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AUTO 2000 : - Division of Applied Mathematics

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1. Figure 18 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 18 4 Three dimensional blow up of the solution curves t at labels 1 dotted and 2 solid line from Figure 3 8 162 1 000 l 030 1 60 10 0 5 99 0 0 970 0 99 0 0 930 1 020 1 000 0 980 0960 0 940 x Figure 18 5 Computed homoclinic orbits approaching the BT point flip 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 18 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 make fifth we approach a Bogdanov Takens point BR PT TY LAB PAR 1 sia PAR 10 ES PAR 33 1 50 EP 13 1 938276E 00 7 523344E 00 6 310810E 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 18 5 Note how the computed homoclinic orbi
2. Or Run Osv Save Cap Append Cp Plot Ocp Files Copy Cmv Files Move cl Files Clean Gdl Files Delete dm Equations Demo Table C 1 Command Mode GUI correspondences The AUTO command Or xxx yyy is given in the GUI as follows click Files Restart and enter yyy as data Then click Run As noted in Section A this will run AUTO with the current equations file xxx c 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 C 4 Customizing the GUI C 4 1 Print button The Misc Print button on the Menu Bar can be customized by editing the file GuiConsts h in directory auto 2000 include 206 C 4 2 GUI colors GUI colors can be customized by creating an X resource file Two model files can be found in directory auto 2000 gui namely Xdefaults 1 and Xdefaults 2 To become effective edit 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 lib X11 rgb txt C 4 3 On line help The file auto 2000 include GuiGlobal h contains on line help on AUTO constants and demos The text can be updated 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 XN110240 0 in auto 2000 gui Makefile by CC cc Wf XN120480 0 On other mach
3. AUTO COMMAND ACTION mkdir lin create an empty work directory cd lin change directory demo lin copy the demo files to the work directory run c lin 1 1st run compute the trivial solution branch and locate eigenvalues sv lin save output files as b lin s lin d lin run c lin 2 s lin 2nd run compute a few steps along the bifurcating branch Constants changed IRS ISW DSMAX ap lin append output files to b lin s lin d lin run c lin 3 s lin 3rd run compute a two parameter curve of eigenvalues Constants changed IRS ISW ICP 2 sv 2p save the output files as b 2p s 2p d 2p Table 10 4 Commands for running demo lin 108 10 5 non A Non Autonomous BVP This demo illustrates the continuation of solutions to the non autonomous boundary value prob lem 10 5 with boundary conditions u1 0 0 wi 1 0 Here z is the independent variable This system is first converted to the following equivalent autonomous system Uy U2 u pe 10 6 uz 1 with boundary conditions u 0 0 w 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 run c non 1 compute the solution branch sv non save output files as b non s non d non Table
4. 1 00 0 10 0 30 0 30 0 70 0 90 x T Figure 21 2 R reversible homoclinic orbits with oscillatory decay as x oo corresponding to label 6 and monotone decay at label 10 179 1 19 UZ 4 1 600000E 00 4 329404E 01 7 769491E 01 1 31 UZ 5 1 000000E 00 4 808488E 01 1 083298E 00 1 86 UZ 6 9 664802E 10 5 158463E 01 1 258650E 00 contains the label of a limit point ILP was set to 1 in c rev 3 which corresponds to a coa lescence of two reversible homoclinic orbits The two solutions on either side of this limit point are displayed in Figure 21 3 The computation ends in a no convergence point The solution here is depicted in Figure 21 4 The lack of convergence is due to the large peak and trough of the solution rapidly moving to the left as P 2 cf Champneys amp Spence 1993 Continuing from the initial solution in the other parameter direction make fourth we obtain the output BR PT TY LAB PAR 1 L2 NORM MAX U 1 Aa 1 7 UZ 8 1 600000E 00 3 701709E 01 3 836833E 01 1 33 UZ 9 9 999980E 01 3 614405E 01 1 775035E 01 1 93 UZ 10 7 819855E 06 3 713007E 01 4 698309E 02 which again ends at a no convergence error for similar reasons 180 Oy 25 0 3 9 0 O FS A 00 0 00 0 20 0 40 Figure 21 3 Two R reversible homoclinic orbits at P 1 6 corresponding to labels 1 smaller amplitude and 5 larger amplitude 1
5. 2y z 18 1 2 ely z with e 0 1 and es 1 Koper 1995 To copy across the demo kpr and compile we type dm kpr 18 2 The Primary Branch of Homoclinics First we locate a homoclinic orbit using the homotopy method The file kpr c already con tains approximate parameter values for a homoclinic orbit namely PAR 1 1 851185 k PAR 2 0 15 The files c kpr 1 and h kpr 1 specify 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 make first Among the output there is the line BR PT TY LAB PERIOD L2 NORM ee PAR 17 1 29 UZ 2 1 900184E 01 1 693817E 00 4 433433E 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 18 1 it can be noted that although the right hand projection boundary condition is satisfied the solution is still quite away from the equilibrium 159 1 000 1 0 0 1 000 0 980 0 960 0 940 1 50 100 0 990 0 970 0 930 x Figure 18 1 Projection on the x y plane of solutions of the boundary value problem with 2T 19 08778 1
6. Table 11 1 Commands for running demo pdl 114 11 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 0u 0t D 0 u 0x p u 1 u uta duo 0t Da Pua 01 us aqua 11 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 D PAR 16 1 The boundary conditions are u 0 u1 L 0 and u2 0 ue 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 ui x u2 z Initial data at time zero are u x sin 1x 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 c 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 and should generally be used only to locate stationary states AUTO COMMAND ACTION
7. Re A2 1 7 Three leading eigenvalues stable Re A1 Re u1 Re u2 1 8 Three leading eigenvalues unstable Re u1 Re A1 Re A2 1 9 Local bifurcation zero eigenvalue or Hopf number of stable eigenvalues decreases Re p1 0 i 10 Local bifurcation zero eigenvalue or Hopf number of unstable eigenvalues de creases Re A1 0 4 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 users may wish to add their own test functions by editing the function PSIHO in autlib5 c It is important to remember that in order to specify activated test functions it 1s 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 15 7 Starting Strategies There are four possible starting procedures for continuation i Data can be read from a previousl
8. 5 continue non orientable orbit restart from s 3 save output files as b 5 s 5 d 5 Table 16 1 Detailed AUTO Commands for running demo san 149 AUTO COMMAND ACTION run c san 6 h sv 6 gt san 6 s san restart and homotopy to PAR 4 1 0 save output files as b 6 s 6 d 6 run c san ap 6 7 h san 7 s 6 homotopy to PAR 5 0 0 restart from s 6 append output files to b 6 s 6 d 6 run c san ap 6 8 h san 8 s 6 homotopy to PAR 1 0 5 restart from s 6 append output files to b 6 s 6 d 6 run c san ap 6 9 h san 9 s 6 homotopy to PAR 2 3 0 restart from s 6 append output files to b 6 s 6 d 6 run c san ap 6 10 h san 10 s 6 homotopy to PAR 7 0 25 restart from s 6 append output files to b 6 s 6 d 6 run c san sv 11 11 h san 11 s 6 continue in PAR 7 to detect orbit flip restart from s 6 save output files as b 11 s 11 d 11 run c san 12 jh san 12 s 11 three parameter continuation of orbit flip restart from s 11 sv 12 save output files as b 12 5 12 d 12 Table 16 2 Detailed AUTO Commands for running demo san u6 I 0 A ee y i mS gt 25 By Le SS N A 5 J
9. ACTION clO dl ab dl 2p remove temporary files of demo ab remove ab data files of demo ab remove 2p data files of demo ab first run PT TY LAB 1 EP 33 LP 70 LP 90 HB 92 EP oP WN EF Saved as ab ab BR 4 4 4 4 4 Appended to ab ab BR 2 2 second run PT TY LAB 30 6 60 7 90 8 120 9 150 EP 10 third run PT TY LAB 27 LP 11 100 EP 12 Saved as 2p ab BR 2 ab BR 4 Appended to 2p fourth run PT TY LAB 35 EP fifth run PT TY LAB 100 EP 11 Table 7 3 Cleaning the demo ab work directory PAR 1 0 00000E 00 1 05739E 01 8 89318E 02 1 30899E 01 1 51241E 01 stationary solutions L2 NORM 0 00000E 00 1 48439E 00 3 28824E 00 4 27186E 00 4 36974E 00 periodic solutions PAR 1 1 19881E 01 1 15303E 01 1 05650E 01 1 05507E 01 1 05507E 01 PAR 1 1 35335E 01 L2 NORM 3 98712E 00 3 14630E 00 2 21917E 00 1 69684E 00 1 60388E 00 L2 NORM 2 06012E 00 U 1 0 00000E 00 3 11023E 01 6 88982E 01 8 95080E 01 9 15589E 01 MAX U 1 91911E 01 99577E 01 99166E 01 99086E 01 99789E 01 WO O O O O a 2 parameter locus of folds U 1 4 99653E 01 1 09381E 08 2 13650E 01 9 53147E 01 U 2 0 00000E 00 1 45144E 00 3 21525E 00 4 17704E 00 4 27275E 00 MAX U 2 02034E 00 95764E 00 36609E 00 29629E 00 28146E 00 oO O ON U 2 1 99861E 00 2 13437E 01 the l
10. Note that h x is not a smooth function 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 x i arctan Kzx 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 ld chu load the problem definition run c chu 1 1st run stationary solutions sv chu save output files as b chu s chu d chu run c chu 2 s chu 2nd run periodic solutions with detection of period doubling constants changed IPS IRS ICP ICP ap chu append the output files to b chu s chu d chu Table 9 11 Commands for running demo chu 102 9 12 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 o a 9 14 uy u l u This equation has a Hopf bifurcation from the trivial solution at A 0 The bifurcating branch of period
11. mkdir pd2 create an empty work directory cd pd2 change directory demo pd2 copy the demo files to the work directory run c pd2 1 time integration towards stationary state sv 1 save output files as b 1 s 1 d 1 run c pd2 2 s 1 continuation of stationary states read restart data from s 1 constants changed IPS IRS ICP etc sv 2 save output files as b 2 s 2 d 2 Table 11 2 Commands for running demo pd2 115 11 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 Ou Ot Ou Ox p paR ur uz pa ua OU2 Ot BO ug dx p p4R ux uz Pr p3 u2 All equation parameters except p3 are fixed throughout AUTO COMMAND ACTION mkdir wav cd wav demo wav create an empty work directory change directory copy the demo files to the work directory run c wav 1 sv ode lst run stationary solutions of the system without diffusion save output files as b ode s ode d ode Cp c wav 2 c wav run c wav 2 s wav sv wav constants changed IPS 2nd run detect bifurcations to wave train solutions Constants changed IPS save output files as b wav
12. sib 7 h sib hbs2 s 6 sv 7 186 BR PT TY LAB PAR 20 L2 NORM ses PAR 21 PAR 5 3 10 18 3 458968E 01 4 468176E 01 7 877123E 07 1 558861E 11 3 20 19 2 736992E 01 4 468176E 01 2 911187E 05 1 639739E 09 3 30 20 1 737196E 01 4 468171E 01 4 422734E 03 3 101671E 05 3 38 EP 21 1 014512E 01 4 467963E 01 2 000000E 01 1 486151E 02 The output 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 22 2 Next we are ready to close this gap by continuing in a u and e while keeping T at a constant value Columns 3 Columns 147 1 00 00 2 00e 01 0 00e 00 0 Figure 22 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 0 3 6 and the Y box must contain 1 4 7 b rn c sib 8 h sib hbs2 s 7 ap 6 BR PT TY LAB PAR 4 L2 NORM baad PAR 5 PAR 21 3 3 UZ 22 7 399999E 02 4 467807E 01 1 431624E 02 1 937464E 01 3 32 EP 23 1 992281E 01 4 465901E 01 6 054949E 03 2 292996E 06 The output is appended to b 6 s 6 and d 6 Now we have obtained a 2 homoclinic orbit at label 24 Ho
13. 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 u1 u2 and uz The correponding parameter values are defined in the user supplied subroutine stpnt The AUTO command us lor then converts the data in lor dat to a labeled AUTO solution with label 1 in a new file s 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 lor Note that the file s dat should be used for restart only Do not append new output files to s dat as the command us lor only creates s dat with no corresponding b dat AUTO COMMAND ACTION mkdir lor create an empty work directory cd lor change directory demo lor copy the demo files to the work directory 1ld lor load the problem definition us lor convert lor dat to AUTO format in s dat run c lor 1 s dat compute a solution branch restart from s dat sv lor save output files as b lor s lor d lor run c lor 2 s lor switch branches at a period doubling de tected in the first run Constants changed IRS ISW NTST ap lor append the output files to b lor s lor d lor Table 9 4 Commands for running demo lor 94 9 5 fre A Periodically Forced System T
14. Table 7 10 Command to run the relabeling program on b 2p and s 2p RELABELING COMMAND ACTION l list the labeled solutions in s 2p r relabel the solutions l list the new solution labeling wW rewrite b 2p and s 2p Table 7 11 Relabeling commands for the files b 2p and s 2p 7 10 Plotting the 2 Parameter Diagram To plot the files b 2p and s 2p enter the command listed in Table 7 12 The saved plot is shown in Figure 7 3 AUTO COMMAND ACTION plot 2p run to graph the contents of b 2p and s 2p Table 7 12 Command to plot the files b 2p and s 2p 86 Column 1 3 008 01 12 2 00e 01 11 1 00e 01 11 i 0 00e 00 1 008 01 1 008 01 P 1 396 17 a 2 008 01 Column 0 Figure 7 3 Loci of folds and Hopf bifurcations for demo ab 87 Chapter 8 AUTO Demos Fixed points 8 1 enz Stationary Solutions of an Enzyme Model The equations that model a two compartment enzyme system Kern vez 1980 are given by s so 81 s2 81 pR s1 8 1 8 so p 82 81 82 pR sa i where es 1 s H482 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 create an empty work directory cd enz change directory demo enz copy the demo files to the work directory ld enz load the problem definition
