 7 87712E 07 1 55885E 11 3 20 19 2 73699E 01 4 46818E 01 2 91119E 05 1 63974E 09 3 30 20 1 73720E 01 4 46817E 01 4 42273E 03 3 10167E 05 3 38 UZ 21 1 01451E 01 4 46796E 01 2 00000E 01 1 48615E 02 251 Columns 3 Columns 147 4 000 01 3 008 01 2 008 01 4 006 01 a 1 00 00 Cotes 2 008 01 4000 01 Pp Soe 1 008 00 Figure 27 2 Behaviour of the second piece of the broken homoclinic orbit when creating a Lin gap a Projection of the broken homoclinic orbit onto the x y plane where e 0 2 To include all the pieces necessary to obtain this figure the X box must contain U 1 U 4 U 7 and the Y box must contain U 2 U 5 U 8 b The outpu
191. hese in two parameters detect whether the Hopf bifurcation is sub or supercritical and detect zero Hopf Bogdanov Takens and generalized Hopf Bautin bifurcations Demos pp3 and ppp Locate folds limit points continue these in two parameters and detect cusp zero Hopf and Bogdanov Takens bifurcations Locate branch points folds period doubling and torus Neimark Sacker bifurcations continue these in two or three parameters and switch branches at branch points and period doubling bifurcations for fixed points of the discrete dynamical system u D f u y Demo dd2 Find extrema of an objective function along solution families and successively continue such extrema in more parameters Demo opt 2 3 Ordinary Differential Equations For the ODE 2 2 the program can Compute families of stable and unstable periodic solutions and compute the Floquet mul tipliers that determine stability along these families Starting data for the computation of periodic orbits are generated automatically at Hopf bifurcation points Demo ab Run 2 Locate folds branch points period doubling bifurcations and bifurcations to tori along families of periodic solutions Branch switching is possible at branch points and at period doubling bifurcations Demos tor lor Continue folds period doubling bifurcations and bifurcations to tori in two parameters detecting 1 1 1 2 1 3 and 1 4 resonances Demos pl
192. ical report Cornell Theory Center Center for Applied Mathematics Cornell University Doedel E J Keller H B amp Kern vez J P 1991a Numerical analysis and control of bifurcation problems I Bifurcation in finite dimensions Int J Bifurcation and Chaos 1 3 493 520 Doedel E J Keller H B amp Kern vez J P 19916 Numerical analysis and control of bifurcation problems II Bifurcation in infinite dimensions Int J Bifurcation and Chaos 1 4 745 772 Doedel E J Kooi B W van Voorn G A K amp Kuznetsov Y A 2008a Continuation of connecting orbits in 3D ODESs I Point to cycle connections Int J Bifurcation and Chaos 18 1889 1903 262 Doedel E J Kooi B W van Voorn G A K amp Kuznetsov Y A 2008b Continuation of connecting orbits in 3D ODEs II Cycle to cycle connections To appear in Int J Bifurcation and Chaos Doedel E J Paffenroth R C Champneys A R Fairgrieve T F Kuznetsov Y A Olde man B E Sandstede B amp Wang X J 2000 AUTO2000 Software for continuation and bifurcation problems in ordinary differential equations Technical report California Institute of Technology Pasadena CA 91125 Fairgrieve T F 1994 The computation and use of Floquet multipliers for bifurcation analysis PhD thesis University of Toronto Fairgrieve T F amp Jepson A D 1991 O K Floquet multipliers SIAM J Num
193. ich end in the suffix auto are called basic scripts and can be run by typing auto scriptname auto The scripts shown in Section and Section are examples of basic scripts Scripts which end in the suffix xauto are called expert scripts and can be run by typing autox scriptname xauto More information on expert scripts can be found in Section See the README file in that directory for more information 4 3 First Example We begin with a simple example of the AUTO CLUI In this example we copy the ab demo from the AUTO installation directory and run it For more information on the ab demo see Section 12 8 The commands listed in Table 4 2 will copy the demo files to your work directory and run the first part of the demo The results of running these commands are shown in Figure Let us examine more closely what action each of the commands performs First demo ab Section 4 14 6 in the reference copies the files in AUTO_DIR demo ab into the work directory Next ab load equation ab Section 4 14 1 in the reference informs the AUTO CLUI that the name of the user defined function file is ab f90 The commands load and the closely related run are two of the most commonly used commands in the AUTO CLUI since they read and parse the user files which are manipulated by other commands The AUTO CLUI stores this setting in the variable ab until it is changed by a command such as another load command The idea of storing
194. igure 4 18 This example shows how to change branch numbers and delete a branch in the ab demo output file Using the coloring method branch setting you can then give one color to all branches of fixed points and one other color to all branches of periodic orbits The source for this script can be found in AUTO_DIR demos python branches auto 42 AUTO gt s data AUTO gt sol s 3 tor s 4 data 4 s HB1 or data HB1 AUTO gt print sol BR PT TY LAB ISW NTST NCOL NDIM IPS IPRIV 2 42 HB 4 1 1 0 3 1 0 Pointset lrz parameterized Independent variable t 0 Coordinates U 1 7 95601972 U 2 7 95601972 U 3 23 73684367 Labels by index Empty Active ICP 1 rldot 0 69738311435 udotps Pointset lrz non parameterized Coordinates UDOT 1 0 11686228 UDOT 2 0 11686228 UDOT 3 0 69738311 Labels by index Empty PAR 1 5 2 4736843668E 01 2 6666666667E 00 1 0000000000E 01 o 0 PAR 6 10 0 0000000000E 00 0 0000000000E 00 0 0000000000E 00 0 0 PAR 11 11 6 5283032822E 01 AUTO gt print sol LAB 4 AUTO gt print sol L2 NORM or sol b L2 NORM 26 268502943 AUTO gt so1 0 u 7 9560197197999996 7 9560197197999996 23 736843667999999 t 0 0 u dot 0 11686227712 0 11686227712 0 69738311435 AUTO gt so1 t array 0 AUTO gt so1 U 1 array 7 95601972 AUTO gt sol PAR 1 or sol PAR 1 24 736843667999999 AUTO gt
195. in the previous chapter AUTO can also be run with the GUI described in Chapter 9 or using the Unix commands described below These Unix commands run both directly in the shell and at the AUTO Python prompt The AUTO aliases must have been activated see Section and an equations file xxx f90 and a corresponding constants file c xxx see Section 3 must be in the current user directory Do not run AUTO in the directory auto 07p or in any of its subdirectories Most commands only need a plain Unix shell but some use Python the commands that depend on Python are cnvc dlb Cdsp klb Cksp lbf 11 Cls Gmb Cpp and sb 5 1 Basic commands r Type r xxx torun AUTO Restart data if needed are expected in s xxx and AUTO constants in c xxx This is the simplest way to run AUTO Type Or xxx yyy to run AUTO with equations file xxx f90 and restart data file s yyy AUTO constants must be in c xxx Type Or xxx yyy zzz to run AUTO with equations file xxx f90 restart data file s yyy and constants file c zzz OR The command CR xxx is equivalent to the command r xxx above Type CR xxx ito run AUTO with equations file xxx f90 constants file c xxx i and if needed restart data file s xxx Type CR xxx i yyy to run AUTO with equations file xxx 90 constants file c xxx i and restart data file s yyy Use R on case insensitive file systems sv Type sv xxx to save the output files fort 7 fort 8 fort 9 as b xxx S
196. ined Newton Chord iterations When this maximum is reached the step will be retried with half the stepsize This is repeated until convergence or until the minimum stepsize is reached In the latter case the computation of the family is discontinued and a message printed in fort 9 The recommended value is ITNW 5 but ITNW 7 may be used for difficult problems for example demos spb chu plp etc 10 5 Continuation Step Size 10 5 1 DS AUTO uses pseudo arclength continuation for following solution families The pseudo arclength stepsize is the distance between the current solution and the next solution on a family By default this distance includes all state variables or state functions and all free parameters The constant DS defines the pseudo arclength stepsize to be used for the first attempted step along any family Note that if IADS gt 0 then DS will automatically be adapted for subsequent steps and for failed steps DS may be chosen positive or negative changing its sign reverses the direction of computation The relation DSMIN lt DS lt DSMAX must be satisfied The precise choice of DS is problem dependent the demos use a value that was found appropriate after some experimentation 10 5 2 DSMIN This is minimum allowable absolute value of the pseudo arclength stepsize DSMIN must be positive It is only effective if the pseudo arclength step is adaptive i e if IADS gt 0 The choice of DSMIN is highly problem depend
197. information is one of the ideas that sets the CLUI apart from the command language described in Section Next ab load ab constants ab 1 parses the AUTO constants file c ab 1 and reads it into memory Note that changes to the file c ab 1 after it has been loaded in will not be used 26 Script Description demol auto The demo script from Section 4 3 demo2 auto demo3 auto and demo4 auto The demo scripts from Section userScript xauto The expert demo script from Figure 4 14 userScript py The loadable expert demo script from Fig ure branches auto The branch manipulating script from Fig ure full Test auto A script which uses the entire AUTO com mand set except for the plotting com mands plotter auto A demonstration of some of the plotting capabilities of AUTO tutorial auto A script which implements the tutorial from Section 12 8 n body compute_lagrange_points_family auto A basic script which computes and plots all of the Lagrange points as a function of the ratio of the masses of the two planets n body compute lagrange points 0 5 auto A basic script which computes all of the Lagrange points for the case where the masses of the two planets are equal and saves the data n body compute periodic family xauto An expert script which starts at a Lagrange point computed by com pute_lagrange_points_0 5 auto and contin
198. ing the user manual i e this document 9 3 Using the GUI AUTO commands are described in Section 5 and illustrated in the demos In Table 9 1 we list the main AUTO commands together with the corresponding GUI button Or Run sv Save ap Append p Plot Ccp Files Copy mv Files Move cl Files Clean 041 Files Delete dm Equations Demo Table 9 1 Command Mode GUI correspondences The AUTO command Cr xxx yyy is given in the GUI as follows click Files Restart and enter yyy as data Then click Run As noted in Section 5 this will run AUTO with the current equations file xxx f90 and the current constants file c xxx while expecting restart data in s yyy The AUTO command ap xxx yyy is given in the GUI by clicking Files Append 9 4 Customizing the GUI 9 4 1 Print button The Misc Print button on the Menu Bar can be customized by editing the file GuiConsts h in directory auto 07p include 9 4 2 GUI colors GUI colors can be customized by creating an X resource file Two model files can be found in directory auto 07p gui namely Xdefaults 1 and Xdefaults 2 To become effective edit 95 one of these if desired and copy it to Xdefaults in your home directory Color names can often be found in the system file usr 1ib X11 rgb txt 9 4 3 On line help The file auto 07p include GuiGlobal h contains on line help on AUTO constants and demos The text can be up
199. ining all necessary files for certain illustrative calculations Each subdirectory say xxx corresponds to a particular equation and contains one equations file xxx f90 f c and one or more constants files c xxx i one for each successive run of the demo You also find Python script files xxx auto and clean auto the command auto xxx auto runs the demo and the command auto clean auto deletes all generated files To see how the equations have been programmed inspect the equations file To understand in detail how AUTO is instructed to carry out a particular task inspect the appropriate constants file and Python script In this chapter we describe the tutorial demo cusp in detail A brief description of other demos is given in later chapters 12 2 cusp A Tutorial Demo This demo illustrates the computation of stationary solutions locating saddle node bifurcations of these solutions and the continuation of a saddle node bifurcation in two parameters The cusp normal form equation is given by i u g r 12 1 12 3 Copying the Demo Files The commands listed in Table will copy the demo files to your work directory Unix COMMAND ACTION auto start the AUTO 07p Command Line User Interface AUTO COMMAND ACTION cd go to main directory or other directory mkdir cusp create an empty work directory Note the 1 is used to signify a command which is sent to the shell cd cusp change to the work directory demo c
200. ints and label 4 The columnname syntax is especially useful for plotting as is illustrated in Figure Here the Python package matplotlib is directly used to plot a branch Of course the command plot may also be used but sometimes more control may be needed Additionally three special keys can be used to query a branch use br BR to obtain the branch number br TY for the branch type and br TY number for the corresponding 37 gt cp AUTO _DIR python demo userScript py gt ls userScript py gt cat userScript py This is an example script for the AUTOO7p command line user interface See the Command Line User Interface chapter in the manual for more details from auto import def myRun demoname demo demoname r run demoname branchpoints r BP for solution in branchpoints bp load solution ISW 1 NTST 50 Compute forwards print Solution label bp LAB forwards fw run bp Compute backwards print Solution label bp LAB backwards bw run bp DS both fw bw merged merge both r r merged r relabel r save r demoname plot r wait gt auto Python 2 5 2 r252 60911 Nov 14 2008 19 46 32 GCC 4 3 2 on linux2 Type help copyright credits or license for more information AUTOInteractiveConsole AUTO gt from userScript import AUTO gt myRun bvp Figure 4 15 This Figure shows the fu
201. ion encouragement and support He is also thankful to AUTO users and research collaborators who have directly or indirectly contributed to its development in particular Jean Pierre Kern vez UTC Compi gne France Don Aronson University of Minnesota Minneapolis Hans Othmer University of Utah and Frank Schilder University of Surrey Some material in this document related to the computation of connecting orbits was developed with Mark Friedman University of Alabama Huntsville Also acknowledged is the work of Nguyen Thanh Long Concordia University Montreal on the graphics program PLAUT Special thanks are due to Sheila Shull California Institute of Technology for her cheerful assistance in the distribution of AUTO over a long period of time Over the years the development of AUTO has been supported by various agencies through the California Institute of Technology and by research grants from NSERC Canada The development of HomCont has benefitted from help and advice from among others W J Beyn Universitat Bielefeld M J Friedman University of Alabama A Rucklidge University of Cambridge M Koper University of Utrecht C J Budd University of Bath and Financial support for this collaboration was also received from the U K Engineering and Physical Science Research Council and the Nuffield Foundation License AUTO is available under the terms of the BSD license Copyright 1979 2007 E J Doedel California Institut
202. ion of Sharp Traveling Waves This demo illustrates the computation of sharp traveling wave front solutions to nonlinear dif fusion problems of the form w A wW wae B w w C w with A w ayw agw B w bo b1w b2w and C w co c w c2w Such equations can have sharp traveling wave fronts as solutions i e solutions of the form w x t u x ct for which there is a zo such that u z 0 for z gt zo u z 4 0 for z lt zo and u z constant as z gt oo These solutions are actually generalized solutions since they need not be differentiable at Zo Specifically in this demo a homotopy path will be computed from an analytically known exact sharp traveling wave solution of 1 Wi 2WWee 2w w 1 w to a corresponding sharp traveling wave of 2 wi 2w w Wee ww w 1 w This problem is also considered in Doedel Keller amp Kern vez 1991b For these two special cases the functions A B C are defined by the coefficients in Table 18 3 ay a2 bo bi b Co C1 C2 Case 1 2 0 2 O 0 0 1 1 Case 2 2 1 0 1 0 0 1 1 Table 18 3 Problem coefficients in demo stw With w x t u x ct z x ct one obtains the reduced system ui 2 U2 uh 2 cuz B ur u3 C ur A ur 18 5 To remove the singularity when u 0 we apply a nonlinear transformation of the independent variable see Aronson 1980 viz d dz
203. it from a previous run Run 4 of demo pp2 an analytically known periodic orbit Run 1 of demo frc or a numerically known periodic orbit Demo lor The sign of PT in fort 7 is used to indicate stability is stable is unstable or unknown IPS 4 A boundary value problem Boundary conditions must be specified in the user supplied routine BCND and integral constraints in ICND The AUTO constants NBC and NINT must be given correct values Demos exp int kar IPS 5 Algebraic optimization problems The objective function must be specified in the user supplied routine FOPT Demo opt IPS 7 A boundary value problem with computation of Floquet multipliers This is a very special option for most boundary value problems one should use IPS 4 Boundary conditions must be specified in the user supplied routine BCND and integral constraints in ICND The AUTO constants NBC and NINT must be given correct values IPS 9 This option is used in connection with the HomCont algorithms described in Chapters for the detection and continuation of homoclinic bifurcations Demos san mtn kpr cir she rev IPS 11 Spatially uniform solutions of a system of parabolic PDEs with detection of traveling wave bifurcations The user need only define the nonlinearity in routine FUNC initialize the wave speed in PAR 10 initialize the diffusion constants in PAR 15 16 and set a free equation parameter in ICP 1 Run 2 of demo wav IPS 12
204. itional free parameters must be specified for such continuations see also Section 10 7 10 8 5 MXBF This constant which is effective for algebraic problems only sets the maximum number of bifur cations to be treated Additional branch points will be noted but the corresponding bifurcating families will not be computed If MXBF is positive then the bifurcating families of the first MXBF branch points will be traced out in both directions If MXBF is negative then the bifurcating families of the first MXBF branch points will be traced out in only one direction 10 8 6 s This constant sets the name of the solution file from which the computation is to be restarted instead of fort 3 if s xxx then the name of the restart file is s xxx 10 8 7 dat This constant where dat xxx sets the name of a user supplied ASCII data file xxx dat from which the contination is to be restarted AUTO automatically sets the period in PAR 11 Other parameter values must be set in STPNT When necessary PAR 11 may also be redefined there The first column in the data file denotes the time which does not need to be rescaled to the interval 0 1 and further columns the coordinates of the solution The constant IRS must be set to 0 Demos lor pen 10 8 8 U This constant where U il x1 i2 x2 changes the value of U i1 to x1 U i2 to x2 and so on with respect to the solution to start from This is only valid for restarting fr
205. izawa 1962 are a simplified version of the Hodgkin Huxley equations Hodgkin amp Huxley 1952 They model nerve axon dynamics and are given by Ut Ure falu w re 27 2 where falu u u a u 1 Travelling wave solutions of the form u w 1 t u w where x ct are solutions of the following ODE system v cv falu Ww 27 3 N W U YW a yw In particular we consider solitary wave solutions of 27 2 These correspond to orbits homoclinic to u v w 0 in system 27 3 In our numerical example we keep y 0 255 We aim to find a 2 homoclinic orbit at a Shil nikov bifurcation All the commands given here are in the file fnb auto First we obtain a homoclinic orbit using a homotopy technique see Friedman Doedel amp Monteiro 1994 using ISTART 3 for the parameter values c 0 21 a 0 2 0 0025 demo fnb ri run fhn sv 1 Among the output we see BR PT TY LAB PERIOD L2 NORM eee PAR 17 1 21 UZ 4 2 91921E 01 2 38053E 01 2 37630E 11 and a zero of PAR 17 means that a zero of an artificial parameter has been located and the right hand end point of the corresponding solution belongs to the plane that is tangent to the stable manifold at the saddle This point still needs to come closer to the equilibrium which we can achieve by further increasing the period to 300 while keeping PAR 17 at 0 r2 run r1 UZ1 c