15. We can now continue the homoclinic plus adjoint in a A PAR 4 PAR 8 by changing the constants stored in c san 3 to read IRS 4 NMX 50 and ICP 1 4 We also add PAR 10 to the list of continuation parameters NICP ICP I I 1 NICP Here PAR 10 isa 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 three ways First we add PAR 21 and PAR 33 to the list of continuation parameters in c san 3 second we set up user defined output at zeros of these parameters in the same file and finally we set NPSI 2 IPSI 1 IPSI 2 1 13 in h san 3 We also add to c san 3 another user zero for detecting when PAR 4 1 0 Running make third reads starting data from s 2 and outputs to the screen BR PT TY LAB PAR 4 PAR 8 PAR 10 PAR 33 1 20 5 7 847219E 01 3 001440E 11 4 268884E 09 1 441124E 01 1 27 UZ 6 1 000000E 00 3 844872E 11 4 460769E 09 5 701675E 00 1 35 UZ 7 1 230857E 00 5 833977E 11 4 530541E 09 9 434843E 06 1 40 8 1 383969E 00 8 133899E 11 4 671817E 09 1 348810E 00 1 50 EP 9 1 695209E 00 1 386324E 10 5 098460E 09 5 311065E 01 Full output is stored in b 3 s 3and d 3 Note that the artificial parame
16. cuz B u u3 C u1 Alur 13 5 To remove the singularity when u 0 we apply a nonlinear transformation of the independent variable see Aronson 1980 viz d d2 A u d dz which changes the above equation into ui 2 A uz ua Sharp traveling waves then correspond to heteroclinic connections in this transformed system 131 Finally we map 0 7 0 1 by the transformation 2 T With this scaling of the independent variable the reduced system becomes E T cuz B ui u3 C us For Case 1 this equation has a known exact solution namely bs i OE 1 exp TE oe 1 exp T ug 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 A 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 run c stw 1 continuation of the sharp traveling wave sv stw save output files as b stw s stw d stw Table 13 4 Commands for running demo stw 132 Chapter 14 AUTO Demos Miscellane
17. which is stored in s 7 That PAR 29 19 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 sign of DS in c kpr 7 The files c kpr 9 and h kpr 9 contain the constants necessary for switching to continuation of the central saddle node homoclinic curve in two parameters starting from the non central saddle node homoclinic orbit stored as label 8 in s 7 make eighth 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 Pans PAR 2 PAR 35 PAR 36 1 38 UZ 13 1 765274E 01 2 405284E 00 9 705426E 03 5 464784E 07 which corresponds to the branch of homoclinic orbits leaving the locus of saddle nodes in a second non central saddle node homoclinic bifurcation a zero of 1 16 Note that the parameter values do not vary much between the two codimension two non central saddle node points labels 8 and 13 However Figure 18 6 shows clearly that between the two codimension two points the homoclinic orb
18. COMMAND ACTION mkdir pvl create an empty work directory cd pvl change directory demo pvl copy the demo files to the work directory run c pvl 1 compute a solution branch sv pvl save output files as b pvl s pvl d pvl Table 14 1 Commands for running demo pvi 134 14 2 ext Spurious Solutions to BVB This demo illustrates the computation of spurious solutions to the boundary value problem ui uz 0 uy Am sin u u u 0 t 0 1 14 2 1iy 0 0 ai 0 Here the differential equation is discretized using a fixed uniform mesh This results in spurious solutions that disappear when an adaptive mesh is used See the AUTO constant IAD in Section 5 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 run c ext 1 detect bifurcations from the trivial solution branch sv ext save output files as b ext s ext d ext run c ext 2 s ext compute a bifurcating branch containing spurious bifurcations Constants changed IRS ISW NUZR apC ext append output files to b ext s ext d ext Table 14 2 Commands for running demo ext 135 143 tim A Test Problem for Timing AUTO This demo is a boundary value problem with variable dimens
19. E J amp 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 Friedman M amp Monteiro A 1993 On locating homoclinic and heteroclinic orbits Technical 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 bifurca tion 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 bi furcation problems II Bifurcation in infinite dimensions Int J Bifurcation and Chaos 1 4 745 772 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 Numer Anal 28 No 5 1446 1462 FitzHugh R 1961 Impulses and physiological states in theoretical models of nerve membrane Biophys J 1 445 446 Freire E Rodriguez Luis A Ga
20. The first run make first 177 starts by copying the files rev c 1 and rev dat 1 to rev c and rev dat 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 h rev 1 via NREV 1 IREV I I 1 NDIM 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 7 UZ 2 1 700002E 00 2 633353E 01 4 179794E 01 1 12 UZ 3 1 800000E 00 2 682659E 01 4 806063E 01 1 15 UZ 4 1 900006E 00 2 493415E 01 4 429364E 01 1 20 EP 5 1 996247E 00 1 111306E 01 1 007111E 01 which is consistent with the theoretical 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 21 1 The second run continues in the other direction of PAR 1 with the test function 4 activated for the detection of saddle to saddle focus transition points make second The output BR PT TY LAB PAR 1 L2 NORM MAX U 1 PAR 22 1 11 UZ 6 1 000005E 00 2 555446E 01 1 767149E 01 3 000005E 00 1 22 UZ 7 1 198325E 07 2 625491E 01 4 697314E 02 2 000000E 00 1 33 UZ 8 1 000000E 00 2 741483E 01 4 3160
21. Use Ch instead of Cr when using HomCont i e when IPS 9 see Chapter 15 Type Ch xxx to run AUTO HomCont Restart data if needed 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 c 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 c restart data file s yyy and constants files c zzz and h zzz OH The command GH xxx is equivalent to the command Ch xxx above Type CH xxx iin order to run AUTO HomCont with equations file xxx c 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 c constants files c xxx i and h xxx i and restart data file s yyy A 0 8 Copying a demo dm Type dm xxx to copy all files from auto 2000 demos xxx to the current user directory Here xxx denotes a demo name e g abc Note that the dm command also copies a Makefile to the current user directory To avoid the overwriting of existing files always run demos in a clean work directory A 0 9 Pendula animation pn Type Cpn xxx to run the pendula animation program with data file s xxx On SGI machine only see demo pen in Section 9 10 and the file auto 2000 pendula README A 0 10 Viewing the manual mn Use Ghostview to view the PostScript version of
22. gt i 5g 6 5 0 2 5 0 0 2 5 5 0 7 5 10 0 u5 Figure 16 1 Second versus third component of the solution to the adjoint equation at labels 5 7 and 9 150 Figure 16 2 Orbits on either side of the orbit flip bifurcation The critical orbit is contained in the x y plane 151 Chapter 17 HomCont Demo mtn 17 1 A Predator Prey Model with Immigration Consider the following system of two equations Scheffer 1995 X X RX 1 sa DK EJ BEPA 17 1 Ai XY A2Z Y Y Ej D Y 5 Bak B2 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 Ag 1 0 Bo 0 5 D S 0 15 The parametric portrait of the system 17 1 on the Z K plane is presented in Figure 17 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 17 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 neg
23. 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 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 Start ing 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 IPS 15 Optimization of periodic solutions The integrand of the objective functional must be specified in the user supplied subroutine fopt Only PAR 1 9 should be used for problem parameters PAR 10 is the value of the objective functional PAR 11 the 71 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 m
24. lt 3 NDIM then the L norm of the IPLT NDIM th solution com ponent 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 subroutine pvls provides a second more general way of defining solution measures see Section 5 7 10 73 5 9 4 NUZR 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 NUZR 0 if no such output is needed Many demos use this setting If NUZR gt 0 then one must enter NUZR pairs Parameter Index Parameter Value with each pair on a separate line 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 terminate at such a solution point 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 5 7 10 74 Chapter 6 Notes on Using AUTO 6 1 Restrictions on the Use of PAR The array PAR in the user supplied subroutines is available
25. run c enz 1 compute stationary solution branches sv enz save output files as b enz s enz d enz Table 8 1 Commands for running demo enz 88 8 2 dd2 Fixed Points of a Discrete Dynamical System This demo illustrates the computation of a solution branch and its bifurcating branches for a discrete dynamical system Also illustrated is the continuation of Naimark Sacker or Hopf bifurcations The equations a discrete predator prey system are er put 1 Fi ut Pr putu 8 2 k l _ k kyk i Uz 1 p3 us paujus In the first run p is free In the second run both p and p 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 1d dd2 load the problem definition run c dd2 1 lst run fixed point solution branches sv dd2 save output files as b dd2 s dd2 d dd2 run c dd2 2 s dd2 2nd run a locus of Naimark Sacker bifur cations Constants changed IRS ISW sv ns save output files as b ns s ns d ns Table 8 2 Commands for running demo dd2 89 Chapter 9 AUTO Demos Periodic solutions 9 1 Irz The Lorenz Equations This demo computes two symmetric homoclinic orbits in the Lorenz equations ui p3 u2 uy US pil Ug 4143 9 1 Uz
26. s wav d wav run c wav 3 s wav ap wav 3rd run wave train solutions of fixed wave speed Constants changed IRS IPS NUZR ILP append output files to b wav s wav d wav run c wav 4 s wav sv rng 4th run wave train solutions of fixed wave length Constants changed IRS IPS NMX ICP NUZR save output files as b rng s rng d rng run c wav 5 s wav sv tim 5th run time evolution computation Con stants changed IPS NMX NPR ICP save output files as b tim s tim d tim Table 11 3 Commands for running demo wav 116 11 2 11 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 sae uz t x Here uz t corresponds to u x z t at the Chebyshev points fork with respect to the interval 0 1 The polynomials TAOL are the Lagrange interpolating coefficients with respect to points anthers where xo 0 and tay 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 inc As an illustrative application we consider the Brusselator Holodniok Knedlik amp Kub ek 1987 u D L Urs uu B 1 u A ve Dy
27. saddle node homoclinic orbits IEQUIB 2 or reversible systems IREV 1 140 Certain test functions are not valid for certain forms of continuation see Section 15 6 below for example PSI 13 and PSI 14 only make sense if ITWIST 1 and PSI 15 and PSI 16 only apply to IEQUIB 2 15 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 the artificial parameters of the homotopy method see Section 15 7 below PAR 21 PAR 36 These parameters are used for storing the test functions see Sec tion 15 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
28. 10 5 Commands for running demo non 109 10 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 i Tug u Tus uz T 2yu4 u3 2uus uz 10 7 de Tus us T 2yu2 24944 2u1u5 with left boundary conditions and asymptotic right boundary conditions foo alfo 1 ua 1 ua 1 7 2 0 al foos 1 Sees u2 1 foo al foo 7 ua 1 us 1 0 10 8 ul 1 fo where 1 2 a feos PLUS 477 fo 10 9 1 2 b foos 7 KL 477 Ae Note that there are five differential equations and six boundary conditions Correspondingly there are two free parameters in the computation of a solution branch namely y and fx 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 run c kar 1 computation of the solution branch sv kar save output files as b kar s kar d kar Table 10 6 Commands for running demo kar 110 10 7 spb A Singularly Perturbed BVP This demo illustrates the use of continuation to compute solutions to the singularly perturbed boundary value probl
29. 19 3 134237E 00 5 614124E 07 3 778288E 00 3 398845E 04 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 T3 PAR 22 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 23 e the parameter A PAR 2 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 rn c kdv 7 h kdv 7 s 6 sv 7 BR PT TY LAB PAR 1 L2 NORM PAR 2 1 2 EP 20 2 336952E 00 7 505830E 00 2 374866E 07 This 3 homoclinic orbit is depicted in Figure 22 6 Columns 0 1 006 01 8 00e 00 6 00e 00 4 00e 00 2 00e 00 Figure 22 6 A 3 homoclinic orbit in a 5th order Hamiltonian Korteweg De Vries model 195 Appendix A Running AUTO using Command Mode AUTO can be run with the interface described in Chapter 4 or with the commands described below The AUTO aliases must have been activated see Section 1 2 and an equations file xxx c and a corresponding constants file c xxx see Section 3 1 must be in the current user directory Do not run AUTO in the directory auto 2000 or in any of its subdire
30. 2 run c kpr 3 h kpr 3 s 2 generate adjoint variables restart from s 2 sv 3 save output files as b 3 s 3 d 3 run c kpr 4 h kpr 4 s 3 continue the homoclinic orbit restart from s 3 ap 3 append output files to b 3 s 3 d 3 run c kpr 5 h kpr 5 s 3 continue in reverse direction restart from s 3 ap 3 append output files to b 3 s 3 d 3 run c kpr 6 h kpr 6 s 2 increase the period restart from s 2 save output files as b 6 s 6 d 6 Table 18 1 Detailed AUTO Commands for running demo kpr 167 Sk Ts 9 eps_1 Figure 18 8 Projection onto the PAR 3 PAR 2 plane of the non central saddle node homo clinic orbit curves labeled 1and 2 and the inclination flip curves labeled 3 and 4 AUTO COMMAND ACTION run c kpr 7 h kpr 7 s gt 6 sv 7 recompute the branch of homoclinic orbits restart from s 6 save output files as b 7 s 7 d 7 run c kpr 8 h kpr 8 s 7 sv 8 continue central saddle node homoclinics restart from s 7 save output files as b 8 s 8 d 8 run c kpr 9 h kpr 9 s gt 8 sv 9 continue homoclinics from codim 2 point restart from s 8 save output files as b 9 s 9 d 9 run c kpr 10 h kpr 10 s 3 sv 10 3 para
31. 2000 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 2000 CLUI name for the command while the left hand side is the desired 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 13 NOTE Defaults for the plotting tool may be included in the autorc file as well The docu mentation for this is under developement but the file AUTO_DIR autorc contains examples of how these options may be set 4 10 Two Dimensional Plotting Tool The two dimensional plotting tool can be run by using the command plot to plot the files fort 7 and fort 8 after a calculation has been run 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 plotter The Options button allows the plotter configuration options to be modified The available options are decribed in Table 4 7 In addition the options can be set from within the CLUI For example the set of commands in Figure 4 15 shows how to create a plotter and change its b
32. 5 9 2 IID This constant controls the amount of diagnostic output printed in fort 9 the greater IID the more detailed the diagnostic output TID 0 Minimal diagnostic output This setting is not recommended TID 2 Regular diagnostic output This is the recommended value of IID 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 every step and for every Newton iteration and should normally be suppressed In particular it can be used to verify whether the starting solution is indeed a solution For this purpose 72 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
33. 781716E 01 6 553641E 03 9 502999E 00 7 203666E 02 The output is appended to b 6 s 6 and d 6 Note that we have found two zeros of PAR 23 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 bifur cation These two 3 homoclinic orbits are depicted in Figure 22 3 b We can follow both of these back to the inclination flip point by setting ITWIST back to 0 rn c sib 13 h sib hom s 6 ap 6 BR PT TY LAB PAR 4 L2 NORM Sue PAR 5 3 13 UZ 36 1 299993E 01 5 048071E 01 2 339037E 03 3 30 EP 37 9 272363E 02 5 065599E 01 2 767140E 04 rn c sib 14 h sib hom s 6 ap 6 BR PT TY LAB PAR 4 L2 NORM od PAR 5 3 4 UZ 37 1 449997E 01 5 473471E 01 4 794005E 03 3 30 EP 39 8 394009E 02 5 526047E 01 7 367526E 05 All the output is appended 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 22 4 It is possible and straightforward to obtain 4 5 6 homoclinic orbits by extending the above strategy 189 2 00e 02 Figure 22 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
34. 8 PAR 31 1 5 UZ 12 2 394737E 07 6 434492E 08 4 133994E 06 at which parameter value the homoclinic orbit is contained in the x y plane see Fig 16 2 Finally we demonstrate that the orbit flip can be continued as three parameters PAR 6 PAR 7 PAR 8 are varied make twelfth BR PT TY LAB PAR 7 Bod PAR 8 PAR 6 1 5 14 5 374538E 19 1 831991E 10 3 250000E 01 1 10 15 6 145911E 19 2 628607E 10 8 250001E 01 1 15 16 4 947133E 19 2 361151E 10 1 325000E 00 1 20 EP 17 5 792940E 19 3 075527E 10 1 825000E 00 The orbit flip continues to be defined by a planar homoclinic orbit at PAR 7 PAR 8 0 148 16 5 Detailed AUTO Commands AUTO COMMAND ACTION mkdir san cd san demo san create an empty work directory change directory copy the demo files to the work directory run c san 1 h san 1 sv 1 continuation in PAR 1 save output files as b 1 s 1 d 1 run c san 2 h san 2 s 1 sv 2 generate adjoint variables restart from s 1 save output files as b 2 s 2 d 2 run c san 3 h san 3 s 2 sv 3 continue homoclinic orbit and adjoint restart from s 2 save output files as b 3 s 3 d 3 run c san 4 h san 4 s 1 sv 4 no convergence without dummy step restart from s 1 save output files as b 4 s 4 d 4 run c san 5 h san 5 s 3 sv
35. BR PT TY LAB 1 10 11 1 20 12 1 30 13 1 40 EP 14 PAR 1 7 114523E 00 9 176810E 00 1 210834E 01 1 503788E 01 PAR 2 7 081751E 02 7 678731E 02 8 543468E 02 9 428036E 02 PAR 29 4 649861E 01 4 684912E 01 4 718871E 01 4 743794E 01 PAR 30 3 183429E 03 1 609294E 02 3 069638E 02 4 144558E 02 The fact that PAR 29 and PAR 30 do not change sign indicates that there are no further non hyperbolic equilibria along this branch Note that restarting in the opposite direction with IRS 11 DS 0 02 make fourth will detect the same codim 2 point D but now as a zero of the test function W409 BR PT TY LAB 1 10 UZ PAR 1 15 6 610459E 00 PAR 2 PAR 29 6 932482E 02 4 636603E 01 PAR 30 1 725013E 09 Note that the values of PAR 1 and PAR 2 differ from that at label 4 only in the sixth significant figure Actually the program runs further and eventually computes the point D and the whole lower branch of P emanating from it however the solutions between D and Dj should be considered as spurious therefore we do not save these data The reliable way to compute the lower branch of P is to restart computation of saddle homoclinic orbits in the other direction from the point D make fifth This gives the lower branch of P approaching the BT point see Figure 17 1 1 The program actually computes the saddle saddle heteroclinic orbit bifurcating from the non central saddle node ho
36. For proper interpretations 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 IID 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 5 9 3 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 L2 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 as 1 NDIM it E nda 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 5 5 5 and Section 5 5 6 affect the definition of the L2 norm If0 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 If 2 NDIM lt IPLT
37. L vrs vv 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 11 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 run c bre 1 compute the stationary solution branch with Hopf bifurcations sv brc save output files as b brc s brc d brc run c bre 2 s brce compute a branch of periodic solutions from the first Hopf point Constants changed IRS IPS ap brc append the output files to b brc s brc d bre run c brc 3 s brc compute a solution branch from a sec ondary periodic bifurcation Constants changed IRS ISW ap brc append the output files to b brc s brc d bre Table 11 4 Commands for running demo brce 117 11 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 s
38. More information on the CLUI may be found in Chapter 4 The new CLUI does not require any additional options to be passed to the configure script To run the new Command Line User Interface CLUI and the old command language you need to set your environment variables correctly Assuming AUTO 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 appropri ate If you are using a sh compatible shell such as sh bash ksh or ash enter the command source HOME auto 2000 cmds auto env sh On the other hand if you are using a csh com patible shell such as csh or tcsh enter the command source HOME auto 2000 cmds auto env csh There is an old and unsupported Graphical User Interface GUI which requires the X Window system and Motif and it is not compiled by default Note that AUTO can be very effectively run in Command Mode i e the GUI is not strictly necessary To compile AUTO with the old GUI type configure enable gui and then make in directory auto 2000 The PostScript conversion command ps will be enabled if the configure script detects the appropriate software but you may have to enter the correct printer name in auto 2000 cmds pr To generate the on line manual type make in auto 2000 doc To prepare AUTO for transfer to another machine type make superclean in directory auto 2000 befo
39. NINT 100 0 0 15 0 100 NMX RLO RL1 A0 A1 100 1028530 NPR MXBF IID ITMX ITNW NWTN JAC 1 e 6 1 e 6 0 0001 EPSL EPSU EPSS 0 01 0 005 0 05 1 DS DSMIN DSMAX IADS 1 NTHL 1 THL 1 1 1 NTHL 11 0 0 NTHU 1 THU I I 1 NTHU 0 NUZR 1 UZR 1I 1I 1 NUZR The significance of the AUTO constants grouped by function is described in the sections below Representative demos that illustrate use of the AUTO constants are also mentioned 5 2 Problem Constants 5 2 1 NDIM Dimension of the system of equations as specified in the user supplied subroutine func 5 2 2 NBC The number of boundary conditions as specified in the user supplied subroutine bend Demos exp kar 61 5 2 3 NINT The number of integral conditions as specified in the user supplied subroutine icnd Demos int lin obv 5 2 4 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 subroutines func bend icnd and fopt where applicable This may be neces sary for sensitive problems It is also recommended for computations in which AUTO gen erates an extended system for example when ISW 2 Demos int dd2 obt plp ops For ISW see Section 5 8 3 5 3 Discretization Constants 5 3 1 NTST The number of mesh intervals u
40. Note also that only special labeled solution points are printed on the screen More detailed results are saved in the data files b ab s ab and d ab The second run traces out the branch of periodic solutions that emanates from the Hopf bifurcation The free parameters are p and the period The detailed results are appended to the existing data files b ab s ab and d ab In the third run one of the folds detected in the first run is followed in the two parameters p and p3 while p remains fixed The fourth run continues this branch in opposite direction Similarly in the fifth run the Hopf bifurcation located in the first run is followed in the two parameters p and p3 In this example this is done in one direction only The detailed results of these continuations are accumulated in the data files b 2p s 2p and d 2p 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 simply that arithmetic differences have accumulated from step to step possibly leading to different step size decisions One could now use the AUTO CLUI to graphically inspect the contents of the data files but we shall do this later However it may be useful to edit these files to view their contents Next reset the work directory by typing the command given in Table 7 3 80 ab BR hp AUTO COMMAND
41. Section 1 2 this command will start the AUTO 2000 CLUI interactive interpretor and provide you with the AUTO 2000 CLUI prompt 21 gt auto Initializing Python 1 5 2 1 Feb 1 2000 16 32 16 GCC egcs 2 91 66 19990314 Linux Copyright 1991 1995 Stichting Mathematisch Centrum Amsterdam AUTOInteractiveConsole AUTO gt Figure 4 1 Typing auto at the Unix shell prompt starts the AUTO 2000 CLUI In addition to the examples in the following sections there are several example scripts which can be found in auto 2000 demos python and are listed in Table 4 1 These scripts are fully annotated and provide good examples of how AUTO 2000 CLUI scripts are written The scripts in auto 2000 demos python n body are espcially lucid examples and preform various related parts of a calculation involving the gravitional N body problem Scripts which end in the suffix auto are called basic scripts and can be run by typing auto scriptname auto The scripts show in Section 4 3 and Section 4 5 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 4 6 See the README file in that directory for more information 4 3 First Example We begin with a simple example of the AUTO 2000 CLUI In this example we copy the ab demo from the AUTO 2000 installation directory and run it For more informa
42. 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 The equations are uv u1 1 ui puus f ay U Uy Potlg P4U1U2 psuzuz p 1 en Pouz _ Uz p3U3 T P5U2U3 Here P2 1 4 Pa 1 2 Pa 3 P5 3 pe 5 and p is the free parameter In the continuation of Hopf points the parameter p4 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 work directory 1d ppp load the problem definition run c ppp 1 compute stationary solutions detect Hopf bifurcations sv ppp save output files as b ppp s ppp d ppp run c ppp 2 s ppp compute a branch of periodic solutions Constants changed IPS IRS ICP ap ppp append the output files to b ppp s ppp d ppp run c ppp 3 s ppp compute Hopf bifurcation curves sv hb save the output files as b hb s hb d hb Table 9 6 Commands for running demo ppp 96 9 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 a fold on a branch of periodic solution in two parameters The calculation of a locus of Hopf bifurcations is also included The equations tha
43. 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 2000 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 2000 CLUI 23 Unix COMMAND ACTION auto start the AUTO 2000 CLUI AUTO 2000 CLUI COMMAND ACTION copy the demo files to the work directory load the filename ab c into memory load the contents of the file r ab 1 into memory copydemo ab load equation ab load constants ab 1 run run AUTO 2000 with the current set of files Table 4 2 Running the demo ab files gt auto Initializing Python 1 5 2 1 Feb egcs on linux i386 Copyright 1991 1995 Stichting Mathematisch Centrum Amsterdam AUTOInteractiveConsole AUTO gt copydemo ab Copying demo ab done AUTO gt load equation ab Runner configured AUTO gt load constants ab 1 Runner configured AUTO gt run gcc 0 DPTHREADS 0 I home amavisitors redrod src auto 2000 include c ab c gcc 0 ab o o ab exe home amavisitors redrod src auto 2000 1ib x o lpthread L home amavisitors redrod src auto 2000 1ib lauto_f2c lm Starting ab 1 2000 16 32 16 GCC egcs 2 91 66 19990314 Linux 1 1 EP 1 0 000000E 00 0 000000E 00 0 000000E 00 0 000000E 00 1 33 LP 2 1 0573