206. les by a selected new name which then becomes the active name 9 2 4 Define button Clicking this button will display the full AUTO constants panel Most of its text fields can be edited but some have restricted input values that can be selected with the right mouse button Some text fields will display a subpanel for entering data To actually apply changes made in the panel click the OK or Apply button at the bottom of the panel 93 9 25 Run button Clicking this button will write the constants file c xxx and run AUTO If the equations file has been edited then it should first be rewritten with the Write button 9 2 6 Save button This pull down menu contains the item Save to save the output files fort 7 fort 8 fort 9 as b XXX S XXX d xxx respectively Here xxx is the active equation name It also contains the item Save As to save the output files under another name Existing data files with the selected name if any will be overwritten 9 2 7 Append button This pull down menu contains the item Append to append the output files fort 7 fort 8 fort 9 to existing data files b xxx s xxx d xxx respectively Here xxx is the active equation name It also contains the item Append To to append the output files to other existing data files 9 2 8 Plot button This pull down menu contains the items Plot to run the plotting program PLAUT for the data files b xxx and s xxx where xxx is the active equation name and
207. ll generated files 5 autorc A file that contains default settings for the 2 dimensional plotting tool PyPLAUT 6 plaut04 rc A file that contains default settings for the 3 dimensional plotting tool PLAUT04 12 4 Executing all Runs Automatically To execute all prepared runs of demo cusp simply type the command given in Table AUTO COMMAND ACTION demofile cusp auto execute all runs of demo cusp interactively Table 12 2 Executing all runs of demo cusp The command in Table 12 2 begins a tutorial which will proceed one step each time the user presses a key Each step consists of a single AUTO command preceded by instructions as to what action the command performs The tutorial script cusp auto performs the demo by reading in a single AUTO constants file and then interactively modifying it to perform each of the demo The essential commands in cusp auto are given in Table Note that there are four separate runs where each run command performs a run In the first run a branch of stationary solutions is traced out Along it one fold LP limit point or in this case a saddle node bifurcation is located The free parameter is u The other parameter A remains fixed in this run Note also that only special labeled solution points are printed on the screen Detailed results are saved in the Python variable mu The second run does the same thing but now in the negative direction of u i e backwards instead of fo
208. ller H B 1980 The Von Karman swirling flows SIAM J Appl Math 38 52 64 Lin X B 1990 Using Melnikov s method to solve Silnikov s problems Proc Royal Soc Edinburgh 116A 295 325 Lorenz J 1982 Nonlinear boundary value problems with turning points and properties of difference schemes in W Eckhaus amp E M de Jager eds Singular Perturbation Theory and Applications Springer Verlag Lutz M 1996 Programming Python O Reilly and Associates Nagumo J Arimoto S amp Yoshizawa S 1962 An active pulse transmission line simulating nerve axon Proc IRE 50 2061 2070 Oldeman B E Champneys A R amp Krauskopf B 2003 Homoclinic branch switching A numerical implementation of Lin s method Int J of Bifurcation and Chaos 10 2977 2999 Rodr guez Luis A J 1991 Bifurcaciones multiparam tricas en osciladores aut nomos PhD thesis Department of Applied Mathematics University of Seville Spain Rucklidge A amp Mathews P 1995 Analysis of the shearing instability in nonlinear convection and magnetoconvection Submitted to Nonlinearity Russell R D amp Christiansen J 1978 Adaptive mesh selection strategies for solving boundary value problems SIAM J Numer Anal 15 59 80 Sandstede B 1993 Verzweigungstheorie homokliner Verdopplungen PhD thesis Universitat Stuttgart 264 Sandstede B 1995a Constructing dyna
209. localization phenomena and the dynamical phase space analogy Proc Roy Soc Lond A 425 245 267 Keller H B 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems in P H Rabinowitz ed Applications of Bifurcation Theory Academic Press pp 359 384 Keller H B 1986 Lectures on Numerical Methods in Bifurcation Problems Springer Verlag Notes by A K Nandakumaran and Mythily Ramaswamy Indian Institute of Science Bangalore 263 Kern vez J P 1980 Enzyme Mathematics North Holland Press Amsterdam Khibnik A I Roose D amp Chua L O 1993 On periodic orbits and homoclinic bifurcations in Chua s circuit with a smooth nonlinearity Int J Bifurcation and Chaos 3 No 2 363 384 Khibnik A Kuznetsov Y Levitin V amp Nikolaev E 1993 Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps Physica D 62 360 371 Koper M 1994 Far from equilibrium phenomena in electrochemical systems PhD thesis Universiteit Utrecht The Netherlands Koper M 1995 Bifurcations of mixed mode oscillations in a three variable autonomous Van der Pol Duffing model with a cross shaped phase diagram Physica D 80 72 94 Krauskopf B amp Rie T 2008 A lin s method approach to finding and continuing heteroclinic connections involving periodic orbits Nonlinearity 21 1655 1690 Lentini M amp Ke
210. loquet multipliers in the info file d xxx Aliases fl commandQueryFloquet 4 14 5 File maintenance relabel Relabel data files Type y relabel x to return the python object x with the solution labels sequentially relabelled starting at 1 as a new object y Type relabel xxx to relabel s xxx and b xxx Backups of the original files are saved Type relabel xxx yyy to relabel the existing data files s xxx and b xxx and save then to s yyy and b yyy d xxx is copied to d yyy Aliases rl commandRelabel double Double a solution Type double to double the solution in fort 8 Type double xxx to double the solution in s xxx Aliases db commandDouble 99 triple Triple a solution Type triple to triple the solution in fort 8 Type triple xxx to triple the solution in s xxx Aliases tr commandTriple us Convert user supplied data files Type us xxx to convert a user supplied data file xxx dat to AUTO format The converted file is called s dat The original file is left unchanged AUTO automatically sets the period in PAR 11 Other parameter values must be set in STPNT When necessary PAR 11 may also be redefined there The constants file file c xxx must be present as the AUTO constants NTST and NCOL are used to define the new mesh Note this technique has been obsoleted by the dat AUTO constant in Section 10 8 7 Aliases userdata commandUse
211. lowed by continuation of this stationary state in a free problem parameter The equation is 2 oY DoS p u 1 u on the space interval 0 L where L PAR 11 10 is fixed throughout as is the diffusion constant D PAR 15 0 1 The boundary conditions are u 0 u L 0 for all time In the first run the continuation parameter is the independent time variable namely PAR 14 while p 1 is fixed The AUTO constants DS DSMIN and DSMAX then control the step size in space time here consisting of PAR 14 and u x Initial data are u x sin rx L at time zero Note that in the subroutine STPNT the initial data must be scaled to the unit interval and that the scaled derivative must also be provided see the equations file pv1 f90 In the second run the continuation parameter is p Euler time integration is only first order accurate so that the time step must be sufficiently small to ensure correct results Indeed this option has been added only as a convenience and should generally be used only to locate stationary states AUTO COMMAND ACTION mkdir pd1 create an empty work directory cd pdl change directory demo pd1 copy the demo files to the work directory ri run e pd1 c pd1 time integration towards stationary state save r1 1 save output files as b 1 s 1 d 1 r2 run r1 IPS 17 ICP 1 NTST 20 continuation of stationary states read NMX 100 RL1 50 NPR 25 DS 0 1 DSMAX 0 5 r
212. lternatively inside the Python CLUI and scripts you can use import os followed by os environ AUTO_COMMAND_PREFIX mpirun np 4 17 Chapter 2 Overview of Capabilities 2 1 Summary AUTO can do a limited bifurcation analysis of algebraic systems f u p 0 fO u E R 2 1 The main algorithms in AUTO however are aimed at the continuation of solutions of systems of ordinary differential equation ODEs of the form u t UD iG Jul R 2 2 subject to boundary including initial conditions and integral constraints Above p denotes one or more free parameters These boundary value algorithms also allow AUTO to do certain stationary solution and wave calculations for the partial differential equation PDE Ut Duzz f u p FC ul ER 2 3 where D denotes a diagonal matrix of diffusion constants The basic algorithms used in AUTO as well as related algorithms can be found in Keller 1977 Keller 1986 Doedel Keller amp Kern vez 1991a Doedel Keller amp Kern vez 19916 Below the basic capabilities of AUTO are specified in more detail Some representative demos are also indicated 2 2 Algebraic Systems Specifically for 2 1 AUTO can Compute solution families Demo ab Run 2 Locate branch points continue these in two or three parameters and automatically com pute bifurcating families Demos pp2 Run 1 and apbp 18 Locate Hopf bifurcation points continue t
213. lue of e i e PAR 6 Note that if PAR 6 becomes large then the manifold may no longer be accurate The free parameters in this run are PAR 3 energy PAR 12 length of the orbit PAR 6 starting distance PAR 22 y coordinate at end point PAR 11 integration time PAR 23 z coordinate at end point sv Hia save Hia H1la Save the results in b Hla s Hla and d Hla R man Hla 2 startHla hetHla run startH1la e man c man Hla 2 Another run starting from a longer initial orbit which computes part of the manifold The free parameters are the same as in the preceding run This computation results in the detection of a connecting orbit sv hetHla save hetHla hetHla Save the results in b hetHla s hetHla and d hetHla Table 14 26 Detailed AUTO shell and Python commands for the Hla demo CR man Hib 3 startHib het2H1b run startH1b e man c man H1b 3 Another run starting from a longer initial orbit which computes part of the manifold The free parameters are the same as in the preceding run This computation results in the detection of another connecting orbit Osv het2H1b save het2H1b het 2H1b Save the results in b het2H1b s het2H1b and d het2H1b Table 14 27 Detailed AUTO shell and Python commands for the H1b demo 158 The instructions below are for the Halo family H1 in AUTO demo Hic Follow the instructions for Lla abov
214. ly that arithmetic differences have accumulated from step to step possibly leading to different step size decisions Next reset the work directory by typing the command given in Table AUTO COMMAND clean delete mu delete cusp ACTION remove temporary files of demo cusp remove mu data files of demo cusp remove cusp data files of demo cusp Table 12 3 Cleaning the demo cusp work directory Run forwards BR PT TY LAB mu L2 NORM X lambda 1 1 EP 1 0 00000E 00 0 00000E 00 0 00000E 00 1 00000E 00 1 14 LP 2 3 84900E 01 5 77360E 01 5 77360E 01 1 00000E 00 1 20 3 1 26582E 01 9 29410E 01 9 29410E 01 1 00000E 00 1 40 4 1 38347E 00 1 40803E 00 1 40803E 00 1 00000E 00 1 47 UZ 5 1 99999E 00 1 52138E 00 1 52138E 00 1 00000E 00 Run backwards BR PT TY LAB mu L2 NORM x lambda 1 1 EP 1 0 00000E 00 0 00000E 00 0 00000E 00 1 00000E 00 1 14 LP 2 3 84900E 01 5 77360E 01 5 77360E 01 1 00000E 00 1 20 3 1 26582E 01 9 29410E 01 9 29410E 01 1 00000E 00 1 40 4 1 38347E 00 1 40803E 00 1 40803E 00 1 00000E 00 1 47 UZ 5 1 99999E 00 1 52138E 00 1 52138E 00 1 00000E 00 Forward continuation of the first fold in two parameters BR PT TY LAB mu L2 NORM x lambda 2 20 11 1 09209E 00 8 17354E 01 8 17354E 01 2 00420E 00 2 34 UZ 12 1 99995E 00 9 99991E 01 9 99991E 01 2 99995E 00 122 Backward continuation of the fold in two parameters BR PT TY LAB mu L2 NORM X lambda 2 20 11 5 42543E 02 3 00470E 01 3
215. ly zero which corresponds to a zero of Yy a non central saddle node homoclinic orbit We shall return to the computation of this codimension two point later Before reaching this point among the output we find two zeroes of PAR 33 test function 1 13 which gives the accurate location of two inclination flip bifurcations BR PT TY LAB PAR 1 gas PAR 2 PAR 10 ee PAR 33 1 6 UZ 7 1 80166E 00 2 00266E 01 7 25140E 07 1 14077E 04 1 12 UZ 8 1 56876E 00 4 39547E 01 2 15617E 07 1 48740E 07 That the test function really does have a regular zero at this point can be checked from the data saved in b 3 plotting PAR 33 as a function of PAR 1 or PAR 2 Figure 23 3 presents solutions t of the modified adjoint variational equation for details see Champneys et al 1996 at parameter values on the homoclinic family before and after the first detected inclination flip 228 Figure 23 3 Projection on the x y plane of solutions t at 1 A 1 825470 k 0 1760749 and 2 A 1 686154 k 0 3183548 Figure 23 4 Three dimensional blow up of the solution curves t at labels 1 dotted and 2 solid line from Figure 3 8 229 1 030 1 010 0 990 0 970 0 950 0 930 1 020 1 000 0 980 0 960 0 940 x Figure 23 5 Computed homoclinic orbits approaching the BT point
216. mCont Demo she 25 1 The following system of five equations Rucklidge amp Mathews 1995 A Heteroclinic Example patry zu SUS 27 pz xu 902 4 1 0 ou 4 0Qu 4n 3 1 0 22 40 u 4 u 4 has been used to describe shearing instabilities in fluid convection The equations possess a rich structure of local and global bifurcations Here we shall reproduce a single curve in the o plane of codimension one heteroclinic orbits connecting a non trivial equilibrium to the origin for Q 0 and 4 The defining problem is contained in equation file she foq and starting data for the orbit at u 0 5 0 163875 are stored in she dat with a truncation interval of PAR 11 85 07 We begin by computing towards u 0 with the option IEQUIB 2 which means that both equilibria are solved for as part of the continuation process 25 1 Ge we E demo she ri run she c she 1 sv 1 This yields the output BR PT TY LAB PAR 3 L2 NORM PAR 1 1 1 EP 1 5 00000E 01 4 05950E 01 1 63875E 01 1 5 2 4 52847E 01 3 72688E 01 1 36505E 01 1 10 3 3 94351E 01 3 30390E 01 1 04419E 01 1 15 4 3 35908E 01 2 87331E 01 7 51623E 02 1 20 5 2 77287E O1 2 43351E 01 4 95320E 02 1 25 6 2 18210E 01 1 98147E 01 2 84629E 02 1 30 EP 7 1 58178E 01 1 51246E 01 1 29327E 02 The last parameter used to store the equilibria PAR 21 is overlaped here with the first test function In
217. main fixed 10 7 9 Internal free parameters The actual continuation scheme in AUTO may use additional free parameters that are auto matically added The simplest example is the computation of periodic solutions and rotations where AUTO automatically puts the period if not specified in PAR 11 The computation of loci of folds Hopf bifurcations and period doublings also requires additional internal contin uation parameters These will be automatically added and their indices will be greater than NPAR Other use depends on IPS see Section 11 1 10 7 10 Parameter overspecification The number of specified parameter indices is allowed to be be greater than the generic number In such case there will be overspecified parameters whose values will appear in the screen and fort 7 output but which are not part of the continuation process A simple example is provided by demo opt where the first three runs have overspecified parameters whose values although constant are printed There is however a more useful application of parameter overspecification In the user supplied routine PVLS one can define solution measures and assign these to otherwise unused parameters Such parameters can then be overspecified in order to print them on the screen and in the fort 7 output It is important to note that such overspecified parameters must appear at the end of the ICP list as they cannot be used as true continuation parameters For an exa
218. mann R F 1983 Numerical computation of periodic solution branches and oscillatory dynamics of the stirred tank reactor with a b c reactions Chem Eng Sci 38 No 9 1493 1499 Doedel E J amp Kern vez J P 1986a AUTO Software for continuation problems in ordinary differential equations with applications Technical report California Institute of Technology Applied Mathematics Doedel E J amp Kern vez J P 19866 A numerical analysis of wave phenomena in a reaction diffusion model in H G Othmer ed Nonlinear Oscillations in Biology and Chemistry Vol 66 Springer Verlag pp 261 273 Doedel E J Wang X J 1995 AUTO94 Software for continuation and bifurcation prob lems in ordinary differential equations Technical report Center for Research on Parallel Computing California Institute of Technology Pasadena CA 91125 CRPC 95 2 Doedel E J Aronson D G amp Othmer H G 1991 The dynamics of coupled current biased Josephson junctions II Int J Bifurcation and Chaos 1 No 1 51 66 Doedel E J Champneys A R Fairgrieve T F Kuznetsov Y A Sandstede B amp Wang X J 1997 AUTO97 Software for continuation and bifurcation problems in ordinary differential equations Technical report California Institute of Technology Pasadena CA 91125 Doedel E J Friedman M amp Monteiro A 1993 On locating homoclinic and heteroclinic orbits Techn
219. mber of collocation point per mesh interval NCOL specified in the constants file c lor 1 AUTO COMMAND ACTION mkdir lor create an empty work directory cd lor change directory demo lor copy the demo files to the work directory lor run lor c lor 1 compute a solution family restart from lor dat save to bifurcation diagram object lor pd run lor PD1 c lor 2 switch branches at a period doubling de tected in the first run Constants changed IRS ISW NTST save lortpd lor save the two runs to b lor s lor d lor Table 14 11 Commands for running demo lor 140 145 fre A Periodically Forced System This demo illustrates the computation of periodic solutions to a periodically forced system In AUTO this can be done by adding a nonlinear oscillator with the desired periodic forcing as one of the solution components An example of such an oscillator is Y 2t py e a y 14 5 y PBx y y a y eee which has the asymptotically stable solution x sin Gt y cos Bt We couple this oscillator to the Fitzhugh Nagumo equations v F v w e w vw dw b rsin Bt ee by replacing sin Gt by z Above F v v v a 1 v and a b and d are fixed The first run is a homotopy from r 0 where a solution is known analytically to r 0 2 Part of the solution family with r 0 2 and varying 3 is computed in the second run For detailed results s
220. mical systems possessing homoclinic bifurcation points of codimension two In preparation Sandstede B 1995b Convergence estimates for the numerical approximation of homoclinic solutions In preparation Sandstede B 1995c Numerical computation of homoclinic flip bifurcations In preparation Scheffer M 1995 Personal communication Uppal A Ray W H amp Poore A B 1974 On the dynamic behaviour of continuous stirred tank reactors Chem Eng Sci 29 967 985 Wang X J 1994 Parallelization and graphical user interface of AUTO94 M Comp Sci Thesis Concordia University Montreal Canada Wang X J amp Doedel E J 1995 AUTO94P An experimental parallel version of AUTO Technical report Center for Research on Parallel Computing California Institute of Tech nology Pasadena CA 91125 CRPC 95 3 Yew A C 2001 Multipulses of nonlinearly coupled Schr dinger equations Journal of Dif ferential Equations 173 1 92 137 265
221. mpared to recent versions of AUTO such as AUTO97 and AUTO2000 15 The main change compared to AUTO97 is that there is now a programmable Python CLUI The constants file can be written using a completely new more flexible syntax but the old syntax is still accepted and files can be converted using the command cnvc see Section B Due to the replacement of EISPACK routines by LAPACK routines for the computation of eigenvalues and eigenvectors the sign of the eigenvectors may have flipped sometimes with respect to earlier versions This affects the sign of some HomCont test functions and the initial direction when using the homotopy method you may have to flip the sign of the starting distance in the routine STPNT Eigenvectors are now normalized to avoid future problems and improve consistency Parameter derivatives in DFDP are now significant when using HomCont If only the deriva tives with respect to phase space variables are specified in DFDU please use the setting JAC 1 When upgrading from AUTO2000 you can continue to use equations files written in C However there is now a strict difference between indexing of the array par in the C file and the references to it using PARO in constants files and output using par 1 PAR I 1 In practise this means that you do not have to change the C file but need to add 1 to all parameter indices in the constant files namely ICP THL and UZR For example the period is referenced by par 10
222. mple of using parameter overspecification for printing user defined solution mea sures see demo pvl This is a boundary value problem Bratu s equation which has only one true continuation parameter namely PAR 1 Three solution measures are defined in the routine PVLS namely the L2 norm of the first solution component the minimum of the sec ond component and the left boundary value of the second component These solution mea sures are assigned to PAR 2 PAR 3 PAR 4 and PAR 5 respectively In the constants file c pvl we have ICP 5 with PAR 1 PAR 5 specified as parameters Thus in this example PAR 2 PAR 5 are overspecified Note that PAR 1 must appear first in the ICP list the other parameters cannot be used as true continuation parameters 10 8 Computation Constants 10 8 1 ILP ILP 0 No detection of folds This choice is recommended ILP 1 Detection of folds To be used if subsequent fold continuation is intended 105 10 8 2 SP This constant controls the detection of bifurcations and adds stopping conditions It is specified as a list of bifurcation type strings followed by an optional number If this number is 0 then the detection of this bifurcation is turned off and if it is missing then the detection is turned on A number n greater than zero specifies that the contination should stop as soon as the nth bifurcation of this type has been reached Examples SP LPO turn off detection of folds
223. n and hence the solution to the equations may have non smooth derivatives However for the orthogonal collocation method to attain its optimal accuracy it is necessary that the solution be sufficiently smooth Moreover the adaptive mesh selection strategy will fail if the solution or one of its lower order derivatives has discontinuities For these reasons we use the smooth approximation 2 ec IE arctan Kz T which get better as K increases In the numerical calculations below we use K 10 The free parameter is amp AUTO COMMAND ACTION mkdir chu create an empty work directory cd chu change directory demo chu copy the demo files to the work directory r1 run e chu c chu lst run stationary solutions r2 run ri HB1 IPS 2 ICP 1 11 2nd run periodic solutions with detection of period doubling save r1 r2 chu save all output to b chu s chu d chu Table 14 19 Commands for running demo chu 150 14 13 phs Effect of the Phase Condition This demo illustrates the effect of the phase condition on the computation of periodic solutions We consider the differential equation u Au ua 14 17 uy u l u This equation has a Hopf bifurcation from the trivial solution at A 0 The bifurcating family of periodic solutions is vertical and along it the period increases monotonically The family terminates in a homoclinic orbit containing the saddle point
224. n b 1 s 1 and d 1 One could now display all data using the AUTO command pp 1 to reproduce the curve P shown in Figure It is worthwhile to compare the homoclinic curves computed above with a curve To const along which the system has a limit cycle of constant large period T 1046 178 which can easily be computed using AUTO or locbif Such a curve is plotted in Figure It obviously approximates well the saddle homoclinic loci of P but demonstrates much bigger deviation from the saddle node homoclinic segment D D This happens because the period of the limit cycle grows to infinity while approaching both types of homoclinic orbit but with different asymptotics as In la a in the saddle homoclinic case and as lla a 1 in the saddle node case 22 4 Three Parameter Continuation Finally we can follow the curve of non central saddle node homoclinic orbits in three parame ters The extra continuation parameter is Dy PAR 3 To achieve this we restart at label 4 corresponding to the codim 2 point D We return to continuation of saddle node homoclinics NUNSTAB 0 IEQUIB 2 but append the defining equation 415 0 to the continuation problem via IFIXED 15 The new continuation problem is specified in c mtn 6 r6 run ri UZ1 c mtn 6 sv 6 Notice that we set ILP 1 and choose PAR 3 as the first continuation parameter so that AUTO can detect limit points with respect to this parameter We also make a
225. n diagram object encapsulates all this information in an easy to use form This object is a list of all of the branches in the appropriate bifurcation diagram file and each branch behaves like an array a PyDSTool Pointset subclass to be precise This array can be viewed as a list of all of the points in the appropriate bifurcation diagram file and each point is a Python dictionary with entries for each piece of data for the point For example the sequence of commands in Figure prints out the label of the first point of a branch in a bifurcation diagram The query able parts of the object are listed in Table The individual elements of the array may be accessed in a number of ways by index of the point using the syntax a column using columnname syntax or by label number or type name plus one based index using the syntax For example assume that the parsed object is contained in a variable data and the first branch is in a variable br data 0 The first point may then be accessed using the command br 0 while the column with label PAR 1 may be accessed using the command br PAR 1 The point with label 57 may be accessed using the command br 57 and the second Hopf bifurcation point using the command br HB2 Using the syntax you can also obtain new lists of points br HB gives a list of all Hopf bifurcation points br 1 4 gives the points with labels 1 and 4 and br UZ 4 gives all user defined po
226. n diagram of demo cusp 125 128 ab A Programmed Demo This demo illustrates the computation of stationary solutions Hopf bifurcations and periodic solutions The equations that model an A B reaction are those from Uppal Ray amp Poore 1974 namely Y u1 p1 1 u1 e Uy u2 pipa l uy e pzua This demo is fully scripted see Table 12 2 AUTO COMMAND ACTION mkdir ab create an empty work directory cd ab change directory demo ab copy the demo files to the work directory auto ab auto run the demo Table 12 6 Commands for running demo ab If you look at the file ab auto you see that the script computes a stationary solution family for certain values of p2 and that a periodic orbit family is computed for each Hopf bifurcation that was found in the stationary solution families 126 Chapter 13 AUTO Demos Fixed points 13 1 enz Stationary Solutions of an Enzyme Model The equations that model a two compartment enzyme system Kern vez 1980 are given by s1 so 1 s2 1 pR s1 13 1 8 so u so 81 s2 pR s2 vy where A eo The free parameter is sy Other parameters are fixed This equation is also considered in Doedel Keller amp Kern vez 1991 a AUTO COMMAND ACTION mkdir enz cd enz demo enz create an empty work directory change directory copy the demo files to the work director
227. n in two param eters append output files as b hen s hen d hen compute last continuation the opposite way append output files to b hen s hen d hen merge hen relabel hen join all forward and backward branches into single branches make all labels unique Table 13 5 Commands for running demo hen 130 13 4 Chapter 14 AUTO Demos Periodic solutions 131 14 1 Irz The Lorenz Equations This demo computes two symmetric homoclinic orbits in the Lorenz equations a o y 2 Y prt Y 2Z 14 1 z a2y Bz Here p is the free parameter and 8 8 3 0 10 The two homoclinic orbits correspond to the final large period orbits on the two periodic solution families AUTO COMMAND ACTION mkdir lrz create an empty work directory cd lrz change directory demo 1rz copy the demo files to the work directory lrz run e 1rz c 1rz compute stationary solutions save lrz save all output to b Irz s Irz d lrz run lrz HB1 IPS 2 ICP rho compute periodic solutions the final orbit gt PERIOD NMX 35 NPR 2 DS 0 5 is near homoclinic append 1rz append all output to b Irz s Irz d lrz run 1rz HB2 IPS 2 ICP rho compute the symmetric periodic solution family PERIOD NMX 35 NPR 2 DS 0 5 append 1rz append all output to b Irz s Irz d lrz Table 14 1 Commands for running demo I
228. n the third run e is fixed at e 001 and the solution is continued in y This run takes more than 1500 continuation steps For a detailed analysis of the solution behavior see Lorenz 1982 AUTO COMMAND ACTION mkdir spb create an empty work directory cd spb change directory demo spb copy the demo files to the work directory ri run e spb c spb 0 1st run homotopy from 0 to A 1 save r1 0 save output files as b 0 s 0 d 0 r2 run ri c spb 1 2nd run let tend to zero restart from the last label of rO constants changed IRS TOP CL NTST DS UZR STOP save r2 1 save the output files as b 1 s 1 d 1 r3 run r2 UZ2 c spb 3 3rd run continuation in y e 0 001 restart from 2nd UZ label of r2 Con stants changed IRS ICP 1 ITNW EPSL EPSU UZR save r3 2 save the output files as b 2 s 2 d 2 Table 15 7 Commands for running demo spb 167 15 8 ezp Complex Bifurcation in a BVP This demo illustrates the computation of a solution family to the complex boundary value problem uy Us ee 15 11 with boundary conditions u 0 0 ui 1 0 Here u and uz are allowed to be complex while the parameter p can only take real values In the real case this is Bratu s equation whose solution family contains a fold see the demo exp It is known Henderson amp Keller 1990 that a simple quadratic fold gives rise to a pit
229. nctional version of the AUTO CLUI from Figure 4 14 being used as an extension to the AUTO CLUI The source code for this script can be found in AUTO_DIR python demo userScript py 38 AUTO gt demo 1rz Copying demo lrz Runner configured AUTO gt data run 1rz done Starting lrz Cag lrz done AUTO gt print data 13 69994168 22 63885621 AUTO gt print br 4 TY number 3 data 15 15715248 PP e 42 16 76398367 24 73684367 27 10974373 index 41 24 736843667999999 26 268502943000001 7 9560197197999996 7 9560197197999996 23 736843667999999 gfortran fopenmp 0 c lrz f o lrz o gfortran fopenmp 0 lrz o o lrz exe home bart auto 07p lib o 18 53533839 29 72303281 section TY name O LAB gt HB 20 48761501 32 60076345 BR PT TY LAB PAR 1 L2 NORM U 1 U 2 1 1 EP 1 0 00000E 00 0 00000E 00 0 00000E 00 0 00000E 00 1 5 BP 2 1 00000E 00 0 00000E 00 0 00000E 00 0 00000E 00 1 13 EP 3 3 16000E 01 0 00000E 00 0 00000E 00 0 00000E 00 BR PT TY LAB PAR 1 L2 NORM U 1 U 2 2 42 HB 4 2 47368E 01 2 62685E 01 7 95602E 00 7 95602E 00 2 45 EP 5 3 26008E 01 3 41635E 01 9 17980E 00 9 17980E 00 BR PT TY LAB PAR 1 L2 NORM U 1 U 2 2 42 HB 6 2 47368E 01 2 62685E 01 7 95602E 00 7 95602E 00 2 45 EP 7 3 26008E 01 3 41635E 01 9 17980E 00 9 17980E 00 AUTO gt br data 1 AUTO gt print br BR PT TY LAB PAR 1 L2 NORM U 1 U 2
230. ndary conditions Demo exp kar ICND A routine ICND that defines the integral conditions Demos int lin FOPT A routine FOPT that defines the objective functional Demos opt ops PVLS A routine PVLS for defining solution measures This routine using a LOGICAL SAVE first TRUE variable can also be used for initialization as it is called first Demo pvl In a C language equation file these routines are written using lowercase letters with Fortran you can use any case 3 4 User Supplied Derivatives If AUTO constant JAC equals 0 then derivatives need not be specified in FUNC BCND ICND and FOPT see Section If JAC 1 then derivatives must be given If JAC 1 then the parameter derivatives may be omitted in FUNC This may be necessary for sensitive problems and is recommended for computations in which AUTO generates an extended system Derivatives are specified as follows where zero entries may be omitted FUNC Derivatives with respect to phase space variables are specified in DFDU 1 NDIM 1 NDIM Parameter derivatives go into DFDP 1 NDIM 1 NPAR BCND Derivatives with respect to the two boundary conditions are specified in DBC 1 NBC 1 NBC and DBC 1 NBC NBC 1 2 NBC respectively Parameter derivatives go into DBC 1 NBC 2 NBC 1 2 NBC NPAR ICND Derivatives with respect to the integral conditions are specified in DINT 1 NINT 1 NINT 23 Parameter derivatives g
231. ndeed in this demo two free parameters are designated namely PAR 1 and PAR 3 10 7 7 Boundary value folds To continue a locus of folds for a general boundary value problem with integral constraints set ICP NBC NINT NDIM 2 and specify this number of parameter indices to designate the free parameters 10 7 8 Optimization problems In algebraic optimization problems one must set ICP 1 10 as AUTO uses PAR 10 as principal continuation parameter to monitor the value of the objective function Furthermore one must designate one free equation parameter in ICP 2 Thus ICP 10 2 in the first run Folds with respect to PAR 10 correspond to extrema of the objective function In a second run one can restart at such a fold with an additional free equation parameter specified in ICP 3 Thus ICP 10 2 3 in the second run The above procedure can be repeated For example folds from the second run can be continued in a third run with three equation parameters specified in addition to PAR 10 Thus 104 ICP 4 in the third run For a simple example see demo opt where a four parameter extremum is located Note that ICP 5 in each of the four constants files of this demo with the indices of PAR 10 and PAR 1 PAR 4 specified in ICP Thus in the first three runs there are overspecified parameters However AUTO will always use the correct number of parameters Although the overspecified parameters will be printed their values will re
232. nitially v 0 9 and 0 5 AUTO COMMAND ACTION mkdir tor create an empty work directory cd tor change directory demo tor copy the demo files to the work directory ri run e tor c tor lst run compute a stationary solution family with Hopf bifurcation r2 run ri HB1 IPS 2 ICP 1 11 compute a family of periodic solutions restart from r1 r3 run r2 BP1i ISW 1 NMX 90 compute a bifurcating family of periodic solutions restart from r2 save ritr2 r3 1 save output to b 1 s 1 d 1 Table 14 17 Commands for running demo tor 147 14 11 pen Rotations of Coupled Pendula This demo illustrates the computation of rotations i e solutions that are periodic modulo a phase gain of an even multiple of 7 AUTO checks the starting data for components with such a phase gain and if present it will automatically adjust the computations accordingly The model equations a system of two coupled pendula Doedel Aronson amp Othmer 1991 are given by ol ed sing I 7 2 1 by eph sin a I y 61 do 14 14 or in equivalent first order form On V1 by Yo Y ep1 sind 1 02 1 14 15 wh eba sin do I y 01 2 Throughout y 0 175 Initially e 0 1 and J 0 4 Numerical data representing one complete rotation are contained in the file pen dat Each row in pen dat contains five real numbe
233. ns save pd save output files as b pd s pd d pd r3 run ri PD1i ISW 1 3rd run the bifurcating period 2 orbit append dd2 append output files to b dd2 s dd2 d dd2 run r3 PD1 4th run the bifurcation period 4 orbit append dd2 gt append output files to b dd2 s dd2 d dd2 Table 13 3 Commands for running demo dd2 128 133 log The Logistic Map This demo shows 5 subsequent periodic doublings in the logistic map k l ya 1 0 13 3 x and approximates the Feigenbaum constant The script log auto shows a Python loop in which values of u for subsequent period doubling bifurcations are compared AUTO COMMAND ACTION mkdir log create an empty work directory cd log change directory demo log copy the demo files to the work directory auto log auto run the script log auto plot log plot the bifurcation diagram Table 13 4 Commands for running demo log 129 13 4 hen The H non Map In this demo a two parameter bifurcation analysis of the H non map k 1 T y k 1 ye e is performed This demo features the detection and continuation of Naimark Sacker period doubling and fold bifurcations in two parameters On these codimension one bifurcation curves certain codimension two bifurcations are detected the 1 1 R1 1 2 R2 1 3 R3 and 1 4 R4 resonance and fold flip LPD bifurcation points After running the script hen
234. ns i e PAR 21 PAR 36 one must have IP 2 NUNSTAB lt 21 If an w is vanished it can be frozen while another dummy or system parameter is allowed to vary in order to make consequently all w 0 The resulting final solution gives the initial homoclinic orbit provided the right hand end point is sufficiently close to the saddle See Chapter 23 for an example however we recommend the homotopy method only for expert users To compute the orientation of a homoclinic orbit in order to detect inclination flip bifur cations it is necessary to compute in tandem a solution to the modified adjoint variational equation by setting ITWIST 1 In order to obtain starting data for such a computation when restarting from a point where just the homoclinic is computed upon increasing ITWIST to 1 AUTO generates trivial data for the adjoint Because the adjoint equations are linear only a single step of Newton s method is required to enable these trivial data to converge to the correct unique bounded solution This can be achieved by making a single continuation step in a trivial parameter i e a parameter that does not appear in the problem Decreasing ITWIST to 0 automatically deletes the data for the adjoint from the continuation problem 20 8 Notes on Running HomCont Demos HomCont demos are given in the following chapters To copy all files of a demo xxx for example san move to a clean directory and type demo xxx Simply t
235. nsion zero connection and terminates at a segment of the codimension one EtoP connection We show this in Figure 15 5 15 14 2 The codimension one point to cycle connection To compute the codimension one point to cycle connection the following steps are used very similar to those in the demo pc1 1 Continue the first equilibrium a at x y 0 y arccos 1v2 2 which undergoes a Hopf bifurcation at v 0 683447 Follow the periodic orbit emanating from the Hopf bifurcation in v and its period T until y 0 74 Extend the system putting the variational equation the eigenfunction into solution co ordinates 4 5 and 6 The trivial 0 eigenvector is continued until we hit a branch point corresponding to where PAR 12 equals the natural logarithm of a Floquet multiplier Switch branches and continue the non trivial eigenvector until its norm h equals 1 Extend the system again from 6 to 9 dimensions to calculate a connection from the periodic orbit to the cross section X given by py m The connection starts at the non trivial eigenvector with respect to the periodic orbit with a distance of 107 and grows forwards in time until it hits the section X at time T Extend the system one last time from 9 to 12 dimensions We calculate a connection backwards in time from the second equilibrium b at x y 0 p arccos 1v3 2 starting at its eigenvector over a distance of e 1076 to the cross secti
236. nts are noted namely those listed in Tables 6 1 and 6 2 The letter codes in the Table are used in the screen output The numerical codes are used internally and in the fort 7 and fort 8 output files described below BP 1 Branch point algebraic systems LP 2 Fold algebraic systems HB 3 Hopf bifurcation 4 User specified regular output point UZ 4 Output at user specified parameter value LP 5 Fold differential equations BP 6 Branch point differential equations PD 7 Period doubling bifurcation TR 8 Torus bifurcation EP 9 End point of family normal termination MX 9 Abnormal termination no convergence Table 6 1 Solution Types fort 7 The fort 7 output file contains the bifurcation diagram Its format is the same as the fort 6 screen output but the fort 7 output is more extensive as every solution point has an output line printed fort 8 The fort 8 output file contains complete graphics and restart data for selected labeled solutions The information per solution is generally much more extensive than that in fort 7 The fort 8 output should normally be kept to a minimum fort 9 Diagnostic messages convergence history eigenvalues and Floquet multipliers are written in fort 9 It is strongly recommended that this output be habitually in 73
237. o bvp bvp r bvp for solution in bvp BP bp 1d solution NTST 50 ISW 1 Compute forwards and backwards bvp bvp mb r bp tr bp DS bvp r1 bvp pl bvp sv bvp bvp wait Figure 4 12 This Figure shows a shorter version of the more complex AUTO CLUI script given above The source for this script can be found in AUTO_DIR demos python demo3 auto demo bvp bvp r bvp for solution in bvp BP bp 1d solution NTST 50 ISW 1 Compute forwards and backwards bvp bvp mb r bp r bp DS bvp r1 bvp pl bvp sv bvp bvp wait Figure 4 13 This Figure shows an even shorter version of the more complex AUTO CLUI script given above The source for this script can be found in AUTO_DIR demos python demo4 auto 35 section we will generalize the script in Figure 4 8 for use with any demo and demonstrate how it can be imported back into the interactive mode to create a new command for the AUTO CLUI Several examples of such expert scripts can be found in auto 07p demos python n body Just as loops and conditionals can be used in Python one can also define functions For example Figure 4 14 is a functional version of script from Figure The changes are actually quite minor The first line from auto import includes the definitions of the AUTO CLUI commands and must be included in all AUTO CLUI scripts which define functions The next