44. 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 This yields the output BR hhh 1 PT TY LAB 5 10 15 20 25 30 EP 7 QORUN e NNUU Ss dm she make first PAR 3 L2 NORM PAR 1 528332E 01 3 726787E 01 1 364973E 01 943370E 01 3 303798E 01 1 044119E 01 358942E 01 2 873213E 01 7 515570E 02 772726E 01 2 433403E 01 4 952636E 02 181955E 01 1 981358E 01 2 845849E 02 581633E 01 1 512340E 01 1 292975E 02 The last parameter used to store the equilibria PAR 21 is overlaped here with the first test function In this example it is harmless since the test functions are irrelevant for heteroclinic continuation 173 Alternatively for this problem there exists an analytic expression for the two equilibria This is specified in the subroutine pvls of she c Re running with IEQUIB 1 make second we obtain the output BR PT TY LAB PAR 3 L2 NORM PAR 1 1 5 2 4 432015E 01 3 657716E 01 1 310559E 01 1 10 3 3 723085E 01 3 142439E 01 9 300982E 02 1 15 4 3 008842E 01 2 611556E 01 5 933966E 02 1 20 5 2 286652E 01 2 062194E 01 3 179939E 02 1 25 6 1 555409E 01 1 491652E 01 1 239897E 02 1 30 EP 7 8 107462E 02 9 143108E 02 2 386616E 03 This output is similar to that above but note that it is obtained slightly more efficiently because the ex
45. ab execute the third run of demo ab Table 7 4 Selected runs of demo ab On the other hand one can use the CLUI to generate the constants file at runtime In the example below the constant file c ab 1 will be read in and the CLUI will be used to make the appropriate changes to perform the same calculation as in Table 7 6 AUTO COMMAND ACTION ld ab load the problem definition ab run c ab 1 execute the run which uses the constants in c ab 1 sv ab save the results of the run into the files b ab s ab and d ab cc IRS 2 start the new calculation from a solution with label 2 cc ICP 0 2 since we are following a locus of folds we require two free parameters cc ISP 0 turn off detection of branch points cc ISW 2 since we start at a fold the ISW parameter indicates we desire to compute a locus of such points cc DSMAX 0 5 increase the maximum allowed step size run s ab execute the third run of demo ab Table 7 5 Selected runs of demo ab 7 6 Using AUTO Commands Next with the commands in Table we execute the first two runs of demo ab again using commands similar Table that one would normally use in an actual application We still use 82 AUTO COMMAND ACTION O remove temporary files of any previous runs of the demo dl ab remove ab data files of any previous runs of the demo d1 2p remove 2p data files of any previous runs of the demo ld ab make
46. 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 4 4 AUTO gt load e ab Runner configured AUTO gt load c ab 1 Runner configured Figure 4 3 Loading two files individually AUTO gt load e ab c ab 1 Runner configured Figure 4 4 Loading two files at the same time 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 only required that ab c and c ab exist s ab and h ab are optional and if they do not exist no error message will be given 4 4 Scripting Section 4 3 showed commands being interactively entered at the AUTO 2000 CLUI prompt but since the AUTO 2000 CLUI is based on Python one has the ability to write scripts for performing 25 sequences of commands automatically A Python script is very similar to the interactive mode shown in Section 4 3 except that the commands are placed in a file and read all at once For example if the commands from Figure 4 2 where placed into the file demol auto in the format shown in Figure 4 5 then the commands could be run all at once by typing auto demol auto See Figure 4 6 for the full output copydemo ab load equati
47. add an additional parameter A that breaks the Hamiltonian structure in this system by introducing artificial friction Thus the actual system of equations that is used for continuation is A1 J VH 0 where x q1 q2 p1 p2 and J is the usual skew symmetric matrix in R4 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 nj It 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 1d kdv rn sv 1 193 BR PT TY LAB PAR O L2 NORM eo PAR 2 1 1 EP 1 3 000000E 00 5 565438E 00 0 000000E 00 1 2 EP 2 3 049592E 00 5 491407E 00 1 807155E 17 Here PAR 0 a PAR 1 b and PAR 2 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 rn c kdv 2 h kdv 2 s 1 sv 2 BR PT TY LAB PAR 1 L2 NORM os PAR 8 1 2 EP 3 2 765575E 00 5 491418E 00 6 250114E 04 We now need to move back to the orbit flip at a 3 rn c kdv 3 h kdv 3 s 2 sv 3 BR PT TY LAB PAR O L2 NORM pee PAR 2 1 14 UZ 5 3 000000E 00 5 476133E 00 1 483821E 09 Now
48. and torus bifurcations ISP 0 This setting disables the detection of branch points period doubling bifurcations and torus bifurcations and the computation of Floquet multipliers ISP 1 Branch points are detected for algebraic equations but not for periodic solutions and boundary value problems Period doubling bifurcations and torus bifurcations are not located either 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 6 4 ISP 3 Branch points will be detected but AUTO will not monitor the Floquet multipliers Period doubling and torus bifurcations will go undetected This option is useful for certain problems with non generic Floquet behavior The Floquet multipliers will be output to the diagnostic file 5 8 3 ISW This constant controls branch switching at branch points for the case of differential equations Note that branch switching is automatic for algebraic equations ISW 1 This is the normal valu
49. cir cd cir demo cir create an empty work directory change directory copy the demo files to the work directory us cir run c cir 1 h cir 1 s dat sv 1 use the starting data in cir dat to create s dat increase the truncation interval restart from s dat save output files as b 1 s 1 d 1 run c cir 2 h cir 2 s 1 sv gt 2 continue saddle focus homoclinic orbit restart from s 1 save output files as b 2 s 2 d 2 run c cir 3 h cir 3 s 2 ap 2 generate adjoint variables restart from s 2 append output files as b 2 s 2 d 2 Table 19 1 Detailed AUTO Commands for running demo cir 172 Chapter 20 HomCont Demo 20 1 she A Heteroclinic Example The following system of five equations Rucklidge amp Mathews 1995 2 2 deL 8 PpI TY 2ZU y 2 40o xu 40 uz 9oz 4ru 4uz 4 1 0 ou 4 0Qu 4n 3 1 0 22 40 u 4 v s 20 1 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 0 1 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 c and starting data for the orbit at u 0 5 0 163875 are stored in she dat with
50. commandShell shell Run a shell command command Triple tr triple Triple a solution commandUserData us userdata Covert user supplied data files command Wait wait Wait for the user to enter a key 40 4 13 Reference 4 13 1 commandAppend Purpose Append data files Description Type commandAppend xxx to append the output files fort 7 fort 8 fort 9 to exist ing data files s xxx b xxx and d xxx if you are using the default filename templates Type commandAppend xxx yyy to append existing data files s xxx b xxx and d xxx to data files s yyy b yyy and d yyy if you are using the default filename templates Aliases ap append 4 13 2 commandCat Purpose Print the contents of a file Description Type commandCat xxx to list the contents of the file xxx This calls the Unix function cat for reading the file Aliases cat 41 4 13 3 commandCd Purpose Change directories Description Type commandCd xxx to change to the directory xxx This command understands both shell variables and home directory expansion Aliases cd 4 13 4 commandClean Purpose Clean the current directory Description Type commandClean to clean the current directory This command will delete all files of the form fort o and exe Aliases clean cl 42 4 13 5 commandCopyAndLoadDemo Purpose Copy a demo
51. constant The output is stored in b 5 s 5 and d 5 BR PT TY LAB PAR 5 L2 NORM spats PAR 8 3 2 EP 15 2 550843E 09 4 018898E 01 1 000000E 02 Here PAR 8 is a dummy unused parameter and u 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 NPSI to 1 IPSI 1 to 13 and monitoring PAR 32 rn c sib 6 h sib if s 5 sv 6 BR PT TY LAB PAR 4 L2 NORM weds PAR 5 PAR 32 3 11 UZ 16 7 117745E 02 4 018899E 01 1 243774E 11 2 366987E 07 The output is stored in b 6 s 6 and d 6 Hence an inclination flip was found at a 0 7117745 Now we are ready to perform homoclinic branch switching using the techniques described in Oldeman et al 2001 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 the solution to the adjoint equation to obtain the Lin vector Since both ingredients are there we can now continue in y 1 and T to obtain the initial Lin gap Recall from Chapter 15 that the Lin gaps correspond to PAR 19 i 2 and the time intervals T correspond to PAR 20 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 rn c
52. converted file is called fig x ps The original file is left unchanged pr Type pr fig x to convert a saved PLAUT figure fig x from compact PLOT10 format to PostScript format and send it to the printer The converted file is called fig x ps The original file is left unchanged A 0 3 File manipulation Ocp Type Ccp xxx yyy to copy the data files b xxx s xxx d xxx C xxx to b yyy s yyy d yyy C yyy respectively Omv Type mv xxx yyy to move the data files b xxx s xxx d xxx c xxx to b yyy s yyy d yyy c yyy respectively df Type Cdf to delete the output files fort 7 fort 8 fort 9 cl Type cl to clean the current directory This command will delete all files of the form fort o and exe dl Type dl xxx to delete the data files b xxx s xxx d xxx A 0 4 Diagnostics lp 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 Type Clp xxx to list the value of the limit point function in the data file d xxx This function vanishes at a limit point fold bp Type Cbp to list the value of the branch point function in the output file fort 9 This function vanishes at a branch point Type Cbp xxx to list the value of the branch point function in the data file d xxx This function vanishes at a branch point Ohb Type Chb to list the value of the Hopf function in the output file fort
53. 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 22 T7 gt 12 03201 22 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 Yoshizawa 1962 are a simplified version of the Hodgkin Huxley equations Hodgkin amp Huxley 1952 They model nerve axon dynamics and are given by Ut Usa falu UW n 22 2 where falu u u a u 1 Travelling wave solutions of the form u w x t u w where x ct are solutions of the following ODE system v cv falu w 22 3 w u YU al yw In particular we consider solitary wave solutions of 22 2 These correspond to orbits homoclinic to u v w 0 in system 22 3 In our numerical example we keep y 0 190 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 sib ld fnb rn sv 1 Among the output we see
54. fsh d fsh Table 13 1 Commands for running demo fsh 129 13 2 nag A Saddle Saddle Connection This demo illustrates the computation of traveling wave front solutions to Nagumo s equation Wt Wee f w a e S00 t gt 0 13 3 f w a w 1 w w a Cast 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 u2 2 13 4 ub 2 cuz z f uil2 4 wee where z xz 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 1 env San et Se EL 2 ioe tie Zz uil 00 lt Z lt 00 U1 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 subroutine stpnt in nag c 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 run c nag 1 compute part of first branch of heteroclinic orbits sv nag save output files as b nag s nag d nag run c nag 2 s nag compute first branch in opp
55. gt t and U i gt U i for alli 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 IREV 0 15 3 7 NFIXED IFIXED Number and labels of test functions that are held fixed E g with NFIXED 1 one can compute a locus in one extra parameter of a singularity defined by test function PSI IFIXED 1 0 15 3 8 NPSI IPSI Number and labels of activated test functions for detecting homoclinic bifurcations see Sec tion 15 6 for a list If a test function is activated then the corresponding parameter IPSI I 20 must be added to the list of continuation parameters NICP ICP I I 1 NICP and zero of this parameter added to the list of user defined output points NUZR 1 PAR I I 1 NUZR in C XXX 15 4 Restrictions on HomCont Constants Note that certain combinations of these constants are not allowed in the present implementation In particular The computation of orientation ITWIST 1 is not implemented for IEQUIB lt O heteroclinic orbits IEQUIB 2 saddle node homoclinics IREV 1 reversible systems ISTART 3 homotopy 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
56. headings in fort 7 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 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 To 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 200 which can be converted to PostScript format with the AUTO commands ps and Opr see Section B 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 QIb to relabel solutions in fort 7 and fort 8 end To end execution of PLAUT B 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 do Use solid curves showing so