238. o convert a saved PLAUT figure fig x from compact PLOT10 format to PostScript format The converted file is called fig x ps The original file is left unchanged File manipulation Type Ccp namel name2 or Ccp namel name2 name3 or Ccp namel name2 name3 name4 to copy the data files dirl c xxx dir1 b xxx dir1 s xxx and dirl d xxx to dir2 c yyy dir2 b yyy dir2 s yyy and dir2 d yyy The values of dir1 xxx and dir2 yyy are as follows depending on whether namel is a directory or name2 is a directory 67 Omv df cl Gdl cnvc rn rc gz uz sr 5 4 lp Ccp namel name2 no directory names namel and name2 namel is a directory name1 name2 and name2 name2 is a directory namel and name2 namel Ccp namel name2 names namel is a directory name1 name2 and name3 name2 is a directory namel and name2 name3 cp namel name2 name3 name4 namel name2 and name3 name4 Type Cmv namel name2 or Cmv namel name2 name3 or Omv namel name2 name3 name4 to move the data files dir1 b xxx dir1 s xxx and dir1 d xxx to dir2 b yyy dir2 s yyy and dir2 d yyy and copy the constants file dir1 c xxx to dir2 c yyy The values of dir1 xxx and dir2 yyy are determined in the same way as for cp above Type Cdf to delete the output files fort 7 fort 8 fort 9 Type cl to clean the current directory This command will delete all files of the form fort 0 and exe
239. o into DINT 1 NINT NINT 1 NINT NPAR Examples of user supplied derivatives can be found in demos dd2 int plp opt and ops 24 Chapter 4 Running AUTO using Python Commands 4 1 Typographical Conventions This chapter uses the following conventions All code examples will be in in the following font AUTO gt copydemo ab Copying demo ab done To distinguish commands which are typed to the Unix shell from those which are typed to the AUTO command line user interface CLUI we will use the following two prompts gt Commands which follow this prompt are for the Unix shell AUTO gt Commands which follow this prompt are for the AUTO CLUL 4 2 General Overview The AUTO command line user interface CLUI is similar to the command language described in Section 5 in that it facilitates the interactive creating and editing of equations files and constants files It differs from the other command language in that it is based on the object oriented scripting language Python see Lutz 1996 and provides extensive programming capabilities This chapter will provide documentation for the AUTO CLUI commands but is not intended as a tutorial for the Python language We will attempt to make this chapter self contained by describing all Python constructs that we use in the examples but for more extensive docu mentation on the Python language including tutorials and pointers to further documentation
240. o that its unstable manifold is 2 dimensional In the computations the independent time variable t is scaled to vary in the unit interval so that the actual integration time T becomes an explicit parameter in the equations see for example demo um2 The calculations can be done by running the Python script um3 auto included in the demo In the first run an orbit is grown by continuation in the integration time T starting from a very small value of T so that a solution that is constant in time is an accurate initial approximation The starting solution is in fact a point on a circle of small radius r 0 03 in the unstable eigenspace of the origin The precise starting point is in the strongly unstable direction namely the y direction The initial orbit is grown until the L2 norm of its endpoint x 1 y 1 lt 1 reaches the value 1 The value of e is fixed at 0 5 in the first run In the second run continuation is used to decrease e to 0 01 The norm of the endpoint x 1 y 1 z 1 is fixed in this run while T is variable In the third run the norm of the endpoint remains fixed but the initial point z 0 y 0 z 0 is allowed to move around the small circle of radius r in the unstable eigenspace of the origin The endpoint thereby moves on the surface of the large sphere of radius 1 The integration time T remains variable The orbits computed in this run generate the local manifold When viewing the orbits computed in
241. oint corresponding to where PAR 12 equals the natural logarithm of a Floquet multiplier 4 Switch branches and continue the non trivial eigenvector until its norm h equals 1 5 Extend the system again from 6 to 9 dimensions to calculate a connection from the cross section J given by x 10 to the periodic orbit The connection starts at the non trivial eigenvector with respect to the periodic orbit with a distance of 1077 and grows backwards in time until it hits the section Y at time T 6 Extend the system one last time from 9 to 12 dimensions We calculate a connection from the equilibrium at 0 starting at its eigenvector over a distance of e 1077 to the cross section X at time T7 The second intersection is the one that is closest to the intersection computed in step 5 and it is the one we want 7 Firstly in the routine PVLS in the equations file pcl f90 the Lin vector a normalized vector between the points of intersection computed in steps 5 and 6 is put in Zz Z and Z PAR 24 PAR 26 Starting data for the Lin gap which measures the distance between these two points of intersection is put in 7 PAR 23 Subsequently close this gap by continuing in 7 p 0 T7 T u and T This process is illustrated in Figure 15 1 8 Continue the point to cycle connection obtained in step 7 in the system parameters p and B together with e T7 T u and T Connections for various values of p 3 are shown in
242. olution is followed in the parameters a PAR 1 PAR 8 up to a 0 25 The output is summarised on the screen as BR PT TY LAB PAR 1 L2 NORM PAR 8 1 1 EP 1 0 00000E 00 4 00000E 01 0 00000E 00 5 UZ 2 2 50000E 01 4 03054F 01 1 85981E 11 214 and saved in more detail in the Python variable r1 Next we want to add a solution to the adjoint equation to the solution obtained at a 0 25 This is achieved by starting from the last label and making the changes ITWIST 1 NMX 2 and ICP 1 9 We also disable any user defined functions UZR The computation so defined is a single step in a trivial parameter PAR 9 namely a parameter that does not appear in the problem The effect is to perform a Newton step to enable AUTO to converge to a solution of the adjoint equation r2 run r1 ICP 9 8 ITWIST 1 NMX 2 UZR The output is stored in the Python variable r2 We can now continue the homoclinic plus adjoint in a A PAR 4 PAR 8 by changing the constants to read NMX 50 and ICP 1 4 We also add PAR 10 to the list of continuation parameters ICP Here PAR 10 is a dummy parameter used in order to make the continuation of the adjoint well posed Theoretically it should be zero if the computation of the adjoint is successful Sandstede 1995a The test functions for detecting resonant bifurcations ISPI 1 1 and inclination flips ISPI 1 13 are also activated Recall that this should be specified in th
243. om algebraic problems or a constant in time solution 107 10 8 9 PAR This constant where PAR i1 x1 i2 x2 changes the value of PAR i1 to x1 PAR i2 to x2 and so on with respect to the solution to start from 10 8 10 IRS This constant sets the label of the solution where the computation is to be restarted IRS 0 This setting is typically used in the first run of a new problem In this case a starting solution must be defined in the user supplied routine STPNT For representative examples of analytical starting solutions see demos ab and frc For starting from unlabeled numerical data see the dat command above and demos lor and pen IRS gt 0 Restart the computation at the previously computed solution with label IRS This solution is normally expected to be in the current data file s xxx see also the r and R commands in Section 5 Various AUTO constants can be modified when restarting IRS lt 0O Restart the computation at the IRSth computed solution in the restart file This is especially useful if you do not want to look up label numbers and know for sure that the solution to continue from is at a fixed position IRS XYn Restart the computation at the nth label of type XY in the restart file for instance HB12 to restart at the twelfth Hopf bifurcation 10 8 11 TY This constant modifies the type from the restart solution This is sometimes useful in conser vative or extended systems declaring a regula
244. omoclinic solution see Figure 23 2 The second stage to obtain a starting solution is to add a solution to the modified adjoint variational equation This is achieved by setting both ITWIST and ISTART to 1 in c kpr 3 which generates a trivial guess for the adjoint equations Because the adjoint equations are linear only a single Newton step by continuation in a trivial parameter is required to provide a solution Rather than choose a parameter that might be used internally by AUTO in c kpr 3 we take the continuation parameter to be PAR 11 which is not quite a trivial parameter but whose affect upon the solution is mild r3 run r2 UZ1 c kpr 3 sv 3 The output at the second point label 6 contains the converged homoclinic solution variables U 1 U 2 U 3 and the adjoint U 4 U 5 U 6 We now have a starting solution and are ready to perform two parameter continuation The fourth run r3 r3 run r3 c kpr 4 ap 3 continues the homoclinic orbit in PAR 1 and PAR 2 Note that several other parameters appear in the output PAR 10 is a dummy parameter that should be zero when the adjoint is being computed correctly PAR 29 PAR 30 PAR 33 correspond to the test functions 49 410 and 413 That these test functions were activated is specified in three places in c kpr 4 as described in Section 20 6 Note that at the end point of the family reached when after NMX 50 steps PAR 29 is approximate
245. on Y at time T The Lin vector a normalized vector between the points of intersection computed in steps 5 and 6 is put in Z Z and Z PAR 24 PAR 26 Starting data for the Lin gap which measures the distance between these two points of intersection is put in 7 PAR 23 Close the gap computed in step 6 by continuing in y 11 0 T T u and T The connection is found at v 7 41189 This process is illustrated in Figure 15 3 Continue the point to cycle connection obtained in step 7 in the system parameters v and va together with e Tt T u and T Connections for various values of v1 v2 are shown in Figure 15 4 175 phigamma phi phi eta 0 3 02 01 0 0 0 1 0 2 0 3 Ss xgamma x x l 0 01 i f 1 fi 1 0 7385 0 7390 0 7395 0 7400 0 7405 0 7410 0 7415 nut Figure 15 3 Closing the Lin gap to obtain the point to cycle connection The left panel is a plot of v versus the gap size y and the right panel shows the corresponding orbit segments To obtain these figures run plot closegap or pp closegap and plot3 closegap or pl closegap 2 0 15 a 5 a 1 0 l 5 oO E E 0 5 3 g 2 A 2 N 2 2 00 1 0 5 10p 0 50 0 25 0 00 Y SAS 15 E 025 0 50 0 25 0 00 z ES 0 25 0 50 xgamma x x 20 1 1 1 1 f f f f 1 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 nut Figure 15 4 Parameter space diagram le
246. ongly recommended because the Jacobian is used extensively for calculating the linearization at the equilibria and 205 hence for evaluating boundary conditions and certain test functions However note that JAC 1 does not necessarily mean that auto will use the analytically specified Jacobian for continuation The earlier HomCont files h xxx are obsolete but still supported You can convert from the old to the new format by running the command cnvc xxx yyy where xxx refers to either or both of the files c xxx and h xxx A new style constants file c yyy is written The command cnvc xxx overwrites c xxx and deletes h xxx 20 3 HomCont Constants An example for the HomCont specific constants in c xxx is listed below NUNSTAB 1 NSTAB 2 IEQUIB 1 ITWIST 1 ISTART 5 IREV IFIXED 13 IPSI 9 10 These constants have the following meaning 20 3 1 NUNSTAB Number of unstable eigenvalues of the left hand equilibrium the equilibrium approached by the orbit as t 00 The default value is 1 which means autodetection In almost all cases autodetection is possible The exception is when starting from a homoclinic to a saddle node equilibrium when TEQUIB41 Examples for this exception are in runs 9 12 and 13 of the demo kpr see Chapter 23 Even in this case only one of NUNSTAB and NSTAB needs to be specified the other one being computed as NDIM minus the specified constant 20 3 2 NSTAB Number of stable eigenvalues o
247. or Prey Model with Immigration 22 2 Continuation of Central Saddle Node Homoclinics 22 3 Switching between Saddle Node and Saddle Homoclinic Orbits 22 4 Three Parameter Continuation 22 5 Detailed AUTO Commands 23 HomCont Demo kpr 23 1 Koper s Extended Van der Pol Model 2 00004 4s 23 2 The Primary Branch of Homoclinics 196 197 198 199 201 202 203 204 205 205 205 206 206 206 206 207 207 208 208 208 208 209 209 211 212 214 214 214 216 216 217 220 220 220 222 223 223 23 3 More Accuracy and Saddle Node Homoclinic Orbits 230 23 4 Three Parameter Continuation ee 233 23 5 Detailed AUTO Commands 4 eco 44 es eae oe ee we wR eS 234 24 HomCont Demo cir 236 24 1 Electronic Circuit of Freire et Gl wc He we entra RY RS 236 24 2 Detailed AUTO Commands cross eh ow eee ee hee A 239 25 HomCont Demo she 240 25 1 A Heteroclinic Fxample 2c 2 cee bee ee ee Re eR RE eR ee 240 25 2 Detailed AUTO Commands 0 e 241 26 HomCont Demo rev 243 26 1 A Reversible System era ar E Ae a 243 26 2 An R Reversible Homoclinic Solution 243 26 3 An Ro Reversible Homoclinic Solution 244 26 4 Detailed AUTO Commands 0 0 00 eee ee 246 27 HomCont Demo Homoclinic branch switching 248 27 1 Branch swi
248. or constrained optimization prob lems by applying it to the following simple problem on the unit sphere in R Coordinate 1 is treated as the state variable Coordinates 2 5 are treated as control parameters For details on the successive continuation procedure see Doedel Keller amp Kern vez 1991a Doedel Keller amp Kern vez 19916 AUTO COMMAND ACTION mkdir opt cd opt demo opt create an empty work directory change directory copy the demo files to the work directory ri run e 0pt c 0pt save ri 1 one free equation parameter save output files as b 1 s 1 d 1 r2 run r1 LP1 save r2 2 two free equation parameters read restart data from r1 save output files as b 2 s 2 d 2 r3 run r2 LP1 save r3 3 three free equation parameters read restart data from r2 save output files as b 3 s 3 d 3 run r3 LP1 save r4 4 four free equation parameters read restart data from r3 save output files as b 4 5 4 d 4 Table 17 1 Commands for running demo opt 189 Find the maximum sum of coordinates 17 2 ops Optimization of Periodic Solutions This demo illustrates the method of successive continuation for the optimization of periodic solutions For a detailed description of the basic method see Doedel Keller amp Kern vez 19910 The illustrative system of autonomous ODEs taken from Rodriguez Luis 1991 is a t
249. or to the Fortran 2003 standard for timing flushing output and accessing command line arguments The con figure script first looks if the F2003 routines are supported src f2003 f90 then checks for a set of routines that are widely implemented across Unix compilers src unix f90 and if that fails too uses a set of dummy replacement routines src compat f90 which could be edited for some obscure installations The PostScript conversion command ps is compiled by default Alternatively you can type make in the directory auto 07p tek2ps To generate the on line manual type make in auto 07p doc which depends on the presence of xfig s transfig fig2dev utility To prepare AUTO for transfer to another machine type make superclean in the directory auto 07p before creating the tar file This will remove all executable object and other non essential files and thereby reduce the size of the package Some LAPACK routines used by AUTO for computing eigenvalues and Floquet multipliers are included in the package Anderson Bai Bischof Blackford Demmel Dongarra Du Croz Greenbaum Hammarling McKenney amp Sorensen 1999 The Python CLUI includes a slightly modified version of the Pointset and Point classes from PyDSTool by R Clewley M D LaMar and E Sherwood http pydstool sourceforge net 1 1 1 Installation on Linux Unix A free Fortran 95 compiler Gfortran is shipped with most recent Linux distributions or can be obtain
250. ot along the Y axis for solutions solution_z The column to plot along the Z axis for solutions stability Turn on or off stability information using dashed curves symbol font The font to use for marker symbols symbol_color The color to use for the marker symbols tick label template A string which defines the format of the tick labels tick length The length of the tick marks top_margin The margin between the graph and the top edge top_title The label for the top title top title fontsize The font size for the top title torus_symbol The symbol to use for torus bifurcation points torus_torus_symbol The symbol to use for torus torus points type The type of the plot either solution or bifurcation use_labels Whether or not to display label numbers in the graph use_symbols Whether or not to display bifurcation symbols in the graph user_point_symbol The symbol to use for user defined output points width Width of the graph xlabel The label for the x axis xlabel_fontsize The font size for the x axis label xticks The number of ticks on the x axis ylabel The label for the y axis ylabel_fontsize The font size for the y axis label 49 yticks The number of ticks on the y axis zero _hopf symbol The symbol to use for zero Hopf points Zlabel The label for the z axis
251. p pp3 tor The continuation of orbits of fixed period is also possible This is the simplest way to compute curves of homoclinic orbits if the period is sufficiently large Demo pp2 The continuation of branch points in two parameters is only possible in non generic prob lems characterized by problem specific symmetries Demo 1cbp Run 2 Generically in non symmetric problems branch points are continued in three parameters Demos 1cbp Run 3 and abcb 19 Do each of the above for rotations 1 e when some of the solution components are periodic modulo a phase gain of a multiple of 27 Demo pen Follow curves of homoclinic orbits and detect and continue various codimension 2 bifur cations using the HomCont algorithms of Champneys amp Kuznetsov 1994 Champneys Kuznetsov amp Sandstede 1996 Demos san mnt kpr cir she rev Locate extrema of an integral objective functional along a family of periodic solutions and successively continue such extrema in more parameters Demo ops Compute curves of solutions to on 0 1 subject to general nonlinear boundary and integral conditions The boundary conditions need not be separated i e they may involve both u 0 and u 1 simultaneously The side conditions may also depend on parameters The number of boundary conditions plus the number of integral conditions need not equal the dimension of the ODE provided there is a corresponding number of addition
252. play u versus time 13 select columns 1 and 3 in s pp2 d display the orbits uz versus time 23 select columns 2 and 3 in s pp2 d phase plane display uz versus u1 sav save plot see Figure fig 2 or fig2 eps upon prompt enter a new file name ex exit from 2D mode end exit from PLAUT PyPLAUT Table 14 10 Plotting commands for demo pp2 138 PAR 1 Figure 14 1 The bifurcation diagram of demo pp2 Figure 14 2 The phase plot of solutions 11 15 19 and 23 in demo pp2 139 14 4 lor Starting an Orbit from Numerical Data This demo illustrates how to start the computation of a family of periodic solutions from nu merical data obtained for example from an initial value solver As an illustrative application we consider the Lorenz equations x o y x Y prt Y 2Z 14 4 z xYy Bz Numerical simulations with a simple initial value solver show the existence of a stable periodic orbit when p 280 6 8 3 0 10 Numerical data representing one complete periodic oscillation are contained in the file lor dat Each row in lor dat contains four real numbers namely the time variable t x y and z The correponding parameter values are defined in the user supplied subroutine STPNT The AUTO constant dat lor then allows for using the data in lor dat where we also specify IRS 0 The mesh will be suitably adapted to the solution using the number of mesh intervals NTST and the nu
253. please see Lutz 1996 or the web page http www python org which contains an excellent tutorial at http www python org doc current tut tut html To use the CLUI for a new equation change to an empty directory For an existing equations file change to its directory Do not activate the CLUI in the directory auto 07p or in any of its subdirectories Then type auto 25 gt auto Python 2 5 2 r252 60911 Nov 14 2008 19 46 32 GCC 4 3 2 on linux2 Type help copyright credits or license for more information AUTOInteractiveConsole AUTO gt Figure 4 1 Typing auto at the Unix shell prompt starts the AUTO CLUI If your command search path has been correctly set see Section L 1 this command will start the AUTO CLUI interactive interpretor and provide you with the AUTO CLUI prompt If you have IPython installed http ipython scipy org then you can get a friendlier interface using the command auto i enabling TAB completion persistent command line his tory and other features In addition to the examples in the following sections there are several example scripts which can be found in auto 07p demos python and are listed in Table These scripts are fully annotated and provide good examples of how AUTO CLUI scripts are written The scripts in auto 07p demos python n body are especially lucid examples and perform various related parts of a calculation involving the gravitational N body problem Scripts wh
254. pposite direction with IRS 15 DS 0 02 r4 run ri c mtn 4 sv 4 will detect the same codim 2 point D but now as a zero of the test function 40 BR PT TY LAB PAR 1 ge PAR 2 PAR 29 PAR 30 1 38 UZ 11 6 61046E 00 6 93248E 02 4 63660E 01 3 13439E 08 Note that the values of PAR 1 and PAR 2 are equal to those at label 4 up to at least six significant figures Actually the program runs further and eventually computes the point D and the whole lower family of P emanating from it however the solutions between D and D should be considered as spurious therefore we do not save these data The reliable way to compute the lower family of P is to restart computation of saddle homoclinic orbits in the other direction from the point D ri ci run ri 0Z3 0 min 5 ap 1 This gives the lower family of P approaching the BT point see Figure 22 1 1 The program actually computes the saddle saddle heteroclinic orbit bifurcating from the non central saddle node homoclinic at the point D see Champneys et al 1996 Fig 2 and continues it to the one emanating from D 222 BR PT TY LAB PAR 1 PAR 2 PAR 29 PAR 30 1 10 16 4 96649E 00 6 29843E 02 4 38247E 01 4 94481E 03 1 20 17 4 92531E 00 7 96087E 02 3 39922E 01 3 28829E 02 1 30 18 7 09217E 00 1 58708E 01 1 69289E 01 3 87631E 02 1 40 EP 19 1 10181E 01 2 80980E 01 3 48294E 02 2 10449EF 02 The data are appended to the stored results i
255. proximate solitary wave solutions to 2 3 Demo wav Run 4 Do time evolution calculations for 2 3 given periodic initial data on the interval 0 L The initial data must be specified on 0 1 and L must be set separately because of internal scaling The initial data may be given analytically or obtained from a previous computa tion of wave trains solitary waves or from a previous evolution calculation Conversely if an evolution calculation results in a stationary wave then this wave can be used as starting data for a wave continuation calculation Demo wav Run 5 Do time evolution calculations for subject to user specified boundary conditions As above the initial data must be specified on 0 1 and the space interval length L must be specified separately Time evolution computations of are adaptive in space and in time Discretization in time is not very accurate only implicit Euler Indeed time integration of has only been included as a convenience and it is not very efficient Demos pd1 pd2 Compute curves of stationary solutions to subject to user specified boundary con ditions The initial data may be given analytically obtained from a previous stationary solution computation or from a time evolution calculation Demos pd1 pd2 In connection with periodic waves note that is just a special case of and that its fixed point analysis is a special case of 2 1 One advantage of the built in capacity of AUTO
256. pt file The Configure button brings up the dialog for setting configuration options Query string Meaning 1_1_resonance_symbol The symbol to use for 1 1 resonance points 47 1_2_resonance_symbol The symbol to use for 1 2 resonance points 1_3_resonance_symbol The symbol to use for 1 3 resonance points 1_4 resonance symbol The symbol to use for 1 4 resonance points azimuth Azimuth of the axes in 3D plots background The background color of the plot bifurcation_column_defaults A set of bifurcation columns the user is likely to use bifurcation_coordnames Names to use instead of PAR 1 for bifurcation diagrams bifurcation_diagram A parsed bifurcation diagram file to plot bifurcation diagram filename The filename of the bifurcation diagram to plot bifurcation symbol The symbol to use for bifurcation points bifurcation_x The column to plot along the X axis for bifurcation diagrams bifurcation_y The column to plot along the Y axis for bifurcation diagrams bifurcation_z The column to plot along the Z axis for bifurcation diagrams bogdanov_takens_symbol The symbol to use for Bogdanov Takens points bottom_margin The margin between the graph and the bottom edge color_list A list of colors to use for multiple plots coloring method color_list index branch BR type TY or curve seq
257. qua tions Type f1 xxx to list the Floquet multipliers in the data file d xxx Differential equa tions Type Cno to show any notes in fort 9 Type Ono xxx to show any notes in d xxx 69 5 0 e7 Ces e9 j7 j8 j9 5 6 rl lb LB fc db tr File editing To use the vi editor to edit the output file fort 7 To use the vi editor to edit the output file fort 8 To use the vi editor to edit the output file fort 9 To use the SGI jot editor to edit the output file fort 7 To use the SGI jot editor to edit the output file fort 8 To use the SGI jot editor to edit the output file fort 9 File maintenance Type Crl to sequentially relabel solutions with the numbers 1 2 3 in the output files fort 7 and fort 8 The original files are backed up as fort 7 and fort 8 Type rl xxx to relabel solutions in the data files b xxx and s xxx The original files are backed up as b xxx and s xxx Type rl xxx yyy to relabel solutions in the data files b xxx and s xxx The modified files are written as b yyy and s yyy Type lb to run an interactive utility program for listing deleting and relabeling solutions and branches in the output files fort 7 and fort 8 The original files are backed up as fort 7 and fort 8 Type lb xxx to list delete and relabel solutions and branches in the data files b xxx and s xxx The original files are backed up as b xxx and s xxx Type lb xxx yy
258. quilibria in higher dimensional phase space In the computations the independent time variable t is scaled to vary in the unit interval so that the actual integration time T becomes an explicit parameter in the equations namely a T ex y y T y r To carry out the calculations run the Python script um2 auto included in the demo In order to better appreciate the power of orbit continuation for computing such manifolds one can also run the script for a smaller value of e e g e 107 107 by changing the value entered on the last line of the constants file c umn 2 One can view the phase portrait by plotting the solutions in the solutions file s 3 in the x y plane In the first run an orbit is grown by continuation in the integration time T starting from a very small value of T so that a solution that is constant in time is an accurate initial approximation The starting solution is in fact a point on a circle of small radius r around the stationary point i e around the origin For illustrative purpose the value of r is 0 1 in this demo but could be smaller if more accuracy is needed The precise starting point is in the strongly unstable direction namely in the y direction which is the direction of the eigenvector associated with the strongly unstable eigenvalue which here has value 1 The growing of the initial orbit is terminated when the norm of its endpoint reaches the value 0 6 i e when x 1
259. r point to be a Hopf bifurcation point TY HB or a branch point TY BP Use TY HB4 to copy the period of the emanating periodic orbit from PAR 4 for example set in the routine PVLS in the equations file to PAR 11 Demo r3b 10 8 12 IPS This constant defines the problem type IPS 0 An algebraic bifurcation problem Hopf bifurcations will not be detected and stability properties will not be indicated in the fort 7 output file IPS 1 Stationary solutions of ODEs with detection of Hopf bifurcations The sign of PT the point number in fort 7 is used to indicate stability is stable is unstable Demo ab IPS 1 Fixed points of the discrete dynamical system u f u p with detection of Hopf bifurcations The sign of PT in fort 7 indicates stability is stable is unstable Demo dd2 108 IPS 2 Time integration using implicit Euler The AUTO constants DS DSMIN DSMAX and ITNW NWTN control the stepsize In fact pseudo arclength is used for continuation in time Note that the time discretization is only first order accurate so that results should be carefully interpreted Indeed this option has been included primarily for the detection of stationary solutions which can then be entered in the user supplied routine STPNT Demo ivp IPS 2 Computation of periodic solutions Starting data can be a Hopf bifurcation point Run 2 of demo ab a periodic orb
260. rData dlb Delete special labels Type dlb x list to delete the special points in list from the Python object x which must be a solution list or a bifurcation diagram Type dlb list xxx to delete from the data files b xxx and s xxx Type dlb list xxx yyy to save to b yyy and s yyy instead of xxx Type dlb list to delete from fort 7 and fort 8 list is a label number or type name code or a list of those such as 1 or 2 3 or UZ or BP LP or it can be None or omitted to mean the special points BP LP HB PD TR EP MX Alternatively a boolean user defined function f that takes a solution can be specified for list such as def f s return s PAR 9 lt 0 where all solutions are deleted that satisfy the given condition or def f s1 s2 return abs s1 L2 NORM s2 L2 NORM lt 1le 4 where all solutions are compared with each other and s2 is deleted if the given condition is satisfied which causes pruning of solutions that are close to each other Type information is kept in the bifurcation diagram for plotting Alias commandDeleteLabels 60 klb Keep special labels Type k1b x list to only keep the special points in list in the Python object x which must be a solution list or a bifurcation diagram Type klb list xxx to keep them in the data files b xxx and s xxx Type klb list xxx yyy to save to b yyy and s yyy in
261. re processing First the command 33 both fw bw concatenates using standard Python list syntax the forwards and backwards branches Subsequently merged merge both merges the two branches into one continuous branch where the backwards branch is flipped Finally the command bvp bvp merged appends the merged branch to the existing results bvp relabel bvp save bvp bvp plot bvp wait Figure 4 11 The third part of the complex AUTO CLUI script The last section of the script extracted into Figure does some postprocessing and plotting First the command bvp relabel bvp relabels so all solutions in the bvp object have unique labels starting at 1 The command save bvp bvp Section 4 14 1 in the reference saves the results of the AUTO runs into files using the base name bvp and the filename extensions in Tablel4 3 For example in this case the bifurcation diagram will be saved as b bvp the solution will be saved as s bvp and the diagnostics will be saved as d bvp Now that the section of script shown in Figure 4 11 has finished computing the bifurcation diagram the command plot bvp brings up a plotting window Section 4 14 2 in the reference and the command wait causes the AUTO CLUI to wait for input You may now exit the AUTO CLUI by pressing any key in the window in which you started the AUTO CLUI For convenience some of these commands have shorter forms For instance the load run
262. re saved Type merge xxx yyy to merge branches in the existing data files s xxx b xxx and d xxx and save them to s yyy b yyy and d yyy Aliases mb commandMergeBranches subtract Subtract branches in data files Type z subtract x y ref to return the python object x where using interpolation the first branch in y is subtracted from all branches in x as a new object z Use ref e g PAR 1 as the reference column in y only the first monotonically increasing or decreasing part is used Type subtract xxx yyy ref to subtract using interpolation the first branch in b yyy from all branches in b xxx and save the result in b xxx A Backup of the original file is saved Use optional arguments branch m and point n to denote the branch and first point on that branch within y or b yyy where m n are in 1 2 3 Aliases sb commandSubtractBranches 62 4 14 6 Copying a demo demo Copy a demo into the current directory and load it Type demo xxx to copy all files from auto 07p demos xxx to the current user directory Here xxx denotes a demo name e g abc To avoid the overwriting of existing files always run demos in a clean work directory NOTE This command automatically performs the load command as well Aliases dm commandCopyAndLoadDemo copydemo Copy a demo into the current directory Type copydemo xxx to copy all files from auto 07p demos xxx to t
263. ree ways First we add PAR 21 and PAR 33 to the list of continuation parameters second we set up user defined output at zeros of these parameters and finally we set IPSI 1 13 We also add another user zero for detecting when PAR 4 1 0 Running r3 run r2 ICP 4 8 10 21 33 IPSI 1 13 NMX 50 NPR 20 UZR 4 1 0 21 0 33 0 save r3 r3 starts from the last and in this case only labelled solution in r2 and outputs to the screen BR PT TY LAB PAR 4 ae PAR 8 PAR 10 pa PAR 33 1 20 4 7 84722E 01 2 72146E 11 4 21812E 09 1 44112E 01 1 27 UZ 5 1 00000E 00 3 91152E 11 4 38659E 09 5 70167E 00 1 35 UZ 6 1 23086E 00 6 18304E 11 4 62672E 09 9 48584E 06 1 40 7 1 38397E 00 8 41993E 11 4 63701E 09 1 34882E 00 1 50 EP 8 1 69521E 00 1 36449E 10 5 35972E 09 5 31105E 01 Full output is stored in b 3 s 3 and d 3 Note that the artificial parameter e PAR 10 is zero within the allowed tolerance At label 6 a zero of test function 413 has been detected which corresponds to an inclination flip with respect to the stable manifold That the orientation of the homoclinic loop changes as the family passes through this point can be read from the information in d 3 However in d 3 the line ORIENTABLE 0 2982090775E 03 at PT 35 would seems to contradict the detection of the inclination flip at this point Nonetheless the important fact is the zero of the test function and note that the value o
264. retical result that the solution tends uniformly to zero as P 0 Note by plotting the data saved in s 1 that only half of the homoclinic orbit is computed up to its point of symmetry See Figure The second run continues in the other direction of PAR 1 with the test function Y2 activated for the detection of saddle to saddle focus transition points ri ri run ri UZ1 c revw 2 ap 1 gt The output BR PT TY LAB PAR 1 L2 NORM MAX U 1 PAR 22 1 11 UZ 6 1 00001E 00 2 81700E 01 1 76625E 01 3 00001E 00 1 22 UZ 7 1 00743E 07 2 89421E 01 4 69706E 02 2 00000E 00 1 33 UZ 8 1 00000E 00 3 02208E 01 4 32654E 03 1 00000E 00 1 44 UZ 9 2 00000E 00 3 16798E 01 1 22616E 11 2 66362E 08 1 55 EP 10 3 09920E 00 3 32927E 01 4 00188E 10 1 09920E 00 shows a saddle to saddle focus transition indicated by a zero of PAR 22 at PAR 1 2 Beyond that label the first component of the solution is negative and up to the point of symmetry monotone decreasing See Figure 26 3 An Ro Reversible Homoclinic Solution r3 run rev c rev 3 sv 3 starts by using the file rev dat 3 via the dat AUTO constant in c rev 3 and runs them with the constants stored in c rev 3 The orbit contained in the data file is a multi pulse homoclinic solution for P 1 6 with truncation half interval PAR 11 47 4464189 which is reversible under R This reversibility is specified in c rev 1 via IREV 1 0 1 0 The o
265. returned as a Python list one would type bd toArray Similarily if one wanted to write out the bifurcation diagram to the file outputfile one would type bd writeRawFilename outputfile To get the solution with label 57 returned as a numpy array one would type bd 57 toarray Similarily if one wanted to write out the solution to the file outputfile one would type bd 57 writeRawFilename outputfile 4 10 The autorc or autorc File Much of the default behavior of the AUTO CLUI can be controlled by the autorc file The autorc file can exist in either the main AUTO directory the users home directory or the current directory In the current directory it can also have the name autorc that is without the dot For any option which is defined in more then one file the autorc file in the current directory if it exists takes precedence followed by the autorc file in the users home directory if it exists and then the autorc file in the main AUTO directory Hence options may be defined on either a per directory per user or global basis 46 The first section of the autorc file begins with the line AUTO command aliases and this section defines short names or aliases for the AUTO CLUI commands Each line thereafter is a definition of a command similiar to branchPoint commandQueryBranchPoint The right hand side of the assignment is the internal AUTO CLUI name for the command while the left hand side is the desire
266. riod doubling and torus bifurcations will go undetected This option is useful for certain problems with non generic Floquet behavior ISP 4 Branch points and Hopf bifurcations are detected for algebraic equations Branch points are not detected for periodic solutions and boundary value problems AUTO will monitor the Floquet multipliers and period doubling and torus bifurcations will be de tected 10 8 4 ISW This constant controls branch switching at branch points for the case of differential equations Note that branch switching is automatic for algebraic equations 106 ISW 1 This is the normal value of ISW ISW 1 If IRS is the label of a branch point or a period doubling bifurcation then branch switching will be done For period doubling bifurcations it is recommended that NTST be increased For examples see Run 2 and Run 3 of demo lor where branch switching is done at period doubling bifurcations and Run 2 and Run 3 of demo bvp where branch switching is done at a transcritical branch point ISW 2 If IRS is the label of a fold a Hopf bifurcation point a period doubling a torus bifurcation or in a non generic symmetric system a branch point then a locus of such points will be computed An additional free parameter must be specified for such contin uations see also Section ISW 3 If IRS is the label of a branch point in a generic non symmetric system then a locus of such points will be computed Two add
267. rnatively see or r research att com tools At the time of writing the first of these is recommended to be able to benefit from parallelization e Python you can install 32 bit Python 2 7 NumPy and Matplotlib dmg files from www python org numpy scipy org and matplotlib sf net respectively For example download from http www python org ftp python 2 7 2 python 2 7 2 macosx10 3 dmg http sourceforge net projects numpy files NumPy 1 6 1 numpy 1 6 1 py2 7 python org macosx10 3 dmg download and http sourceforge net projects matplotlib files matplotlib matplotlib 1 1 0 matplotlib 1 1 0 py2 7 python org macosx10 3 dmg download 13 The default system Python in Mac OS X is usable but may be too old for Matplotlib and NumPy There are other alternatives for instance the Enthought Python Distribution at Fink should work but native graphics provided by the previous alternatives seem to work better To be able to plot in the Python CLUI in some versions of OS X AUTO uses pythonw instead of python This should happen automatically e PLAUTO4 Get a Qt dmg from http qt nokia com downloads qt for open source cpp development on mac os x Similarly you can get a binary Coin package from After that you can compile SoQt at least version 1 5 0 from the source code at For example down load http ftp coin3d org coin bin macosx all Coin 3 1 3 gcc4 dmg http get qt nokia com qt source qt mac opensource 4 8
268. rs namely the time variable t da Y and Y The correponding parameter values are defined in the user supplied subroutine STPNT Actually in this example a scaled time variable t is given in pen dat For this reason the period PAR 11 is also set in STPNT Normally AUTO would automatically set the period according to the data in pen dat The AUTO constant dat pen in c pen 1 causes AUTO to start from the data in pen dat The mesh will be suitably adapted to the solution using the number of mesh intervals NTST and the number of collocation point per mesh interval NCOL specified in the constants file c pen 1 The first run with J as free problem parameter starts from the solution here IRS 0 in pen dat A period doubling bifurcation is located and the period doubled family is computed in the second run Two branch points are located and the bifurcating families are traced out in the third and fourth run respectively The fifth run generates starting data for the subsequent computation of a locus of period doubling bifurcations The actual computation is done in the sixth run with e and J as free problem parameters 148 AUTO COMMAND ACTION mkdir pen cd pen demo pen create an empty work directory change directory copy the demo files to the work directory pen run pen c pen 1 locate a period doubling bifurcation restart from pen dat pen pentrun pen