57. 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 Doedel E J 1981 AUTO a program for the automatic bifurcation analysis of autonomous systems Cong Numer 30 265 384 208 Doedel E J 1984 The computer aided bifurcation analysis of predator prey models J Math Biol 20 1 14 Doedel E J amp Heinemann 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
58. 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 Ban galore 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 Uni versiteit 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 Lentini M amp Keller 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 Ed inburgh 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 Sin
59. in the G v plane but first we perform continuation in T v to obtain a better approximation to a homoclinic orbit make first yields the output BR PT TY LAB PERIOD L2 NORM ips 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 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 focus with a one dimensional unstable manifold We now restart at LAB 3 corresponding to a time interval 27 200 and change the principal continuation parameters to be v 8 The new constants defining the continuation are given in 169 c cir 2 and h cir 2 We also activate the test functions pertinent to codimension two singularities which may be encountered along a branch of saddle focus homoclinic orbits viz Ya Va Ws Wo and 10 This must be specified in three ways by choosing NPSI 5 and appropriate IPSI I in h 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 y indices and finally adding USZR functions defining zeros of these parameters in c cir 2 Running make second
60. into the current directory and load it Description Type commandCopyAndLoadDemo xxx to copy all files from auto 2000 demos xxx to the current user directory Here xxx denotes a demo name e g abc Note that the dm command also copies a Makefile to the current user directory To avoid the overwriting of existing files always run demos in a clean work directory NOTE This command automatically performs the commandRunnerLoadName command as well Aliases dm demo 4 13 6 commandCopyDataFiles Purpose Copy data files Description Type commandCopyDataFiles 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 if you are using the default filename templates Aliases copy Cp 43 4 13 7 commandCopyDemo Purpose Copy a demo into the current directory Description Type commandCopyDemo xxx to copy all files from auto 2000 demos xxx to the current user directory Here xxx denotes a demo name e g abc Note that the dm command also copies a Makefile to the current user directory To avoid the overwriting of existing files always run demos in a clean work directory Aliases copydemo 4 13 8 commandCopyFortFiles Purpose Save data files Description Type commandCopyFortFiles xxx to save the output files fort 7 fort 8 fort 9 to b xxx s xxx d xxx if you are using the defau
61. iso s iso d iso Table 9 7 Commands for running demo plp 97 9 8 pp3 Period Doubling Continuation This demo illustrates the computation of stationary solutions Hopf bifurcations and periodic solutions branch switching at a period doubling bifurcation and the computation of a locus of period doubling bifurcations The equations model a 3D predator prey system with harvesting Doedel 1984 Uy Un uz free uj 1 uy P4U Ua P2U3 Pau Us psuguz pi l e Pe P3U3 T P5sUgUz The free parameter is p except in the period doubling continuation where both p and pa are AUTO COMMAND ACTION mkdir pp3 create an empty work directory cd pp3 change directory demo pp3 copy the demo files to the work directory 1d pp3 load the problem definition run c pp3 1 lst run stationary solutions sv pp3 save output files as b pp3 s pp3 d pp3 ap pp3 run c pp3 2 s pp3 compute a branch of periodic solutions Constants changed IRS IPS NMX append output files to b pp3 s pp3 d pp3 ap pp3 run c pp3 3 s pp3 compute the branch bifurcating at the period doubling Constants changed IRS ISW NTST append output files to b pp3 s pp3 d pp3 sv tmp run c pp3 4 s pp3 generate starting data for the period doubling continuation C
62. 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 13 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 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 166 The projection onto the e k plane of all four of these codimension two curves is given in Figure 18 8 The intersection of the inclination flip lines with one of the non central saddle node homo clinic lines is apparent Note that the two non central saddle node homoclinic orbit curves are make twelfth make thirteenth almost overlaid but that as in Figure 18 6 the orbits look quite distinct in phase space 18 5 Detailed AUTO Commands AUTO COMMAND ACTION sv 6 mkdir kpr create an empty work directory cd kpr change directory demo kpr copy the demo files to the work directory run c kpr 1 h kpr 1 continuation in the time length parameter PAR 11 sv 1 save output files as b 1 s 1 d 1 run c kpr 2 h kpr 2 s 1 locate the homoclinic orbit restart from s 1 sv 2 save output files as b 2 s 2 d
63. non orientable resonant bifurcations see Sand stede 1995a for details and proofs The starting point for all calculations is a 0 b 1 where there exists an explicit solution given by COMO Oe 5 40 0 This solution is specified in the routine stpnt 16 2 Inclination Flip We start by copying the demo to the current work directory and running the first step dm san make first This computation starts from the analytic solution above with a 0 b 1 c 2 a 0 8 1 and y u 0 The homoclinic solution 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 000000E 00 4 000000E 01 0 000000E 00 1 5 UZ 2 2 500000E 01 4 030545E 01 3 620329E 11 1 10 EP 3 7 384434E 01 4 339575E 01 9 038826E 09 145 and saved in more detail as b 1 s l and d 1 Next we want to add a solution to the adjoint equation to the solution obtained at a 0 25 This is achieved by making the change ITWIST 1 saved in h san 2 and IRS 2 NMX 2and ICP 1 9 saved in c san 2 We also disable any user defined functions NUZR 0 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 make second The output is stored in b 2 s 2 and d 2
64. 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 space consisting of time here PAR 14 and the state vector here u1 u2 AUTO COMMAND ACTION mkdir ivp create an empty work directory cd ivp change directory demo ivp copy the demo files to the work directory ld ivp load the problem definition run c ivp 1 time integration sv ivp save output files as b ivp s ivp d ivp Table 9 13 Commands for running demo ivp 104 Chapter 10 AUTO Demos BVP 10 1 exp Bratu s Equation This demo illustrates the computation of a solution branch to the boundary value problem f Uy U2 ie ape 10 1 with boundary conditions u 0 0 ui 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 run c exp 1 lst run compute solution branch containing fold sv exp save output files as b exp s exp d exp run c exp 2 s exp 2nd run restart at a labeled solution using increased accuracy Constants changed IRS NTST A1 DSMAX vspace0 2cm ap exp append output files to b exp s exp d exp Table 10 1 Commands f
65. possible options Long name Short name Description equation e The equations file constants C The AUTO constants file solution s The restart solution file h homcont The Homcont parameter file Options which are not explicitly set retain their previous value For example one may type commandRunnerLoadName e ab c ab 1 to use ab c as the equations file and c ab 1 as the constants file if you are using the default filename templates Type commandRunnerLoadName name load all files with base name This does the same thing as running commandRunnerLoad Name e name c name s name h name Aliases Id load 57 4 13 34 commandRunnerPrintFort2 Purpose Print continuation parameters Description Type commandRunnerPrintFort2 to print all the parameters Type commandRun nerPrintFort2 xxx to return the parameter xxx Aliases pc pr printconstant 4 13 35 commandShell Purpose Run a shell command Description Type shell xxx to run the command xxx in the Unix shell and display the results in the AUTO command line user interface Aliases shell 58 4 13 36 commandTriple Purpose Triple a solution Description Type commandTriple to triple the solution in fort 7 and fort 8 Type command Triple xxx to triple the solution in b xxx and s xxx if you are using the default filename templates Aliase
66. 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 NFIXED 1 IFIXED 1 15 The new continuation problem is specified in c mtn 6 and h mtn 6 make sixth 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 user defined function NUZR 1 to detect intersections with the plane Dy 0 01 We get among other output BR PT TY LAB PAR 3 L2 NORM sis PAR 1 PAR 2 1 22 LP 19 1 081212E 02 5 325894E 00 5 673631E 00 6 608184E 02 1 31 UZ 20 1 000000E 02 4 819681E 00 5 180317E 00 6 385503E 02 the first line of which represents the D value at which the homoclinic curve P has a tangency with the branch 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 Dz The results of this computation and a similar one starting from D in the opposite direction with DS 0 01 are displayed in Figure 17 3 155 17 5 Detailed AUTO Commands AUTO COMMAND ACTION mkdir mtn create an empty work directory run c mind A nin 1s dat su 1 cd mtn change directory demo mtn copy the demo files to the work directory us mtn use the starting data
67. 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 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 0 375 183 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 1d sib rn sv 1 The equilibrium is followed in a until a or PAR 1 is at our desired value 0 375 BR PT TY LAB PAR 1 TN U 1 U 2 U 3 1 1 EP 1 0 000000E 00 6 666667E 01 0 000000E 00 0 000000E 00 1 6 EP 2 3 750000E 01 6 666667E 01 1 333333E 01 0 000000E 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 rn c sib 2 s 1 sv 2 or alternatively cc IRS 2 cc ICP 4 rn s 1 sv 2 BR PT TY LAB PAR 4 poy U 1 U 2 U 3 1 18 HB 3 3 184290E 01 6 543750E 01 1 347543E 01 7 701025E 02 The output is saved i
68. she 3 h she 3 s 2 ap 2 continue in reverse direction restart from s 2 append output files to b 2 5 2 d 2 Table 20 1 Detailed AUTO Commands for running demo she 175 a s7 p S ge ee r X oa aS Figure 20 1 Projections into x y z space of the family of heteroclinic orbits 176 Chapter 21 HomCont Demo rev 21 1 A Reversible System The fourth order differential equation u Pu u u 0 arises in a number of contexts e g as the travelling wave equation for a nonlinear Schrodinger 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 Uy U2 eG 21 1 ug Us UA Pug u u3 Note that 21 1 is invariant under two separate reversibilities R uz U2 U3 U4 t gt uz U2 U3 Ua4 t 21 2 and Ra u1 U2 U3 Us t b gt u1 U2 U3 U4 t 21 3 First we copy the demo into a new directory dm 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 Ro 21 2 An R amp Reversible Homoclinic Solution
69. this manual 199 Appendix B The Graphics Program PLAUT PLAUT 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 the p command defined in Section A The PLAUT window a Tektronix window will appear in which PLAUT commands can be entered FIXME This is not correct anymore For examples of using PLAUT see the tutorial demo ab in particular Sections 7 7 and 7 10 See also demo pp2 in Section 9 3 B 1 Basic PLAUT Commands The principal PLAUT commands are bdo bd ax 2d sav 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 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 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
70. to be taken along any branch 5 6 2 RLO The lower bound on the principal continuation parameter This is the parameter which appears first in the ICP list see Section 5 7 1 65 5 6 3 RL1 The upper bound on the principal continuation parameter 5 6 4 AO The lower bound on the principal solution measure By default if IPLT 0 the principal solution measure is the L2 norm of the state vector or state vector function See the AUTO constant IPLT in Section 5 9 3 for choosing another principal solution measure 5 6 5 Al The upper bound on the principal solution measure 5 7 Free Parameters 5 7 1 NICP 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 constant NICP indicates how many free parameters have been specified while the array ICP actually 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 5 7 2 Fixed points The simplest case is the continuation of a solution branch to the system f u p 0 where F u R cf 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 he
71. to the work directory 1ld abc load the problem definition run c abc 1 compute the stationary solution branch with Hopf bifurcations sv abc save output files as b abc s abc d abc ap abc run c abc 2 s abc compute a branch of periodic solutions from the first Hopf point Constants changed IRS IPS NICP ICP append the output files to b abc s abc d abc ap abc run c abc 3 s abc compute a branch of periodic solutions from the second Hopf point Constants changed IRS NMX append the output files to b abc s abc d abc Table 9 2 Commands for running demo abc 92 93 pp2 A 2D Predator Prey Model This demo illustrates a variety of calculations The equations which model a predator prey system with harvesting are pill oe uy p2u 1 u Uy U2 p4uyug u1U2 9 3 Here p is the principal continuation parameter p 5 pa 3 and initially p 3 For two parameter computations pz is also free AUTO COMMAND ACTION mkdir pp2 create an empty work directory cd pp2 change directory demo pp2 copy the demo files to the work directory 1d pp2 load the problem definition run c pp2 1 1st run stationary solutions sv pp2 save output files as b pp2 s pp2 d pp2 run c pp2 2 s pp2 ap pp2 2nd run restart at a labeled solution Con st