269. rsed object is contained in a variable s data The first solution may be accessed using the command s 0 while the solution with label 57 may be accessed using the command s 57 Individual solutions can be also be obtained directly from the bifurcation diagram object data in the same way as for individual points above by using s data label For example the solution with label 57 may be accessed using the command data 57 and the second Hopf bifurcation solution using the command data HB2 All individual solutions can then be used as a starting point for a new run Similarly using the syntax you can also obtain new lists of solutions data HB gives a list of all Hopf bifurcation solutions data 1 4 gives the solutions with labels 1 and 4 and data UZ 4 gives all user defined solutions and label 4 A for loop can then iterate through 41 get and run the ab demo demo ab auto ab auto load data from b ab s ab and d ab ab loadbd ab change the branch number to either 1 or 2 depending on IPS for branch in ab branch BR branch c IPS delete the last branch del ab 1 subtract the first branch from all other branches with respect to PAR 1 ab subtract ab ab PAR 1 plot the branches coloring by branch number plot ab coloring_method branch color_list black red wait save data to b abnew s abnew and d abnew save ab abnew F
270. rwards The backwards continuation is appended to the forwards continuation in the data files Afterwards we perform a relabelling to make sure that we have unique labels for each special solution Next the relabelled result is saved to the data files b mu s mu and d mu The results are then plotted on the screen Pressing the enter key at the command line causes an automatic y vs x display that shows the two fold points at labels 2 and 7 In the third run the fold detected in the first run is followed in the two parameters u and A We know that label 2 with solution mu 2 is the right solution to start from However 121 we did not know this number in advance and moreover in sensitive cases it can be different on different computer types Another way to specify the starting label is to use the notation mu LP1 this specifies the first LP labelled solution of all solutions in mu Furthermore the command that accomplishes this must change the constant ISW of the constants file it must be set to 2 to cause a two parameter continuation The fourth run continues this branch in opposite direction The detailed results of these continuations are saved in the data files b cusp s cusp and d cusp Finally a plot of the cusp is produced The numerical results are given below in somewhat abbreviated form Some differences in output are to be expected on different machines This does not mean that the results have different accuracy but simp
271. rz 132 142 abc The A B C Reaction This demo illustrates the computation of stationary solutions Hopf bifurcations and periodic solutions in the A B gt C reaction Doedel amp Heinemann 1983 uy u p1 1 uj e uy uz pje 1 uy psu 14 2 uz U3 p3uz pipae 1 uy p2p5uz with po 1 p 1 55 p4 8 and ps 0 04 The free parameter is p The equations as programmed in the equations file abc f90 appear in Table The starting point an equilibrium of the equations is also defined in the equations file abc 90 as shown in Table The equations file abc 90 also contains the skeletons of some other routines which must be supplied but which are not used in this application A more advanced version that continues branch points in three parameters is provided by the demo abcb In the constants file c abc 1 for the first run as shown in Table 14 4 we note the following IPS 1 a family of stationary solutions is computed IRS 0 the starting point defined in STPNT is to be used see Table 14 3 ICP 1 the continuation parameter is PAR 1 UZR 1 0 4 there is one user output point namely at PAR 1 0 4 Moreover since the index 1 in the last line of the constants file c abc 1 is negative the calculation will terminate when the calculation reaches the value PAR 1 0 4 In the constants file c abc 2 for the second run as shown in Tabl
272. s 1 NDIM 2 Ux x dz Note that the interval of integration is 0 1 the standard interval used by AUTO For periodic solutions the independent variable is transformed to range from 0 to 1 before the norm is computed The AUTO constants THL and THU see Section and Section 10 5 6 affect the definition of the L gt 2 norm If 0 lt IPLT lt NDIM then the maximum of the IPLT th solution component is printed If NDIM lt IPLT lt 0 then the minimum of the IPLT th solution component is printed Demo fsh If NDIM lt IPLT lt 2 NDIM then the integral of the IPLT NDIM th solution component is printed Demos exp lor 112 If 2 NDIM lt IPLT lt 3 NDIM then the L2 norm of the IPLT NDIM th solution component is printed Demo frc Note that for algebraic problems the maximum and the minimum are identical Also for ODEs the maximum and the minimum of a solution component are generally much less accurate than the L2 norm and component integrals Note also that the routine PVLS provides a second more general way of defining solution measures see Section 10 9 11 UZR This constant allows the setting of parameter values at which labeled plotting and restart infor mation is to be written in the fort 8 output file Optionally it also allows the computation to terminate at such a point Set UZR if no such output is needed Many demos use this setting Else one must enter pairs
273. s in fact located and hence an extremum of the objective functional with respect to both A2 and A3 has been found Note that in general 7 is the value of the functional that vanishes at an extremum with respect to the system parameter A Run 6 In the final run the above found two parameter extremum is continued in three system parameters here A A2 and A3 toward A 0 Again given the particular choice of objective functional this final continuation has an alternate significance here it also represents a three parameter family of transcritical secondary periodic bifurcations points Although not illustrated here one can restart an ordinary continuation of periodic solutions using IPS 2 or IPS 3 from a labeled solution point on a family computed with IPS 15 191 The free scalar variables specified in the AUTO constants files for Run 3 and Run 4 are shown in Table 17 2 Index 3 11 12 22 22 23 31 Variable A3 T a 72 A2 As T Table 17 2 Runs 3 and 4 files c ops 3 and c ops 4 The parameter a which is the norm of the adjoint variables becomes nonzero after branch switching in Run 4 The negative indices 22 23 and 31 set the active optimality functionals namely for A2 A3 and T respectively with corresponding variables 72 73 and To respectively These should be set in the first run with IPS 15 and remain unchanged in all subsequent runs
274. s some control over the fort 6 screen and fort 7 output via the AUTO constant IPLT Section 10 9 10 Furthermore the routine PVLS can be used to define solution measures which can then be printed by parameter overspecification see Section 10 7 10 For an example see demo pvl The AUTO commands sv sv Cap ap and df df can be used to manipulate the output files fort 7 fort 8 and fort 9 Furthermore the AUTO command 1b r1 can be used to delete and relabel solutions and branches simultaneously in fort 7 and fort 8 For details see Section The graphics programs PLAUT PLAUT04 and the Python CLUI command plot can be used to graphically inspect the data in fort 7 and fort 8 see Chapters 7 and 75 Chapter 7 The Graphics Programs PLAUT and PyPLAUT PLAUT and PyPLAUT can be used to extract graphical information from the AUTO output files fort 7 and fort 8 or from the corresponding data files b xxx and s xxx To invoke PLAUT use the p command defined in Section 5 The PLAUT window a Tektronix window will appear in which PLAUT commands can be entered To invoke PyPLAUT use the Cpp command The same plotting window as you get by using plot in the Python interface appears see Section 4 11 but you can also manipulate it by typing PLAUT commands in the terminal in which you typed Cpp For examples of using PLAUT and PyPLAUT see the tutorial demos pp2 and pp3 in sections 14 3 and 14 9 respectively The files
275. save Type b loadbd Loptions to load output files or output data There are three possible options Long name Short name Description bifurcationdiagram b The bifurcation diagram file solution s The solution file or list of solutions diagnostics d The diagnostics file Type loadbd name to load all files with base name This does the same thing as running loadbd b name s name d name plot b will then plot the b and s components Returns a bifurcation diagram object representing the files in b Aliases bd commandParseOutputFiles Save data files Type save x xxx to save bifurcation diagram x to the files b xxx s xxx d xxx Existing files with these names will be overwritten If x is a solution a list of solutions or does not contain any bifurcation diagram or diagnostics data then only the file s xxx is saved to Type save xxx to save the output files fort 7 fort 8 fort 9 to b xxx s xxx d xxx Ex isting files with these names will be overwritten Aliases commandCopyFortFiles append Append data files Type append x xxx to append bifurcation diagram x to the data files b xxx s xxx and d xxx This is equivalent to the command save xtload xxx xxx Type append xxx xxx to append existing data files s xxx b xxx and d xxx to bifurca tion diagram x This is equivalent to the command x load xxx x Type append xxx to
276. save the displayed plot in a file You will be asked to enter a file name Each plot must be stored in a separate new file The plot is stored in compact PLOT10 format which can be converted to PostScript format with the AUTO commands Cps see Section 7 4 cl To clear the graphics window lab To list the labels of all solutions stored in fort 8 Note that PLAUT requires all labels to be distinct In case of multiple labels you can use the AUTO command 1b to relabel solutions in fort 7 and fort 8 end To end execution of PLAUT 7 2 Default Options After entering the commands bd0 bd or 2d you will be asked whether you want solution labels grid lines titles or axes labels For quick plotting it is convenient to bypass these selections This can be done by the default commands d0 d1 d2 d3 or d4 below These can be entered as a single command or they can be entered as prefixes in the bd0 and bd commands Thus for example one can enter the command d1bd0 d0 Use solid curves showing symbols but no solution labels di Use solid curves except use dashed curves for unstable solutions and for solutions of unknown stability Show solution labels and symbols d2 As d1 but without solution labels d3 As d1 but with grid lines d4 As d2 but with grid lines If no default option d0 d1 d2 d3 or d4 has been selected or if you want to override a default feature then the following commands can be used These can be
277. sed first 5 Some variables can only be set to Yes or No They cannot be assigned other values 6 No variable name should be modified 85 It is strongly recommended that the default resource file is used as a template when writing a custom resource file Below is a copy of the default resource file version 0 0 Line colors are represented by RGB values from O to 1 0 DEFAULT color is also used when animationLabel 0 i e when showing all solutions and animating the solution change Point Type RED GREEN BLUE PATTERN DEFAULT 1 0 1 0 1 0 Oxffff BP ALG 1 0 0 0 0 0 Oxffff LP ALG 0 0 1 0 0 0 Oxffff HB 0 0 0 0 1 0 Oxffff UZ4 1 0 1 0 0 0 Oxffff UZ 4 0 5 0 5 0 0 Oxffff LP DIF 0 0 0 0 0 5 Oxffff BP DIF 0 0 0 5 0 5 Oxffff PD 1 0 0 0 1 0 Oxffff TR 0 0 1 0 1 0 Oxffff EP 0 3 0 0 0 3 Oxffff MX 0 6 0 0 0 6 Oxffff OTHERS 1 0 1 0 1 0 Oxffff o o Initialize the line pattern for showing stability UNSTABLE LINE PATTERN Oxffff STABLE LINE PATTERN Oxffff Initialize the default options Draw Reference Plane No Draw Reference Sphere No Orbit Animation No Satellite Animation No Draw Primaries No Draw Libration Points No Normalize Data No Draw Labels No Show Label Numbers No Draw Background No Draw Legend No Initialize the default coordinate axes 0 None 1 at origin 2 at left and
278. set by the user in the file brf f90 As an illustrative application we consider the Brusselator Holodniok Knedlik amp Kub ek 1987 us D L Ugg uv B 1 u A ve Dy L vrs uv Bu with boundary conditions u 0 t u 1 t A and v 0 t v 1 t B A Note that given the non adaptive spatial discretization the computational procedure here is not appropriate for PDEs with solutions that rapidly vary in space and care must be taken to recognize spurious solutions and bifurcations 16 4 AUTO COMMAND ACTION mkdir brf create an empty work directory cd brf change directory demo brf copy the demo files to the work directory ri run e brf c brf compute the stationary solution family with Hopf bifurcations r2 run r1 HB1 IPS 2 ICP 5 11 compute a family of periodic solutions NMX 120 EPSL 1e 8 from the first Hopf point r3 run r2 BP1 ISW 1 compute a solution family from a sec NMX 100 EPSL 1e 7 ondary periodic bifurcation save ritr2 r3 brf save all output to b brf s brf d brf Table 16 5 Commands for running demo brf 186 16 6 bru Euler Time Integration the Brusselator This demo illustrates the use of Euler s method for time integration of a nonlinear parabolic PDE The example is the Brusselator Holodniok Knedlik amp Kub ek 1987 given by up D L Urs uu B 1 u A 2 2 16 5 Y Dy L Vrs u v Bu
279. side the unit circle two are on the unit circle and the other two are inside the unit circle 81 File Style Type Options Center Help Dialog inf 8 851312E 01 3 0568203E 01 5 522057E 01 6 902421E 01 1 061813E 00 1 432000E 01 6 696635E 01 Floquet multipliers 1 000000E 00 0 000000E 00 1 000000E 00 0 000000E 00 1 234718E 02 1 102920E 02 a Motion X Motion Y MRE 7 Figure 8 7 Picking a point Line fio me Sat f Thickness Orbit p eS MOIS 04 010e 00 plot3d File Style Type Options Center Help Label ALL v Number of Periods Labels To Be Shown Z Axis Coloring Method Line Tube Thickness attelite Animation Speed Figure 8 8 Menu bar layout 82 Orbit Animation Speed 8 1 9 Choosing the variables AUTO can generate large amounts of data The CR3BP for example has 6 variables t e 1 Yy 2 1 y z and time One can choose to draw any combination of these variables in 2 or 3 dimensions using PLAUTO4 On the list bar we can see three dropdown lists with label X Y and Z See Figure 8 8 Each of these three lists has the exact number of choices namely the number of variables of the system plus one In our case these lists have 7 choices which are represented by the integers 0 to 6 0 represents time 1 to 6 stand for x y z 2 y and z respectively 1 is selected for X which indicates that
280. splus direction Continuation in any other system parameter is easily obtained by editing the ICP entry The script plot auto produces the plots we show here in Figures 15 6 15 7 and 0 0 0 55 0 60 E 1 0 l L i 1 0 5 0 4 s 0 40 0 45 0 50 0 55 0 60 0 65 h Figure 15 6 Repelling and attracting slow manifold curves where they match up left and their intersection curves at s 0 2797 in the h v plane right 179 i 1 fi 1 28 00 0 01 0 02 0 03 0 04 0 05 0 06 0 07 35 3 0 2 5 2 0 1 5 1 0 0 5 s 0 0 Figure 15 7 Continuation of canard orbits the AUTO L norm as a function of e for all six orbits left and a projection of labels 4 8 11 and 16 black red blue green of the orbit s on the v s plane right 0 0 0 0 0 5 H 0 5F m 0 0 Y 0 5 b 4 10b gt 10 10 1 5 Aihe 1 5L 20 i i i 2 i i ji j i il _a io jo io i N i 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 20 0 25 0 30 0 35 0 40 045 0 50 0 55 0 20 0 25 0 30 0 35 0 40 0 45 0 50 0 55 Ss Ss Ss Figure 15 8 Continuation of canard orbits projection of all canard orbits on the v s plane for e 0 015 left e 1074 middle and e 10 right 180 Chapter 16 AUTO Demos Parabolic PDEs 181 16 1 pdl Stationary States 1D Problem This demo uses Euler s method to locate a stationary solution of a nonlinear parabolic PDE fol
281. stead of xxx Type k1b list to keep them in fort 7 and fort 8 list is a label number or type name code or a list of those such as 1 or 2 3 or UZ or BP LP or it can be None or omitted to mean BP LP HB PD TR EP MX deleting UZ and regular points Alternatively a boolean user defined function f that takes a solution can be specified for list such as def f s return s PAR 9 lt 0 where only solutions are kept that satisfy the given condition Type information is kept in the bifurcation diagram for plotting Alias commandKeepLabels dsp Delete special points Type dsp x list to delete the special points in list from the Python object x which must be a solution list or a bifurcation diagram Type dsp list xxx to delete from the data files b xxx and s xxx Type dsp list xxx yyy to save to b yyy and s yyy instead of xxx Type dsp list to delete from fort 7 and fort 8 list is a label number or type name code or a list of those such as 1 or 2 3 or UZ or BP LP or it can be None or omitted to mean the special points BP LP HB PD TR EP MX Alternatively a boolean user defined function f that takes a solution can be specified for list such as def f s return s PAR 9 lt 0 where all solutions are deleted that satisfy the given condition or def f s1 s2 return abs s1 L2 NORM
282. sv 1 Among the output there is the line BR PT TY LAB PERIOD L2 NORM pd PAR 17 1 29 UZ 2 1 90018E 01 1 69382E 00 4 46147E 09 which indicates that a zero of the artificial parameter w has been located This means that the right hand end point of the solution belongs to the plane that is tangent to the stable manifold at the saddle The output is stored in files b 1 s 1 d 1 Upon plotting the data at label 2 see Figure 23 1 it can be noted that although the right hand projection boundary condition is satisfied the solution is still quite away from the equilibrium 226 s1020 1 000 0 880 09 60 0 940 1 010 0 990 0 970 0 950 x Figure 23 1 Projection on the x y plane of solutions of the boundary value problem with 2T 19 08778 1 020 1 000 0 980 0 960 0 940 1 010 0 990 0 970 0 950 x Figure 23 2 Projection on the x y plane of solutions of the boundary value problem with 2T 60 0 227 The right hand endpoint can be made to approach the equilibrium by performing a further continuation in T with the right hand projection condition satisfied PAR 17 fixed but with A allowed to vary r2 run r1 UZ1 c kpr 2 sv 2 the output at label 4 stored in s 2 BR PT TY LAB PERIOD L2 NORM as PAR 1 1 35 UZ 4 6 00000E 01 1 67281E 00 1 85119E 00 provides a good approximation to a h
283. t is stored in b 7 s 7 and d 7 Here we see that T the time it takes to make the first loop with respect to the Poincar section decreases This is illustrated in Figure Next we are ready to close this gap by continuing in a u and 1 while keeping T at a constant value r8 run r7 ICP 4 5 22 NPR 310 DS 0 01 DSMAX 0 01 UZR 22 0 0 4 0 074 r6 r6 r8 BR PT TY LAB PAR 4 L2 NORM a PAR 5 PAR 22 3 3 UZ 22 7 40000E 02 4 46781E 01 1 43162E 02 1 93746E 01 3 32 UZ 23 1 98414E 01 4 46590E 01 6 05495E 03 2 29300E 06 The output is appended to the Python variable r6 Now we have obtained a 2 homoclinic orbit at label 23 However the homoclinic orbit is still split in three parts We can switch back to a normal orbit by setting ITWIST back to 0 and continuing in the usual way Here we continue back to the inclination flip point in a and pu r9 run r8 ICP 4 5 NMX 30 DS DSMAX 0 1 UZR 4 0 15 ISTART 1 ITWIST 0 r6 r6 r9 BR PT TY LAB PAR 4 L2 NORM er PAR 5 3 7 UZ 24 1 50000E 01 4 94490E 01 3 60248E 03 3 30 EP 25 7 61403E 02 4 98746E 01 2 64847E 06 So the 2 homoclinic orbit converges back to the 1 homoclinic orbit at the inclination flip bifur cation The output is appended to the python variable r6 The resulting 2 homoclinic orbits can be seen using 252 4 000 01 NE v 0 Columns t 4 000 01 0 008 00 1 008 00 Columns t Figure
284. t of values for user defined output SP STOP list of bifurcations to check and bifurcation stop conditions NUNSTAB NSTAB IEQUIB ITWIST ISTART IREV IFIXED IPSI HomCont unstable and stable manifold dimensions HomCont control solution types adjoint starting HomCont control reversibility fixed parameters test functions 115 Chapter 11 Notes on Using AUTO 11 1 Restrictions on the Use of PAR The array PAR in the user supplied routines is available for equation parameters that the user wants to vary at some point in the computations In any particular computation the free parameter s must be designated in ICP see Section 10 7 The following restrictions apply The default maximum number of parameters NPAR has a pre defined value in auto 07p include auto h of NPARX 36 Any change NPARX must be followed by recompilation of AUTO Generally one should avoid certain parameters for equation parameters as AUTO may need those internally as follows IPS 0 4 No additional parameters with indices less than or equal to NPAR are reserved IPS 1 1 2 7 AUTO reserves PAR 11 to store the period for stationary solutions at Hopf bifurcations only and continuously for periodic orbits For IPS 2 and IPS 7 AUTO also reserves PAR 12 to store the angle of the torus at torus bifurcations IPS 2 The integration time is stored in PAR 14 IPS 5 The value of the objective functional is stored in PAR 10 I
285. tailed example see demo ops IPS 16 This option is similar to IPS 14 except that the user supplies the boundary conditions Thus this option can be used for time integration of parabolic systems subject to user defined boundary conditions For examples see the first runs of demos pd1 pd2 and bru Note that the space derivatives of the initial conditions must also be supplied in the user supplied routine STPNT The initial conditions must satisfy the boundary conditions This option is mainly intended for the detecting stationary solutions IPS 17 This option can be used to continue stationary solutions of parabolic systems obtained from an evolution run with IPS 16 For examples see the second runs of demos pdi and pd2 10 9 Output Control 10 9 1 unames This constant where unames i1 s1 i2 s2 changes the names in all output from U il to s1 from U i2 to s2 and so on You can also refer to these strings instead of the corresponding indices in the constants U and THU 10 9 2 parnames This constant where parnames il s1 i2 s2 changes the names in all output from PAR i1 to s1 from PAR i2 to s2 and so on You can also refer to these strings instead of the corresponding indices in the constants ICP THL and UZR 10 93 e This constant where e xxx is only for use by post processors it denotes the name of the equations file and is stored in the bifurcation diagram file fort 7 so restarts in the Python in
286. tching at an inclination flip in Sandstede s model 248 27 2 Branch switching for a Shil nikov type homoclinic orbit in the Fitz Hugh Nagumo de Ae ds dob denn tee io A woe te ge Aeon de de 255 27 3 Branch switching to a 3 homoclinic orbit in a 5th order Korteweg De Vries model 258 Preface This is a guide to the software package AUTO for continuation and bifurcation problems in ordi nary differential equations Earlier versions of AUTO were described in Doedel 1981 Doedel amp Kern vez 1986a Doedel amp Wang 1995 Wang amp Doedel 1995 Doedel Champneys Fairgrieve Kuznetsov Sandstede amp Wang 1997 Doedel Paffenroth Champneys Fairgrieve Kuznetsov Oldeman Sandstede amp Wang 2000 For a description of the basic algorithms see Doedel Keller amp Kern vez 1991 a Doedel Keller amp Kern vez 1991b AUTO incorporates the HomCont algorithms of Champneys amp Kuznetsov 1994 Champneys Kuznetsov amp Sandstede 1996 for the bifurcation analysis of homoclinic orbits and the BPcont algorithms of Dercole 2008 for the continuation of branch points in both symmetric and non symmetric problems The Floquet multiplier algorithms were written by Fairgrieve 1994 Fairgrieve amp Jepson 1991 A GUI was written by Wang 1994 The Python CLUI is the work of Randy Paffenroth Acknowledgments The first author is much indebted to H B Keller of the California Institute of Technology for his inspirat
287. terface are possible without needing to specify the equations file 10 9 4 sv This constant specifies a string to write the output to instead of fort 7 fort 8 and fort 9 if sv xxx then the output files are b xxx s xxx and d xxx 110 10 9 5 NPR This constant can be used to regularly write fort 8 plotting and restart data IF NPR gt 0 then such output is written every NPR steps IF NPR 0 or if NPR gt NMX then no such output is written Note that special solutions such as branch points folds end points etc are always written in fort 8 Furthermore one can specify parameter values where plotting and restart data is to be written see Section For these reasons and to limit the output volume it is recommended that NPR output be kept to a minimum 10 9 6 IBR This constant specifies the initial branch number BR that is used The default IBR 0 means that that this number is determined automatically 10 9 7 LAB This constant specifies the initial label number LAB that is used The default LAB 0 means that that this number is determined automatically Using LAB 1 means you do not need to relabel after a non appended continuation if this is desired 10 9 8 IIS This constant controls the amount of information printed in fort 8 the greater IIS the more solutions contain the corresponding vector giving the direction of the branch The direction of the branch is necessary for restart points when switching branches but make the
288. thb research project globif index_main html for background information 171 15 13 pcl Lorenz Point to cycle connections with Lin s method This demo computes a point to cycle connection or EtoP connection for equilibrium to periodic orbit in the Lorenz equations r oly 2 y pr y Tz 15 15 z xy Pz using Lin s method as described in Krauskopf amp Rief 2008 Initially we fix 6 8 3 0 10 and let p vary starting from 0 Here we have a transition from simple to chaotic dynamics For approximate values of p in the interval 13 9265 24 0579 one finds preturbulence organized by a pair of symmetrically related periodic orbits that emanate from a homoclinic bifurcation at p 13 9265 For p 24 0579 there exist two symmetric point to cycle connections that mark the appearance of a chaotic attractor for higher values of p The computation to find one of these two connections uses the following steps 1 Find the secondary equilibria emanating from the pitchfork bifurcation of the equilibrium at 0 and their Hopf bifurcations similarly to the demo Irz 2 Follow the periodic orbit emanating from the Hopf bifurcation in p and its period T until p 24 0579 a value close to where a point to cycle connection is known to exist 3 Extend the system putting the variational equation the eigenfunction into solution co ordinates 4 5 and 6 The trivial 0 eigenvector is continued until we hit a branch p
289. the artificial parameters of the homotopy method see Section below PAR 21 PAR 36 These parameters are used for storing the test functions see Sec tion 20 6 The output is in an identical format to AUTO except that additional information at each computed point is written in fort 9 This information comprises the eigenvalues of the left hand equilibrium the values of each activated test function and if ITWIST 1 whether the saddle homoclinic loop is orientable or not Note that the statement about orientability is only meaningful if the leading eigenvalues are not complex and the homoclinic solution is not in a flip configuration that is none of the test functions y for i 11 12 13 14 is zero or close to zero see Section 20 6 Finally the values of the IPSI activated test functions are written 20 6 Test Functions Codimension two homoclinic orbits are detected along branches of codim 1 homoclinics by lo cating zeroes of certain test functions The test functions that are switched on during any continuation are given by the choice of the labels and are specified by the parameters IPSI in c xxx Here IPSI IPSI 1 IPSI NPSI gives the labels of the test functions numbers between 1 and 16 A zero of each labeled test function defines a certain codimension two ho moclinic singularity specified as follows The notation used for eigenvalues is the same as that in Champneys amp Kuznetsov 1994 or Champn
290. the given type as in dlb UZ xxx otherwise only labels with type UZ and regular labels are kept Type information is kept in the bifurcation diagram for plotting Type Ck1b to keep all special labels in fort 7 and fort 8 Backups are made Type klb xxx to keep all special labels in b xxx and s xxx Backups are made Type klb xxx yyy to keep all special labels in b xxx and s xxx The output is written to b yyy and s yyy Optionally give an argument of the form UZ HB LP EP PD TR BP MX or RG to keep all labels with the given type as in klb UZ xxx and remove all others otherwise all labels are kept except for labels with type UZ and regular labels Type information is kept in the bifurcation diagram for plotting The command Cdsp is equivalent to the command Cdlb above except that type infor mation is not kept in the bifurcation diagram for plotting The command Cksp is equivalent to the command Ck1b above except that type infor mation is not kept in the bifurcation diagram for plotting The command Cdlp is equivalent to the command dsp LP above The command Ckbp is equivalent to the command Cksp BP above The command klp is equivalent to the command ksp LP above The command Ckuz is equivalent to the command ksp UZ above Type Crd to reduce the solution in fort 7 and fort 8 e Ord xxx to reduce the solution in b xxx and s xxx Reducing means that all even regular point solutions from NP
291. tics file fort 9 11 5 Floquet Multipliers AUTO extracts an approximation to the linearized Poincar map from the Jacobian of the linearized collocation system that arises in Newton s method This procedure is very efficient the map is computed at negligible extra cost The linear equations solver of AUTO is described in Doedel Keller amp Kern vez 19916 The actual Floquet multiplier solver was written by Fairgrieve 1994 For a detailed description of the algorithm see Fairgrieve amp Jepson 1991 For periodic solutions the exact linearized Poincar map always has a multiplier z 1 A good accuracy check is to inspect this multiplier in the diagnostics output file fort 9 If this multiplier becomes inaccurate then the automatic detection of potential secondary periodic bifurcations if ISP 2 is discontinued and a warning is printed in fort 9 It is strongly recommended that the contents of this file be habitually inspected in particular to verify whether solutions labeled as BP or TR cf Table 6 1 have indeed been correctly classified 11 6 Memory Requirements The run time memory requirements depend on the values the user set in the constants file and are roughly proportional to the value NTST x NDIMx NCOL 1 NBC x NDIM x NCOL NINT 1 118 Chapter 12 AUTO Demos Tutorial 119 12 1 Introduction The directory auto 07p demos has a large number of subdirectories for example ab pp2 exp etc each conta
292. ting and stopping PLAUTO4 26 5204 644 8 A 8 1 2 Changing the Type f oaaae 8 1 3 Changing the Style oaoa aaa eed ae eudid una 8 1 4 Coordinate axes 2 2 8 19 PONG s s a lt a ton ede A GS eae ee oe ete ee SE ees te ev eee ee ee ee a Ole SHOWN i sk ot oO ae eS ODES a Bods amp G 8 1 8 Picking a point in the diagram s 42 s lt sh2 ae eee we G4 05 8 1 9 Choosing the variables lt 2 4444 24 3 bk Rw RRR RE 8 1 10 Choosing labels accio ea ew Ge ew A a AMARE E AIF 8 1 12 Number of periods to be animated o 84 a ee ae ae eee ee 85 8 1 14 Changing the animation speed o 85 A a Le Gs o oe 85 E eee hs oe Bn oe EE a a 85 oe Ree Peele a Pe oe De ee AA glee gs ace 89 92 9 1 General Overview 2 e e a a a 92 rgb ead Se ae Ge ee ac ee ee ee 92 9 12 The Define Constants buttons 0 0 0 0 0 0 0 0 000000 0084 92 9 13 The Load Constants buttons 0 0 0 0 0 0 0 000000 8G 93 9 1 4 The Stop and Exit buttons occiso crees 93 9 2 The Menu Bar 2 00 2 a 93 O22 Equations button easier ebb be ES EGG Be eS SE Ke 93 nee Ee ek we ee we ee ee as ae ee 93 9 2 3 Write button 2 2 a a a 93 9 2 4 Define button 2 a a a a 93 9 2 5 Run button ooa aaa 94 9 2 6 Save button aaa aa 94 9 2 7 Append button ese pde 2 dw wie ene taa ew es 94 9 2 8 Plot button 0 000000 00 94 9 2 9 Files button 2 a 94 no we Se De S
293. tion is now computed from label 3 with step size 107 The detailed commands follow the ones for Hla above except that there is one extra run see Table 14 27 155 mkdir r3b mkdir r3b cd r3b cd r3b Cdm r3b demo r3b Copy the r3b demo to the local directory r3b auto r3b auto auto r3b auto Generate the CR3BP AUTO data files autox ext py L1 3 1e 5 import ext sext ext get L1 3 1e 5 Convert the data for a selected labeled solution from s L1 adding a zero adjoint variable The solution label is 3 and the initial step size into the unstable manifold is 107 The ext py script looks for the relevant Floquet multiplier in d L1 The converted solution will be written in the file s ext or stored in sext Or flq ext flgq run sext c flq e fl1q Compute the Floquet eigenfunction Free scalar variables in this run see c flq are PAR 1 unfolding parameter PAR 4 multiplier PAR 5 norm of eigenfunction If this run is successful then PAR 5 should become nonzero in fact PAR 5 should reach the value 1 If the run is not successful then see REMARK 1 below sv flq save flq flq Save the results in b flq s flq and d flq autox data py import data startman data get flq UZ1 Extract data for a selected orbit from s flq These data are for both the orbit and its Floquet eigenfunction It is assumed that s flq contains only one labeled solution with label
294. to a 3 homoclinic orbit in a 5th order Korteweg De Vries model In Champneys amp Groves 1997 the following water wave model was considered 2 1 dl br ar on U i rr 0 27 4 It represents solitary wave solutions r x at r 0 as x 00 of the 5th order PDE Tt F rel xam T OF sig Sr F PA F TT rra 05 15 where a is the wave speed The ODE corresponds to a Hamiltonian system with Hamiltonian 1 1 1 15 1 H 50 500 Pie 50 TP 500 and 2 mi 2 M a QSr P Tip For r P2 ig gt System 27 4 is also reversible under the transformation to q q2 P1 P2 m q q2 p p2 but we do not exploit the reversible structure IREV 0 and instead use it as an example of Hamiltonian system This system exhibits an orbit flip for a reversible Hamiltonian system In Hamiltonian systems homoclinic orbits are codimension zero phenomena and we have to add an additional parameter that breaks the Hamiltonian structure in this system by introducing artificial friction Thus the actual system of equations that is used for continuation is t AM J VH z where x q1 q2 P1 p2 and J is the usual skew symmetric matrix in Rt It is now possible to continue a homoclinic orbit in HomCont in two parameters A and either a or b see also Beyn 1990 An explicit solution exists for a 3 5 2b 1 b 2 b gt 1 2 and it is given by r t 3 b 5 sech Ger Di It
295. to create a new command as in Figure 4 15 First we create a file called userScript py which contains the script from Figure with one minor modification We want the function only to run when we use it interactively not when the file userScript py is read in so we remove the last line where the function is called We start the AUTO CLUI with the Unix command auto and once the AUTO CLUI is running we use the command from userScript import to import the file userScript py into the AUTO CLUI The import command makes all functions in that file available for our use in this case myRun is the only one It is important to note that from userScript import does not use the py extension on the file name After importing our new function we may use it just like any other function in the AUTO CLUI for example by typing myRun bvp 4 7 Bifurcation Diagram Objects The run and loadbd commands see Section 4 14 1 in the reference for details return a Python structure which we refer to as a bifurcation diagram object It represents the information that is also stored in AUTO s output files fort 7 fort 8 and fort 9 or b s and d and in AUTO s constant files For example the command loadbd ab returns an object corresponding to the files b ab s ab and d ab if you are using the standard filename translations from Table 4 3 The command run ab returns an object corresponding to the output of the run The bifurcatio
296. toring certain inner products One of these is further continued in three equation parameters in the final run where a three parameter extremum is located AUTO COMMAND ACTION mkdir obv create an empty work directory cd obv change directory demo obv copy the demo files to the work directory ri run e obv c obv locate 1 parameter extrema as branch points save r1 obv save output files as b obv s obv d obv r2 run r1 BP1 ISW 1 NMX 5 compute a few step on the first bifurcating family save r2 1 save the output files as b 1 s 1 d 1 r3 run r2 ICP 10 1 2 17 18 13 14 151 locate 2 parameter extremum restart from ISW 1 NMX 100 r2 save r3 2 save the output files as b 2 s 2 d 2 r4 run r3 UZ2 locate 3 parameter extremum restart from ICP 10 1 2 3 18 13 14 15 NMX 25 r3 save r4 3 save the output files as b 3 s 3 d 3 Table 17 6 Commands for running demo obv 195 Chapter 18 AUTO Demos Connecting orbits 196 18 1 fsh A Saddle Node Connection This demo illustrates the computation of travelling wave front solutions to the Fisher equation Wt Wee f w o0o lt x lt oo t gt 0 f w w 1 w et We look for solutions of the form w x t u x ct where c is the wave speed This gives the first order system uy 2 ua z us 2 cualz fual Its fixed point 0 0 has two positive eigenvalues when c gt
297. ts with a py suffix For example the data generated for the Lyapunov family L1 will consist of b L1 the bifurcation diagram data s L1 a selection of periodic orbits d L1 diagnostic data including Floquet multipliers The necessary labeled starting solutions are first computed and stored in the file s start Each starting solution is an equilibrium libration point and its data also contains the period of a bifurcating family of periodic orbits The Table below shows the label of each of the starting solutions in s start indicating which libration point it corresponds to and which family of periodic orbits it will generate 153 Label Libration Pt Family 1 L1 L1 2 L1 v1 3 L2 L2 4 L2 V2 5 L3 L3 6 L3 V3 7 L4 V4 8 L5 V5 Note by looking at the constant files c r3b that actually only the starting solutions labeled 1 and 2 are used in the current calculations as executed by the Python script Starting solution for other values of u can be generated using the script compute_lps py for instance by running autox compute_lps py 0 05 import compute_lps compute_lps compute 0 05 After that it is necessary to run r3b auto again to regenerate the families The demos Lla Hla H1b Hic Vla V1b can be run subsequent to the r3b demo to compute 2D unstable manifolds of selected periodic orbits that belong to the L1 V1 and H1 families 14 15 2 Computing Unstable Manifolds of Periodic Orbits in the
298. u helps to complete this change This menu includes two items Solution and Bifurcation There is a marker beside the current diagram For example if the current diagram is the solution diagram but we want to change to the bifurcation diagram we can do so by clicking Type Bifurcation to switch to the bifurcation diagram Line Tube Surface 2 Solution Mesh Points Bifurcation All Points Figure 8 1 The Type Menu Figure 8 2 The Style Menu 8 1 3 Changing the Style PLAUTO4 provides four ways to draw the graphics 1 e using curves tubes points or as a surface One can select the style from the Style menu The Style menu is shown in Figure 8 1 4 Coordinate axes Figure 8 3 shows the selections of the Coord menu One may use this menu to select to show or not to show coordinate axes and the type of coordinate axes in the graphics 8 1 5 Options The Options Menu provides functions to add or remove widgets from the graphics It also allows to start stop solution or orbit animation The normalize data normalizes the raw data to 0 1 Preference lets us set preferences for the GUI see Figure 8 4 8 1 6 CR3BP animation The Center Menu allows to animate the motion of the three bodies in different coordinate systems We can animate the motion in a large primary centered inertial coordinate system or in a small primary centered inertial system or in the bary c
299. ulation in ph1 f90 is therefore S T so s an pR s a gt a Tla ay a pR s a prin with boundary conditions s 0 s 1 0 a 0 a 1 0 Note that the AUTO parameter PAR 9 defined in the subroutine PVLS in ph1 f90 is used to monitor the value of s 0 Since there are two constraints the third run requires only one free parameter namely T PAR 11 Note that to numerical accuracy T does not change during this run Alternatively one can use the free parameter p PAR 4 in the third run In this case to numerical accuracy p does not change during the run The third run terminates when PAR 9 reached the value 30 as specified in the equations file c ph1 14 11 144 14 9 pp3 Periodic Families and Loci of Hopf Points This demo illustrates the computation of stationary solution families that contain Hopf bifurca tions and the computation of the emanating families of periodic solutions In this example the periodic solution families intersect at a secondary bifurcation point a branch point It it also shown how to compute a locus of Hopf bifurcation points in two parameters In this example the locus contains branch points which lead to another locus The equations which model a 3D predator prey system with harvesting Doedel 1984 are ui uy 1 u1 pauu Uy P2U2 Pau Ug PsUguz pi 1 e Pot2 14 12 Us P3U3z T P5sUuguz The free parameter is p while the other