72. value range 5 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 5 4 3 EPSS Relative arclength convergence criterion for the detection of special solutions Most demos use EPSS 107 or EPSS 107 which is the recommended value range Generally EPSS should be approximately 100 to 1000 times the value of EPSL EPSU 5 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 Miiller s method with bracketing The recom mended value is ITMX 8 used in most demos 5 4 5 NWTN After NWTN Newton iterations the Jacobian is frozen i e AUTO uses full Newton for the first NWTN iterations and the Chord method for iterations NWTN 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 5 4 6 ITNW The maximum number of combined 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 branch is discontinued and a message printed in fort 9 The recommended value is ITNW 5 but IT
73. 0 distribution bzipped Postscript manual auto2000 0 9 6 ps bz2 ezipped Postscript manual auto2000 0 9 6 ps gz compressed Postscript manual auto2000 0 9 6 ps Z tarred and gzipped source code auto2000 0 9 6 tgz tarred and bzipped source code auto2000 0 9 6 tbz2 tarred and compressed source code auto2000 0 9 6 tar Z zipped source code auto2000 0 9 6 zip Below it is assumed that you are using the Unix shell csh and that the file auto2000 0 9 6 tar Z is in your main directory While in your main directory enter the commands uncompress auto2000 0 9 6 tar Z followed by tar xvfo auto2000 0 9 6 tar This will result in a directory auto with one subdirectory auto 2000 Type cd auto 2000 to change directory to auto 2000 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 There is a new CLUI under development which includes some of the capabilities of the old GUI and will eventually be the recommend way to run AUTO
74. 0 is a stationary solution 79 7 4 Executing all Runs Automatically To execute all prepared runs of demo ab simply type one or both of the command given in Table 7 2 AUTO COMMAND ACTION demofile ab_old auto execute all runs of demo ab interactively using a new constants file for each run demofile ab_new auto execute all runs of demo ab interactively by modifying the constants file before each run Table 7 2 Executing all runs of demo ab Each of the commands in Table 7 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 ab_old auto performs the demo by reading in a sequence of AUTO constants files each of which corresponds to a step of the demo The tutorial script ab_new auto performs the demo by reading in a single AUTO constants file and then interactively modifying it to perform each of the demo Both are valid and effective methods for running AUTO with ab old auto being similar to the way AUTO was used before the advent of the CLUI and ab_new auto using new functionality provided by the CLUI Note that there are five separate runs In the first run a branch of stationary solutions is traced out Along it two folds LP and one Hopf bifurcation HB are located The free parameter is p The other parameters remain fixed in this run
75. 00 20 x T Ls 3 0 00 0 20 0 40 1 00 90 x T Figure 21 4 An Ro reversible homoclinic orbit at label 8 181 21 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 cp rev c 1 rev c cp rev dat 1 rev dat us rev get equations file to rev c get the starting data to rev dat use the starting data in rev dat to create s dat ap 3 run c rev 1 h rev 1 s dat increase PAR 1 ev 1 save output files as b 1 s 1 d 1 run c rev 2 h rev 2 s 1 continue in reverse direction restart from s 1 ap 1 append output files to b 1 s 1 d 1 cp rev c 3 rev c get equations file with new value of PAR 11 cp rev dat 3 rev dat get starting data with different reversibility us rev use the starting data in rev dat to create s dat run c rev 3 h rev 3 s dat restart with different reversibility sv 3 save output files as b 3 s 3 d 3 run c rev 4 h rev 4 s 3 continue in reverse direction restart from s 3 append output files to b 3 s 3 d 3 Table 21 1 Detailed AUTO Commands for running demo rev 182 Chapter 22 HomCont Demo Homoclinic branch switching This demo illustrates homoclinic branch swit
76. 07E 03 1 000000E 00 1 44 UZ 9 2 000000E 00 2 873838E 01 1 245735E 11 2 318248E 08 1 55 EP 10 3 099341E 00 3 020172E 01 2 749454E 11 1 099341E 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 21 2 21 3 An Ro gt Reversible Homoclinic Solution make third Copies the files rev c 3 and rev dat 3 to rev c and rev dat and runs them with the constants stored in c rev 3 and h 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 h rev 1 via NREV 1 IREV I I 1 NDIM 1 0 1 0 The output BR PT TY LAB 1 15 UZ 2 1 16 LP 3 PAR 1 L2 NORM MAX U 1 1 700000E 00 3 836401E 01 4 890015E 01 1 711574E 00 3 922135E 01 5 442385E 01 178 0 SU 0 00 25 50 TS 0 00 1 00 Qe TO 0 30 0 50 0 70 0 90 x T Figure 21 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 ampli tude 20 50 TE 00 29 50 0 00
77. 2 A 2D Predator rey Model 3 00 coe de aes Se AR Ae te A 9 4 lor Starting an Orbit from Numerical Data yal A ae BA eS 9 5 fre A Periodically Forced System a a a A 9 6 ppp Continuation of Hopf Bifurcations 4 4 a a 4545 44 9 7 plp Fold Continuation for Periodic Solutions 9 8 pp3 Period Doubling Continuation 2 44 4 04 2 baw ORS ode ae EES 9 9 tor Detection of Torus Bifurcations 2 68 4 hee A 9 10 pen Rotations of Coupled Pendula 2120 ic a 9 11 chu A Non Smooth System Chua s Circuit o a 9 12 phs Effect of the Phase Condition o0 a a 626 a S44 es ra 9 13 ivp Time Integration with Euler s Method 10 AUTO Demos BVP exp Brabus Equation iva Ll a A a int Boundary and Integral Constraints o o bvp A Nonlinear ODE Eigenvalue Problem lin A Linear ODE Eigenvalue PLOT A LAA A non A Non Autonomous BVP LA A Sere ead RI kar The Von Karman Swirling Flows oaa a e spb A Singularly Perturbed BVP ooa aaa 0000000250 ezp Complex Bifurcation ana BN Peis oo os eet ee PES ood ee eS 11 12 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 AUTO Demos Parabolic PDEs pal Stationary States 1D Problem e a03 biog 4 3 ae ais me Ae ald 8 EA ole pd2 Stationary States 2D Problem boo cer eee a A wav Periodic Wayess AAN ARE Bas oe bre Chebyshev Collocation in Space cor cmi
78. 3 20 27 2 736987E 01 2 911260E 05 6 515911E 07 1 636554E 09 3 30 28 1 737189E 01 4 422894E 03 1 440898E 04 3 101882E 05 3 38 EP 29 1 014512F 01 2 000000E 01 6 974453E 02 1 486151E 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 rn c sib 11 h sib hbs3 s 10 ap 6 188 BR PT TY LAB PAR 4 rhe PAR 5 PAR 21 PAR 23 3 6 UZ 30 8 199998E 02 1 297904E 02 1 769949E 01 6 371836E 02 3 32 EP 31 1 984145E 01 6 054949E 03 2 307164E 06 3 624489E 02 The output is appended to b 6 s 6 and d 6 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 to obtain a 3 homoclinic orbit We replace the continuation parameter e by Tz because T PAR 22 still has to be decreased from its high value 35 and e needs to stay at 0 rn c sib 12 h sib hbs3 s 6 ap 6 BR PT TY LAB PAR 4 Fani PAR 5 PAR 22 PAR 23 3 16 UZ 32 1 983953E 01 6 055361E 03 2 013107E 01 1 824909E 08 3 24 UZ 33 1 800000E 01 6 502928E 03 1 275539E 01 3 142935E 02 3 30 UZ 34 1 669900E 01 6 892692E 03 9 417449E 00 1 031790E 06 3 32 EP 35 1
79. 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 191 rn c fnb 5 h fnb 5 s 4 sv 5 BR PT TY LAB PAR 8 L2 NORM a ane PAR 2 1 2 EP 28 1 000000E 00 1 561579E 02 2 500000E 03 Next we continue in the time 7 PAR 20 the gap e PAR 21 and c PAR 0 and by setting ISTART 2 we try to locate a 2 homoclinic orbit rn c fnb 6 h fnb 6 s 5 sv 6 In fact we find many of them exactly as is predicted by the theory BR PT TY LAB PAR 20 pas PAR 0 PAR 21 1 175 UZ 45 1 647952E 02 2 742181E 01 2 313522E 11 1 179 UZ 46 1 448063E 02 2 742181E 01 1 481383E 11 1 183 UZ 47 1 248379E 02 2 742181E 01 2 171338E 16 1 188 UZ 48 1 048192E 02 2 742181E 01 5 215295E 11 1 192 UZ 49 8 487422E 01 2 742181E 01 3 106887E 15 1 197 UZ 50 6 463349E 01 2 742181E 01 1 803730E 10 Each of these homoclinic orbits differ by about 20 in the value T 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 th
80. 7 7 Plotting the Results with AUTO The bifurcation diagram computed in the runs above is stored in the file b ab while each labeled solution is fully stored in s ab To use AUTO to graphically inspect these data files type the AUTO command given in Table 7 8 The saved plots are shown in Figure 7 1 and in Figure 7 2 Figure 7 1 shows the default view of the plotting tool which consists of a representation of the bifurcation diagram Step by step instructions for creating Figure 7 2 are given below The plotting window consists of a menubar at the top a plotting area and a control panel with four control widgets at the bottom The first step in creating Figure 7 2 is to 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 83 the plotting window will appear a plot of the first labeled solution in s ab Unfortunately this is an equilibrium solution so only a single point is plotted Since we wish to plot the periodic solutions we modify the widget marked Label by changing its value from 1 to 6 7 10 don t forget to hit the return key when you are done modifying the value This signifies that instead of plotting the solution with label 1 we want to plot the solutions with labels 6 7 and 10 simultaneously In the plotting window we now have thr
81. 7000 F 020 1 000 0 980 0 960 0 940 Tel Q 290 0 970 0 950 x Figure 18 2 Projection on the x y plane of solutions of the boundary value problem with 2T 60 0 160 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 make second the output at label 4 stored in kpr 2 BR PT TY LAB PERIOD L2 NORM ee PAR 1 1 35 UZ 4 6 000000E 01 1 672806E 00 1 851185E 00 provides a good approximation to a homoclinic solution see Figure 18 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 h 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 make third 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 pa
82. 9 This function vanishes at a Hopf bifurcation point Type Chb xxx to list the value of the Hopf function in the data file d xxx This function vanishes at a Hopf bifurcation point 197 sp 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 it Type Cit to list the number of Newton iterations per continuation step in fort 9 Type Cit xxx to list the number of Newton iterations per continuation step in d xxx Ost 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 dev Type Gev 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 f1 Type f1 to list the Floquet multipliers in the output file fort 9 Differential equations Type f1 xxx to list the Floquet multipliers in the data file d xxx Differential equations A 0 5 File editing e7 To use the vi editor to edit the output file fort 7 e8 To use the vi editor to edit the output file fort 8 e9 To use the vi editor to edit the output file
83. 90E 01 1 484391E 00 3 110230E 01 1 451441E 00 1 70 LP 3 8 893185E 02 3 288241E 00 6 889822E 01 3 215250E 00 1 90 HB 4 1 308998E 01 4 271867E 00 8 950803E 01 4 177042E 00 1 92 EP 5 1 512417E 01 4 369748E 00 9 155894E 01 4 272750E 00 Total Time 9 502E 02 ab done AUTO gt Figure 4 2 Typing auto at the Unix shell prompt starts the AUTO 2000 CLUI The rest of the commands are interpreted by the AUTO 2000 CLUI 24 Note that the name given to the 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 2000 CLUI into the standard names used by AUTO 2000 The standard filename transformations are show in Table 4 3 Long name Short name Name entered Transformed file name equation e foo 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 2000 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 4 3 For example the command load constants ab 1 can be shortened to load c
84. AUTO 2000 CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS with HomCont Eusebius J Doedel Randy C Paffenroth California Institute of Technology California Institute of Technology Pasadena California USA Pasadena California USA Alan R Champneys Thomas F Fairgrieve University of Bristol Ryerson Polytechnic University United Kingdom Toronto Canada Yuri A Kuznetsov Bart E Oldeman Bjorn Sandstede Universiteit Utrecht University of Bristol Ohio State University The Netherlands United Kingdom Columbus Ohio USA Xianjun Wang Concordia University Montreal Canada July 30 2002 On leave from Concordia University Montreal Canada Contents 1 Installing AUTO 1 1 Typographical Conventions eb Seb aaa A e as ka cae O oe e Sa el eS ee a A a el og 1 3 Restrictions on Problem Size sia aa oe ae Re Oe eS 1 4 Compatibility with Older Versions eva a de A aves LS P rallel Nero s enn at d a e ii e a rt S 2 Overview of Capabilities E O O A oS 2 2 Algebraic Systems ls A anh amp oe ak eS Sods Vik Rea Ae oS 2 3 Ordinary Differential Equations 2 020022 ee ee Zo Parabole PDES El ancy eos A PS Re eB eS dr A Brea IN hg og A tol a cn eth O ee ta i 3 Howto Run AUTO 3E User Mpplicd Files ay sete lo aan ERAS oy RU el te BE RE e EER SS te deed A 3 1 1 The equations file OCC A a ee SE a Sa OS 31 2 The constants fle GK 406 ok ae AA a ee eg 3 2 User Supplied Subroutines 4 3 8 6 ea
85. BR PT TY LAB PERIOD L2 NORM nee PAR 16 1 20 UZ 3 2 922565E 01 2 379162E 01 1 680003E 09 and a zero of PAR 16 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 16 at 0 rn c fnb 2 h fnb 1 s 1 sv 2 BR PT TY LAB PERIOD L2 NORM ned PAR 1 1 190 UZ 10 3 000000E 02 7 379317E 02 1 792864E 01 Next we stop using the homotopy technique and increase the period even further to 1000 rn c fnb 3 h fnb 3 s 2 sv 3 BR PT TY LAB PERIOD L2 NORM eed PAR 1 1 80 UZ 13 1 000000E 03 4 041827E 02 1 792865E 01 A continuation in PAR 1 a and PAR 0 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 21 to locate Belyakov points rn c fnb 4 h fnb 4 s 3 sv 4 BR PT TY LAB PAR 1 L2 NORM eae PAR O PAR 21 1 6 UZ 15 1 318124E 01 3 287104E 02 2 171656E 01 6 312189E 06 1 23 UZ 19 8 545741E 08 1 561579E 02 2 742181E 01 9 887718E 02 Hence there exists a Belyakov point at a c 0 1318124 0 217656 At label 19 we have a lower value of a than at the Belyakov point and by inspection of the file d
86. Commands for running demo bru 119 Chapter 12 AUTO Demos Optimization 120 12 1 opt A Model Algebraic Optimization Problem This demo illustrates the method of successive continuation for constrained optimization problems by applying it to the following simple problem 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 create an empty work directory cd opt change directory demo opt copy the demo files to the work directory run c opt 1 one free equation parameter ev Cd save output files as b 1 s 1 d 1 run c opt 2 s 1 sv 2 two free equation parameters read restart data from s 1 Constants changed IRS save output files as b 2 s 2 d 2 run c opt 3 s 2 sv 3 three free equation parameters read restart data from s 2 Constants changed IRS save output files as b 3 s 3 d 3 run c opt 4 s 3 sv 4 four free equation parameters read restart data from s 3 Constants changed IRS save output files as b 4 s 4 d 4 Table 12 1 Commands for running demo opt 121 Find the maximum sum of coordinates on the 12 2 ops Optimization of Periodic Solution
87. Euler s method for time integration of a nonlinear parabolic PDE The example is the Brusselator Holodniok Knedlik amp Kubi ek 1987 given by up D L Urs uu B 1 u A 9 9 11 5 Y DILO ut0 Bu 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 rx Note that in the subroutine stpnt the space derivatives of u and v must also be provided see the equations file bru c 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 run c bru 1 time integration sv bru save output files as b bru s bru d bru Table 11 6
88. 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 com mand 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 defines the number of computational processes is np Accordingly if you wanted to run the MPI version of AUTO 2000 on four processors with the above external program you would type mpirun np 4 auto exe m Please see your local MPI documentation for more detail As with the Pthreads library if you were to try and run the above command on a machine which did not have MPI the command would exit with an error and inform you that MPI is not available The commands in the auto 2000 cmds directory and described in Chapter 3 may be used with the parallel version as well by setting the AUTO_COMMAND_PREFIX and AUTO_COMMAND_ARGS environment variables For example to the run AUTO 2000 in parallel using the Pthreads li brary on 4 processors just type setenv AUTO_COMMAND_ARGS t both 4 and then use the commands in auto 2000 cmds normally To run AUTO 97 in parallel using the MPI library on 4 processors just type setenv AUTO_COMMAND_ARGS m and setenv AUTO_COMMAND PREFIX mpirun np 4 and then use the commands in auto 2000 cmds normally The previous ex amples assumed you are using the csh s
89. NW 7 may be used for difficult problems for example demos spb chu plp etc 63 5 5 Continuation Step Size 5 5 1 DS AUTO uses pseudo arclength continuation for following solution branches The pseudo arclength stepsize is the distance between the current solution and the next solution on a branch 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 branch 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 5 0 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 dependent most demos use a value that was found appropriate after some experimentation See also the discussion in Section 6 2 5 5 3 DSMAX The maximum allowable absolute value of the pseudo arclength stepsize DSMAX must be pos itive It is only effective if the pseudo arclength step is adaptive i e if IADS gt 0 Th
90. OMMAND ACTION mkdir bvp create an empty work directory cd bvp change directory demo bvp copy the demo files to the work directory run c bvp 1 compute the trivial solution branch and locate eigenvalues sv bvp save output files as b bvp s bvp d bvp run c bvp 2 s bvp compute the first bifurcating branch Constants changed IRS ISW NPR DSMAX ap bvp append output files to b bvp s bvp d bvp run c bvp 3 s bvp compute the first bifurcating branch in op posite direction Constants changed DS ap bvp append output files to b bvp s bvp d bvp Table 10 3 Commands for running demo bvp 107 10 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 branch 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 ti u 10 4 u pitt oa with boundary conditions u1 0 pg 0 and u 1 0 We add the integral constraint 1 u1 6 dt pz 0 0 Then ps is simply the L2 norm of the first solution component In the first two runs pa is fixed while p and p3 are free In the third run ps is fixed while p and p are free
91. PrintFort2 root dea ie ere ois Metts ei asia 58 4 19 35c0ommandShell 6 en Soe AS a Sp ele ee Se Ele ds eee ss 58 4 13 36 commsnd TAple edades EAS dei ee 59 4 13 37 command UCLA a cioe s a sd ie aes Oe RR I BM ee 59 413 38 command Walt i dde a el ee da Bee Sid Basa ls 60 5 Description of AUTO Constants 61 5 1 The AUTO Constants File da RAS E AER 61 5 22 Problem Constants ud pee be eae he R E the A EO eS o de 61 elt NDE Rd E ok hs elena As ithe AE 61 5 3 5 4 5 9 5 6 5 7 5 8 9 22 NBG dose ene eth etme Sas Ae ay ae Se A a oe Be Ges Peas ee ee 61 5 2 3 NINT La Sah de a Ee a ee ES 62 E Ge e las Re herald ips ke detuned Sse hE Sey ga LY ate 62 Discretization Constants sus sl a o A oe ee RR 62 ASS NIST Pima o Se hee Get he tte atk the A SOA a Ny ra ty 62 52 INCOM nn ad a atthe pile bad a ete Sade AI IS hae Bs 62 Boog TADA ope ih pee a oly ok dere ay oe be Se ee ge es ae 62 TFole ranceg Sa II ae pln Pu i 25 63 Bab EPSE ae Gee te os sh ig os Sa ht whe de e e de POA Seed oe en th hed Sa E 63 SA2 EPSV S ce nt ee eh A A Qe A 63 Ph Re ai cet A Oe ee Ate Gon hea A A te eG 63 O A E I a O A A so 63 SAO NW cb efile da da Beis Bisa Poh A E a a a Be ea 63 O LIN o e o ae e a eee he EONS e eee dl dee 63 Continuation Step Size ee a darte ers it a a E arto 64 O A A ee OD AN E 64 sas DMUs o efi s Sengur O a Bag Pars oh Bate Ba e e es a 64 Dio ADOMAX estic Ate tie sl Soi te Th ay Bee Ph ok ta Sd ee f
92. R python demo userScript py gt ls userScript py gt cat userScript py This is an example script for the AUTO2000 command line user interface See the Command Line User Interface chapter in the manual for more details from AUTOclui import def myRun demo copydemo demo 1d demo run sv demo 1d s demo data sl demo ch NTST 50 for solution in data if solution Type name BP ch IRS solution Label ch ISW 1 Compute forward run ap demo Compute back ch DS pr DS run ap demo plot demo wait gt auto Initializing Python 1 5 2 1 Feb 1 2000 16 32 16 GCC egcs 2 91 66 19990314 Linux egcs on linux i386 Copyright 1991 1995 Stichting Mathematisch Centrum Amsterdam AUTOInteractiveConsole AUTO gt from userScript import AUTO gt myRun bvp Figure 4 12 This Figure shows the functional version of the AUTO 2000 CLUI from Figure 4 11 being used as an extension to the AUTO 2000 CLUI The source code for this script can be found in AUTO_DIR python demo userScript py 32 AUTO gt data dg ab Parsed file b ab AUTO gt print data 0 LAB 6 TY name EP data 0 0 0 0 0 0 0 0 section 12 BR 2 PT 1 TY number 9 AUTO gt print data 0 LAB 6 AUTO gt Figure 4 13 This figure shows an example of parsing a bifurcation diagram The first command
93. S run ap bvp Figure 4 10 The second part of the complex AUTO 2000 CLUI script Now that the section of script shown in Figure 4 10 has finished computing the bifurcation diagram the command plot bvp brings up a plotting window Section 4 13 20 in the ref erence and the command wait causes the AUTO 2000 CLUI to wait for input You may now exit the AUTO 2000 CLUI by pressing any key in the window in which you started the AUTO 2000 CLUI 4 6 Extending the AUTO 2000 CLUI The code in Figure 4 7 performed a very useful and common procedure it started an AUTO 2000 cal culation and performed additional continuations at every point which AUTO 2000 detected as a bifurcation Unfortunately the script as written can only be used for the bvp demo In this sec tion we will generalize the script in Figure 4 7 for use with any demo and demonstrate how it can 29 be imported back into the interactive mode to create a new command for the AUTO 2000 CLUI Several examples of such expert scripts can be found in auto 2000 demos python n body Just as loops and conditionals can be used in Python one can also define functions For ex ample Figure 4 11 is a functional version of script from Figure 4 7 The changes are actually quite minor The first line from AUTOclui import includes the definitions of the AUTO 2000 CLUI commands and must be included in all AUTO 2000 CLUI scripts which define functions The next line
94. S ap 1 append output files to b 1 s 1 d 1 run c tor 3 s 1 compute a bifurcating branch of periodic solutions restart from s 1 Constants changed IRS ISW NMX ap 1 append output files to b 1 s 1 d 1 Table 9 9 Commands for running demo tor 99 9 10 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 m 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 o epi sin dy I 7 2 61 by eph sind I b1 2 9 11 or in equivalent first order form On Vi p Va Y eb1 sing 1 7 2 01 9 12 wh eb3 sin gg I 7 d1 2 Throughout y 0 175 Initially e 0 1 and 0 4 Numerical data representing one complete rotation are contained in the file pen dat Each row in pen dat contains five real numbers namely the time variable t 2 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
95. Type commandParseSolutionFile xxx to get a parsed version of the solution file s xxx if you are using the default filename templates Aliases sl solutionget 4 13 20 commandPlotter Purpose 2D plotting of data Description Type commandPlotter xxx to run the graphics program for the graphical inspection of the data files b xxx and s xxx if you are using the default filename templates The return value will be the handle for the graphics window Type commandPlotter to run the graphics program for the graphical inspection of the output files fort 7 and fort 8 The return value will be the handle for the graphics window Aliases p2 pl plot 50 4 13 21 commandPlotter3D Purpose 3D plotting of data Description Type commandPlotter3D xxx to run the graphics program for the graphical inspec tion of the data files b xxx and s xxx if you are using the default filename templates The return value will be the handle for the graphics window Type commandPlotter3D to run the graphics program for the graphical inspection of the output files fort 7 and fort 8 The return value will be the handle for the graphics window Aliases plot3 p3 4 13 22 commandQueryBranchPoint Purpose Print the branch point function Description Type commandQueryBranchPoint to list the value of the branch point function in the output file fort 9 This functi
96. U1U2 P2U3 Here p is the free parameter and py 8 3 p 10 The two homoclinic orbits correspond to the final large period orbits on the two periodic solution branches AUTO COMMAND ACTION mkdir lrz create an empty work directory cd lrz change directory demo 1rz copy the demo files to the work directory ld lrz load the problem definition run c 1lrz 1 compute stationary solutions sv Irz save output files as b lrz s lrz d lrz run c lrz 2 s lrz compute periodic solutions the final orbit is near homoclinic Constants changed IPS IRS NICP ICP NMX NPR DS ap lrz append the output files to b Irz s Irz d lrz run c 1rz 3 s lrz compute the symmetric periodic solution branch Constants changed IRS ap 1lrz append the output files to b Irz s Irz d lrz Table 9 1 Commands for running demo Irz 91 9 2 abc The A B C Reaction This demo illustrates the computation of stationary solutions Hopf bifurcations and periodic solutions in the A B C reaction Doedel amp Heinemann 1983 ut Us Us u pi 1 uje u2 pie 1 u1 psua 9 2 U3 p3u3 p p4e 1 uy popsu with po 1 p 1 55 p4 8 and ps 0 04 The free parameter is p AUTO COMMAND ACTION mkdir abc create an empty work directory cd abc change directory demo abc copy the demo files
97. a E 44 4 13 8 commandCopyFortFiles o o e e 44 4 13 9 commandCreateG GUI A onde ee Se he Soe ye Ee 45 4 13 10commandDeleteDataFiles 0 e 45 4 13 11 commandDeleteFortFiles ae AA e 46 4 13 12 comma ndDouble s o d a s E ARE 46 4 13 13 commandlInteractiveHelp 014 a ad e ek eee De AA 47 ALTO VA command LS E AIDA ERA EA A a A 47 4 13 15 commandMoveFiles a aoaaa a do hh Bede o 48 4 13 16 commandParseConstantsFile oia we eA ee ee 48 4 13 17 commandParseDiagramAndSolutionFile 49 4 13 18 commandParseDiagramPil oidor dra A 49 4 13 19commandParseSolutionFile o o e eee eee 50 4 1320 command Plotter hs anal al A a de A AS 50 4 13 21 commandPlotter3D aaa ea as 51 4 13 22 commandQueryBranchPoint aaa ere he be a el ds 51 4 13 23 commandQueryEigenvalue a aoa 4 4 23 444 7 465444 44464 52 4 13 24 commandQueryFloquet a O A ef a 52 4 13 25 c0ommandQuery Hopf a a pa a et ae ee ae ew ky a hae R 53 4 13 26 commandQuery terations cite dl ae A ee 53 4 13 27 commandQueryLimitpoint o 54 4 13 28 commandQueryNote 44 a Re CRE ER ee 54 4 13 29 commandQuerySecondaryPeriod oa a a 00002 ee 55 4 13 30 commandQueryStepsize aos do ee A e is EES 55 4 13 31 commandRun A os a O BS pln eat 56 4 13 32 commandRunnerConfigFort2 4 es eke BP e Heo 56 4 13 33 commidnd Rammer load Name cars ee Re ee a A 57 4 13 34 commandRunner