300. un san ICP 4 8 UZR 4 1 restart and homotopy to PAR 4 1 0 r7 run r6 ICP 5 8 UZR 5 0 DS homotopy to PAR 5 0 0 r8 run r7 ICP 1 8 UZR 1 0 5 D5 gt gt homotopy to PAR 1 0 5 r9 run r8 ICP 2 8 UZR 2 3 0 homotopy to PAR 2 3 0 r10 run r9 ICP 7 8 UZR 7 0 25 homotopy to PAR 7 0 25 rii run r10 ICP 7 8 31 IPSI 11 UZR 31 0 0 7 0 5 DS gt gt save r11 11 continue in PAR 7 to detect orbit flip save output files as b 11 s 11 d 11 of r11 UZ1 r12 run of ICP 7 8 6 IPSI NPR 5 NMX 20 IFIXED 11 UZR DS save r1i2 12 select first UZ labelled point of r11 to start from three parameter continuation of orbit flip save output files as b 12 s 12 d 12 Table 21 2 Detailed AUTO Commands for running demo san u6 0 aa 1 e R Yo SS 230 gt Se 4 gt 5 7 N 65 5 0 2 5 0 0 2 5 0 7 5 Figure 21 1 6 and 8 218 10 0 u5 Second versus third component of the solution to the adjoint equation at labels 4 Figure 21 2 Orbits on either side of the orbit flip bifurcation The critical orbit is contained in the x y plane 219 Chapter 22 HomCont Demo mtn 22 1 A Predator Prey Model with Immigration Consider the following system of two equations Scheffer
301. undary conditions we always keep the starting point v 0 A 0 s 0 on S and s 1 fixed to 0 2797 The steps are as follows 1 Homotopy step 1 grow the orbit segment in 7 continuing also in v 0 A 0 and s 0 where the starting point is kept on the folded node F until s 0 0 6 2 Homotopy step 2 the extra boundary condition for F is dropped and we now instead fix s 0 0 6 We continue in v 0 h 0 and T until h 0 6 0 3 Actual computation we continue in v 0 s 0 and T fixing h 0 6 0 The end point coordinates v 1 and h 1 are monitored so they can be matched with starting points of the repelling manifold These matches were found manually and are now indicated at specific values of T as UZ labels For the repelling manifold v h s we always keep the end point v 1 h 1 s 1 on S and s 0 fixed to 0 2797 The steps are now as follows 1 Homotopy step 1 grow the orbit segment in 7 until s 1 0 05 178 2 Homotopy step 2 the extra boundary condition for F is dropped and we now instead fix s 1 0 05 We continue until v 1 0 3 Actual computation we continue in s 0 and T fixing v 1 0 The starting point coordinates v 0 and h 0 are now monitored for matches with the attracting manifold The demo contains one folder attr for the attracting slow manifold and one folder rep for the repelling slow manifold In each of th
302. usp copy the demo files to the work directory Table 12 1 Copying the demo cusp files Typing 1s reveals the existence of 5 files 1 cusp f90 This file contains the differential equations and the initial values If you inspect it you will see that only two routines are used The subroutine FUNC specifies the actual differential equation The routine STPNT gives AUTO the initial values of PAR 1 A and PAR 2 u which are 1 0 and 0 0 and the initial value of x which is 0 For your own models you would generally copy another equation file and then only change the pieces that actually define the equation 120 2 c cusp The initial computational constants are stored in this file Most importantly you see that the dimension of the problem NDIM is set to 1 and the problem type IPS is set to 1 to specify continuation of a stationary solution The constants given by ICP specify the parameters that are used for the continuation In this case these are muw for u and lambda for A which correspond to the indices 2 and 1 respectively using the constant parnames Since initially we only really continue in one parameter u the second parameter A is overspecified Another important constant is the initial step size DS as it is positive we initally continue in the positive y direction 3 cusp auto A script with Python CLUI commands that steer the calculation 4 clean auto A script that cleans the directory of a
303. ution is colored in red When using time to color the diagram 0 is set to blue while 1 is set to red a Coloring by Type b Coloring by Label c Coloring by Time Figure 8 10 Coloring We can set the default value in the PLAUTO4 resource file 8 1 12 Number of periods to be animated Generally only one period is animated when we animate the solution in the inertial frame However the SpinBox allows us to change the default value This is a specially designed function for the CR3BP It is useful when we animate the motion in the three bodies in the inertial frame 84 8 1 13 Changing the line tube thickness The Line Thickness spinbox allows us to increase or decrease the line tube thickness in the diagram The PLAUTO4 resource file also provides a way to change the default values of the line tube thickness 8 1 14 Changing the animation speed The Sat and Orbit scale bar allow us to change the animation speed Their Maximum and Minimum value can be set in the resource file 8 1 15 Changing the background picture A user can set the background with his favorite picture To do this a user should copy the picture to the directory SAUTO_DIR plaut04 widgets and then change the name of the file to background rgb 8 2 Setting up the resource file The PLAUTO4 resource file sets default values for almost all controls of PLAUTO4 PLAUTO4 al lows us to write our own resources files an
304. utput BR PT TY LAB PAR 1 L2 NORM MAX U 1 1 1 EP 1 1 60000E 00 3 69766E 01 3 83942E 01 1 7 UZ 2 1 70000E 00 3 83640E 01 4 89066E 01 244 0 00 0 20 0 40 0 60 0 80 1 00 0 10 0 30 0 50 0 70 0 90 x T Figure 26 1 R Reversible homoclinic solutions on the half interval x T 0 1 where T 39 0448429 for P approaching 2 solutions with labels 1 5 respectively have decreasing amplitude 0 00 0 20 0 40 0 60 0 80 1 00 0 10 0 30 0 50 0 70 0 90 x T Figure 26 2 R reversible homoclinic orbits with oscillatory decay as x oo corresponding to label 6 and monotone decay at label 10 245 1 9 LP 3 1 71157E 00 3 92475E 01 5 46080E 01 1 11 UZ 4 1 69884E 00 4 04207E 01 6 10428E 01 1 14 UZ 5 1 60000E 00 4 32940E 01 7 77395E 01 1 26 UZ 6 1 00000E 00 4 80849E 01 1 08252E 00 1 49 UZ 7 b 38706E 08 5 15846E 01 1 25863E 00 1 128 MX 8 9 15462E 01 5 44202E 01 1 32395E 00 contains the label of a limit point ILP was set to 1 in c rev 3 which corresponds to a coales cence of two reversible homoclinic orbits The two solutions on either side of this limit point are displayed in Figure The computation ends in a no convergence point The solution here is depicted in Figure The lack of convergence is due to the large peak and trough of the solution rapidly moving to the left as P 2 cf Ch
305. where e g the file ab ab f90 defines a two dimensional dynamical system and the file exp exp f90 defines a boundary value problem The simplest way to create a new equations file is to copy an appro priate demo file For a fully documented equations file see auto 07p demos cusp cusp f90 or auto 07p gui aut f90 In GUI mode this file can be directly loaded with the GUI button Equations New see Section 9 2 The equations file can either be written in fixed form old style Fortran f free form For tran f90 or in C c 3 2 The Constants File c xxx AUTO constants for xxx f 90 c are normally expected in a corresponding file c xxx Spe cific examples include ab c ab and exp c exp in auto 07p demos See Chapter for the significance of each constant 22 3 3 User Supplied Routines The purpose of each of the user supplied routines in the file xxx 90 f is described below FUNC defines the function f u p in 2 1 2 2 or 2 3 STPNT This routine is called only if IRS 0 see Section 10 8 10 for IRS which typically is the case for the first run or when a system is manually extended A system is extended if NDIM see Section 10 2 1 increases between runs It defines a starting solution u p of or 2 2 The starting solution should not be a branch point Demos ab exp frc lor It extends an existing solution into higher dimensions in the demos p2c and c2c BCND A routine BCND that defines the bou
306. with boundary conditions u 0 t u 1 t A and v 0 t v 1 t B A All parameters are given fixed values for which a stable periodic solution is known to exist The continuation parameter is the independent time variable namely PAR 14 The AUTO constants DS DSMIN and DSMAX then control the step size in space time here consisting of PAR 14 and u x v x Initial data at time zero are u x A 0 5sin rx and v x B A 0 7sin za Note that in the subroutine STPNT the space derivatives of u and v must also be provided see the equations file bru f90 Euler time integration is only first order accurate so that the time step must be sufficiently small to ensure correct results This option has been added only as a convenience and should generally be used only to locate stationary states Indeed in the case of the asymptotic periodic state of this demo the number of required steps is very large and use of a better time integrator is advisable AUTO COMMAND ACTION mkdir bru create an empty work directory cd bru change directory demo bru copy the demo files to the work directory ri run e bru c bru time integration save r1 bru save output files as b bru s bru d bru Table 16 6 Commands for running demo bru 187 Chapter 17 AUTO Demos Optimization 188 17 1 opt A Model Algebraic Optimization Problem This demo illustrates the method of successive continuation f
307. x but without the diagnostics in d xxx Aliases diagramandsolutionget commandParseDiagramAndSolutionFile ch Modify continuation constants Type ch xxx yyy to change the constant xxx to have value yyy This is equivalent to the command s load s xxx yyy where s is a solution Aliases changeconstant cc commandRunnerConfigFort2 hch Modify HomCont continuation constants Type hch xxx yyy to change the HomCont constant xxx to have value yyy This is equivalent to the command s load s xxx yyy where s is a solution Aliases commandRunnerConfigFort12 pr Print continuation parameters Type pr to print all the parameters Type pr xxx to return the parameter xxx These commands are equivalent to the commands print s c print s c xxx where s is a solution Aliases pc pr printconstant commandRunnerPrintFort2 hpr Print HomCont continuation parameters Type hpr to print all the HomCont parameters Type hpr xxx to return the Hom Cont parameter xxx These commands are equivalent to the commands print s c print s c xxx where s is a solution Aliases commandRunnerPrintFort12 splabs Return special labels Type splabs xxx typename to get a list of labels with the specified typename where typename can be one of EP MX BP LP UZ HB PD TR or RG This is equivalent to the command load xxx typenam
308. x ux wy ar By sing 2 y x d 2 cosy 14 Fy y nyt wa ay Px sin p x y y f 2cos p va 15 16 py 1v2 s 1 4 y 2cos p cla y using Lin s method as described in Krauskopf and Rie8 2008 This system describes the dynamics near a saddle node Hopf bifurcation with global reinjection as discussed in Krauskopf and Oldeman 2006 We keep the following parameters fixed throughout w 1 a 1 0 6 0 s 1 c 0 d 0 01 and md Initially we fix va 1 46 since for that value of 12 close to 1 0 74 there exists a codimension one connection from a periodic orbit T to an equilibrium b Together with a codimension zero connection back to I it forms a heteroclinic cycle In this example the flow is such that the codimension one connecting orbit is an EtoP orbit where the flow is from the periodic orbit to the equilibrium A homoclinic orbit to b also approaches this cycle Below we compute all three connecting orbits The below sequences of calculations can be carried out by running the Python script snh auto included in the demo The individual connections can be computed by running the scripts hlb auto homoclinic orbit cb auto codimension one EtoP and tb auto codimension zero EtoP See the scripts and the Fortran file snh f90 for details on how all parameters are mapped and which precise AUTO constants are changed at every step 15 14 1 The homo
309. x of the free problem parameter as AUTO will automatically add PAR 11 However in this case the period will not appear in the screen output and in the fort 7 output file For fixed period orbits one must specify two free problem parameters For example in Run 7 of demo pp2 we have ICP 1 2 with PAR 1 and PAR 2 specified as free problem parameters The period PAR 11 is fixed in this run If the period is large then such a continuation provides a simple and effective method for computing a locus of homoclinic orbits 10 7 4 Folds and Hopf bifurcations The continuation of folds for algebraic problems and the continuation of Hopf bifurcations requires two free problem parameters For example to continue a fold in Run 3 of demo ab we have ICP 1 3 with PAR 1 and PAR 3 specified as free parameters Note that one must set ISW 2 for computing such loci of special solutions Also note that in the continuation of folds the principal continuation parameter must be the one with respect to which the fold was located 103 10 7 5 Folds and period doublings The continuation of folds for periodic orbits and rotations and the continuation of period doubling and torus bifurcations require two free problem parameters plus the free period Thus one would normally specify three parameters For example in Run 6 of demo pen where a locus of period doubling bifurcations is computed for rotations we have ICP 2 3 11 with PAR 2 PAR 3 and PAR
310. xtended system is fully programmed here in the user supplied routines in obv f90 For the case of periodic solutions the optimality system can be generated automatically see the demo ops Consider the system u t ux t u t Ayer u1 42 43 where p u1 Az Az u1 Agu Azu with boundary conditions The objective functional is w f o par The successive continuation equations are given by u t ux t un e AzePlrA2As wy t AeA walt 2y ur t 1 w t wilt where Puy oP 1 24941 4A3u Out with ui 0 0 w 0 B1 0 w 0 0 u 1 0 w1 1 T Ba 0 wa 1 0 1 1 3 w n 20 Sv dt 0 k 1 dl aan 0 Jr er uz Az 32 t al A1 dt 0 J dpeP 1 A238 211 wo t EYA2 Ta dt 0 e Jo Mi D a din 17 3 17 4 17 5 17 6 17 7 In the first run the free equation parameter is A All adjoint variables are zero Three extrema of the objective function are located These correspond to branch points and in the second run branch switching is done at one of these Along the bifurcating family the adjoint 194 variables become nonzero while state variables and A remain constant Any such non trivial solution point can be used for continuation in two equation parameters after fixing the L23 norm of one of the adjoint variables This is done in the third run Along the resulting family several two parameter extrema are located by mono
311. y run enz save enz compute stationary solution families save output files as b enz s enz d enz Table 13 1 Python commands for running demo enz AUTO COMMAND ACTION mkdir enz create an empty work directory cd enz change directory dm enz copy the demo files to the work directory r enz compute stationary solution families sv enz save output files as b enz s enz d enz Table 13 2 Shell commands for running demo enz 127 13 2 dd2 Fixed Points of a Discrete Dynamical System This demo illustrates the computation of a solution family and its bifurcating families for a discrete dynamical system Also illustrated are the continuation of period doubling bifurcations and branch switching at such points The equations a discrete predator prey system are ujt piu 1 ul T3 Paurus 13 2 k l __ k k k S us 1 p3 u3 paujus In the first third and fourth run p is free In the second run both p and po are free The remaining equation parameter p3 is fixed in both runs AUTO COMMAND ACTION mkdir dd2 create an empty work directory cd dd2 change directory demo dd2 copy the demo files to the work directory ri run e dd2 c dd2 lst run fixed point solution branches save dd2 save output files as b dd2 s dd2 d dd2 run ri PD1 ICP p1 p2 ISW 2 2nd run a locus of period doubling bifur catio
312. y to list delete and relabel solutions in the data files b xxx and s xxx The modified files are written as b yyy and s yyy Type CLB or LB on case insensitive file systems is equivalent to 1b above but moves instead of copies files so that it is much quicker but interrupts may be harmful Type fc xxx to convert a user supplied data file xxx dat to AUTO format The converted file is called s dat The original file is left unchanged AUTO automatically sets the period in PAR 11 Other parameter values must be set in STPNT When necessary PAR 11 may also be redefined there The constants file file c xxx must be present as the AUTO constants NTST and NCOL Sections and are used to define the new mesh Note this technique has been obsoleted by the dat AUTO constant in Section 10 8 7 Type db to double the solution in fort 8 Type db xxx to double the solution in s xxx Type Ctr to triple the solution in fort 8 Type tr xxx to triple the solution in s xxx 70 Cdlb Ck1b Cdsp ksp Cdlp Ckbp klp kuz rd RD mb Type Cdlb to delete all special labels in fort 7 and fort 8 Backups are made Type dlb xxx to delete all special labels in b xxx and s xxx Backups are made Type dlb xxx yyy to delete all special labels in b xxx and s xxx The output is written to b yyy and s yyy Optionally give an argument of the form UZ HB LP EP PD TR BP MX or RG to remove all labels with
313. y varying in e and T similar to the method described before but only if the unstable manifold in one dimensional Because this method is more limited than the method using Lin vectors we do not recommend it for normal usage To switch back to a normal homoclinic orbit set ISTART back to a positive value such as 1 Now HomCont has lost all the information about the adjoint so if ITWIST is set to 0 HomCont does a normal continuation without the adjoint and if ITWIST is set to 1 one needs to do a Newton dummy step first to recalculate the adhoint 20 3 6 IREV If IREVA then it is assumed that the system is reversible under the transformation t t and U i U i for all i with IREV i gt 0 Then only half the homoclinic solution is solved for with right hand boundary conditions specifying that the solution is symmetric under the reversibility see Champneys amp Spence 1993 The number of free parameters is then reduced by one Otherwise the default applies IREV 20 3 7 IFIXED Labels of test functions that are held fixed E g with IFIXED n one can compute a locus in one extra parameter of a singularity defined by test function PSI n 0 The default is IFIXED 20 3 8 IPSI Labels of activated test functions for detecting homoclinic bifurcations see Section for a list If a test function is activated then the corresponding parameter IPSI 1 20 must be added to the list of continuation parameters ICP and zero of
314. you can also click on the floppy disk icon to save using a variety of file formats Further information on the plotting tool can be found in Section 4 11 AUTO COMMAND ACTION plot mu run AUTO to graph the contents of mu plot mu run AUTO to graph the contents of b mu and s mu this is the same as plot loadbd mu Table 12 5 Commands for plotting or the bifurcation diagram and solutions of the Python variable mu and the files b mu and s mu 12 6 Plotting the Results with AUTO in 3D Whilst not very useful for this simple example you can also plot your results in 3D using the plot3 command PLAUT04 for example plot3 mu Unlike the PyPLAUT tool by default this shows the stable and unstable parts blue is stable and red is unstable You can also spin the bifurcation diagram around and zoom in using the mouse 12 7 Exporting the Results for different plotters It is often useful to use other plotting programs or general purpose tools to work with AUTO s data The writeRawFilename method see also Section can be used for this In this tutorial we can for instance export the bifurcation diagram using cusp writeRawFilename cusp dat and then use the command plot cusp dat using 1 4 wi li to plot the bifurcation diagram in GNUPlot 124 2 0 T T A O A Figure 12 1 The first bifurcation diagram of demo cusp lambda Figure 12 2 The second bifurcatio
315. yping auto xxx auto will then automatically execute all runs of the demo At each step the user is encouraged to plot the data saved by using the command plot e g plot r plots the data saved in the Python variable r Of course in a real application the runs will not have been prepared in advance and AUTO commands must be used Such commands can be found in a table at the end of each chapter A sequence of detailed AUTO commands will be given in these tables as illustrated in Table and Table 20 2 for two representative runs of HomCont demo san The user is encouraged to copy the format of one of these demos when constructing new examples The output of the HomCont demos reproduced in the following chapters is somewhat machine dependent as already noted in Section In exceptional circumstances AUTO may reach its maximum number of steps NMX before a certain output point or the label of an output point may change In such case the user may have to make appropriate changes in the AUTO constants files 212 COMMAND ACTION san load san IPS 9 NDIM 3 ISP 0 ILP 0 ITNW 7 JAC 1 NTST 35 IEQUIB 0 DS 0 05 set common AUTO constants ri run san ICP 1 8 UZR 1 0 25 run AUTO using the specified constants save r1 6 save output files as b 6 s 6 d 6 CR san 1 use the constants file c san 1 Osv 6 Table 20 1 These two sets of AUTO Commands are equivalent COMMAND ACTION r9 run r8 I

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