98. a discussion of the specific application considered here is given in Doedel Keller amp Kern vez 1991b The required extended system is fully programmed here in the user supplied subroutines in obv c For the case of periodic solutions the optimality system can be generated automatically see the demo ops Consider the system uilt u t O cas 12 3 where p w1 A2 Az u1 Agu Azut with boundary conditions 1 1 ui 0 0 AN 0 12 4 The objective functional is 1 w 9 1 par Ea 0 The successive continuation equations are given by u t u t uh t zeP urA2A 12 5 wilt Meen w t 27 u1 t 1 l wt wy t where Puy oe 24241 4A3u with u1 0 0 w1 0 a Bi 0 wa 0 0 12 6 ul 1 0 w1 1 Bo 0 wa 1 0 1 3 2 2 2 lw w t 1 T So dg dt 0 k 1 1 f wie ao de o 0 ha eP e Ey A1 dt 0 ha Aer 12 4 t w t y Az 12 dt 0 12 7 y fal pra te a t TES dt 0 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 branch the adjoint 126 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 L2 n
99. a of an objective function along solution branches and successively continue such extrema in more parameters Demo opt Ordinary Differential Equations For the ODE 2 2 the program can Compute branches of stable and unstable periodic solutions and compute the Floquet mul tipliers that determine stability along these branches 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 branches of periodic solutions Branch switching is possible at branch points and at period doubling bifurcations Demos tor lor Continue folds and period doubling bifurcations in two parameters Demos plp pp3 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 Do each of the above for rotations i 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 branch of periodic solutions and successively continue s
100. ackground color to black The demo script auto 2000 demo python plotter py contains several examples of changing options in plotters 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 func tion 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 Postscript file The Configure button brings up the dialog for setting configuration options 36 AUTO gt plot pl1 Created plotter AUTO gt plot config bg black AUTO gt Figure 4 15 This example shows how a plotter is created and how the background color may be changed to black 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 Query string Meaning background The background color of the plot bifurcation_column_ defaults A set of bifurcation columns the user i
101. addition to PAR 10 Thus NICP 4 in the third run For a simple example see demo opt where a four parameter extremum is located Note that NICP 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 remain fixed 5 7 9 Internal free parameters The actual continuation scheme in AUTO may use additional free parameters that are automati cally added The simplest example is the computation of periodic solutions and rotations where AUTO automatically adds the period if not specified The computation of loci of folds Hopf bi furcations and period doublings also requires additional internal continuation parameters These will be automatically added and their indices will be greater than 10 5 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
102. all preparations are done to start homoclinic branch switching This is very similar to the technique used in Sandstede s model in Section 22 1 to find a 3 homoclinic orbit we open 2 Lin gaps until 7 3 5 while also varying A PAR 2 rn c kdv 4 h kdv 4 s 3 sv 4 BR PT TY LAB PAR 2 ss sant PAR 20 PAR 21 PAR 23 1 10 8 5 797610E 10 1 672717E 01 8 381610E 08 6 988443E 07 1 19 UZ 9 1 399137E 09 1 012493E 01 6 452744E 12 1 379764E 07 1 20 10 2 122922E 09 9 001030E 00 1 032750E 07 4 022729E 07 ab 29 EP 11 2 154196E 06 3 499999E 00 7 959776E 04 3 999453E 04 We then look for an orbit with a lt 3 and close the gap corresponding to PAR 21 for decreasing a rn c kdv 5 h kdv 5 s 4 sv 5 BR PT TY LAB PAR 1 se Acts PAR 2 PAR 21 PAR 23 1 10 12 2 579042E 00 2 154861E 06 7 659464E 04 3 829183E 04 1 13 UZ 13 2 320452E 00 3 933752E 11 1 088379E 10 1 552594E 08 1 20 EP 14 1 906119E 01 1 022044E 03 7 600151E 01 3 446967E 01 and finally close the gap corresponding to 9 PAR 23 194 rn c kdv 6 h kdv 6 s 5 sv 6 BR PT TY LAB PAR 1 PAR 2 PAR 22 PAR 23 1 23 UZ 15 2 320450E 00 2 198310E 12 1 487623E 01 4 392295E 10 1 30 16 2 320380E 00 1 004669E 09 1 027163E 01 5 060989E 07 1 51 UZ 17 2 336952E 00 2 374866E 07 3 482932E 00 1 195914E 04 1 58 UZ 18 3 080847E 00 2 673602E 12 3 500044E 00 1 934478E 10 1 60 EP
103. ameters 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 5 7 5 Folds and period doublings The continuation of folds for periodic orbits and rotations and the continuation of period doubling bifurcations require two free problem parameters plus the free period Thus one would normally set NICP 3 For example in Run 6 of demo pen where a locus of period doubling bifurcations is computed for rotations we have NICP 3 with PAR 2 PAR 3 and PAR 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 set NICP 2 and 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 NICP 2 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 or period doublings with fixed period simply set NICP 3 and specify three problem parameters not including PAR 11 5 7 6 Boundary value problems The simplest case is that of boundary value
104. and can read into memory namely the user defined function file and the AUTO constants file Section 3 1 There are two other files types that can be read in using the load command and they are the restart solution file Section 3 5 and the HomCont parameter file Section 15 2 22 Script Description demol auto The demo script from Section 4 3 demo2 auto The demo script from Section 4 5 userScript xauto The expert demo script from Figure 4 11 userScript py The loadable expert demo script from Fig ure 4 12 fullTest auto A script which uses the entire AUTO 2000 command set except for the plotting commands plotter auto A demonstration of some of the plotting capabilities of AUTO 2000 full Test auto A script which implements the tutorial from Section 7 2 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 ues in the ratio of the masses until a spec ified mass ratio is reached It then com putes
105. andLs to run the system ls command in the current directory This command will accept whatever arguments are accepted by the Unix command ls Aliases Is 47 4 13 15 commandMoveFiles Purpose Move data files to a new name Description Type command MoveFiles xxx yyy to move the data files b xxx s xxx d xxx and c xxx to b yyy s yyy d yyy and c yyy if you are using the default filename tem plates Aliases move mv 4 13 16 commandParseConstantsFile Purpose Get the current continuation constants Description Type commandParseConstantsFile xxx to get a parsed version of the constants file c xxx if you are using the default filename templates Aliases cn constantsget 48 4 13 17 commandParseDiagramAndSolutionFile Purpose Parse both bifurcation diagram and solution Description Type commandParseDiagramAndSolutionFile xxx to get a parsed version of the diagram file b xxx and solution file s xxx if you are using the default filename tem plates Aliases bt diagramandsolutionget 4 13 18 commandParseDiagramFile Purpose Parse a bifurcation diagram Description Type commandParseDiagramFile xxx to get a parsed version of the diagram file b xxx if you are using the default filename templates Aliases dg diagramget 49 4 13 19 commandParseSolutionFile Purpose Parse solution file Description
106. ants changed IRS RL1 append output files to b pp2 s pp2 d pp2 run c pp2 3 s pp2 ap pp2 3rd run periodic solutions Constants changed IRS IPS ILP append output files to b pp2 s pp2 d pp2 run c pp2 4 s pp2 ap pp2 4th run restart at a labeled periodic solu tion Constants changed IRS NTST append output files to b pp2 s pp2 d pp2 run c pp2 5 s pp2 sv lp 5th run continuation of folds Constants changed IRS IPS ISW ICP save output files as b lp s lp d lp run c pp2 6 s pp2 sv hb 6th run continuation of Hopf bifurcations Constants changed IRS save output files as b hb s hb d hb run c pp2 7 s pp2 sv hom 7th run continuation of homoclinic orbits Constants changed IRS IPS ISP save output files as b hom s hom d hom Table 9 3 Commands for running demo pp2 93 9 4 lor Starting an Orbit from Numerical Data This demo illustrates how to start the computation of a branch of periodic solutions from nu merical data obtained for example from an initial value solver As an illustrative application we consider the Lorenz equations ug p3 uz ur Uy pu Ug UU 9 4 Us UU Pots Numerical simulations with a simple initial value solver show the existence of a stable periodic orbit when p 280 p 8 3 p
107. ar PAR 2 PAR 35 PAR 36 1 27 UZ 5 6 10437E 00 6 932475E 02 6 782898E 07 8 203437E 02 indicating that a zero of the test function IPSI 1 15 This means that at D K Z 6 6104 0 069325 the homoclinic orbit to the saddle node becomes non central namely it returns to the equilibrium 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 make second one obtains BR PT TY LAB PAR 1 ya PAR 2 PAR 35 PAR 36 1 34 UZ 9 5 180323E 00 6 385506E 02 3 349720E 09 9 361957E 02 which means another non central saddle node homoclinic bifurcation occurs at D K Z 6 1808 65 068855 le 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 153 17 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 D make third starts from D Note that now NUNSTAB 1 IEQUIB i has been specified in h mtn 3 Also test functions Yy and 419 have been activated in order to monitor for non hyperbolic equilibria along the homoclinic locus We get the following output
108. as a convenience and it is not very efficient Demos pdl pd2 Compute curves of stationary solutions to 2 3 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 pdl pd2 In connection with periodic waves note that 2 4 is just a special case of 2 2 and that its fixed point analysis is a special case of 2 1 One advantage of the built in capacity of AUTO to deal with problem 2 3 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
109. ative 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 make first 152 The file mtn c contains approximate parameter values K PAR 1 6 0 Z PAR 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 Ty PAR 11 1046 178 Since a homoclinic orbit to a saddle node is being followed we have also made the choices IEQUIB 2 NUNSTAB O NSTAB 1 in h mtn 1 The two test functions 415 and 416 to detect non central saddle node homoclinic orbits are also activated which must be specified in three ways Firstly in h mtn 1 NPSI is set to 2 and the active test functions IPSI I I 1 2 are chosen as 15 and 16 This sets up the monitoring of these test functions Secondly in c mtn 1 user defined functions NUZR 2 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 NICP ICP 1I I 1 NICP Among the output there is a line BR PT TY LAB PAR 1 o
110. bifurcation point Accordingly the function of this loop and conditional is to examine every solution in the fort 8 file and run the following commands if the solution is a bifurcation point The next line is ch IRS solution Label which changes the in memory version of the AUTO 2000 constants file to set IRS see Section 5 8 5 equal to the label of the bifurcation point We then use ch ISW 1 to change the AUTO 2000 constant ISW to 1 which indicates a branch switch see Section 5 8 3 We then use a run command to perform the calculation of the bifurcating branch and then append the data to the s bvp b bvp and d bvp files with the ap bvp command Section 4 13 1 in the reference In addition as can be seen in Figure 4 10 the character is the Python com ment character When the Python interpretor encounters a character it ignores everything from that character to the end of the line Finally we us ch DS pr DS to change the AUTO 2000 initial step size from positive to negative which allows us to compute the bifurcating branch in the other direction see Sec tion 5 5 1 Running the AUTO 2000 calculation with the run command and appending the data the appropriate files with the ap bvp command completes the body of the loop for solution in data if solution Type name BP ch IRS solution Label ch ISW 1 Compute forward run ap bvp Compute back ch DS pr D
111. bit 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 asymmetry in the differential equations Koper 1994 note also that the equilibrium stored as PAR 12 PAR 13 PAR 14 in d 9 approaches the origin as we approach the figure of eight homoclinic 18 4 Three Parameter Continuation We now consider curves in three parameters of each of the 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 make tenth Note that ITWIST 1 in h 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 Yg where the neutrally twisted homoclinic orbit collides with the saddle node curve BR PT TY LAB PAR 1 was PAR 2 PAR 3 PAR 29 1 28 UZ 14 1 282702E 01 2 519325E 00 5 744770E 01 4 347113E 09 The other detected inclination flip at label 8 in s 3 is continued similarly make eleventh giving among its output another codim 3 saddle node inclination flip point BR PT TY LAB PAR 1 rae PAR 2 PAR 3 PAR 29 1 27 UZ 14 1 535420E 01 2 458100E 00 1 171705E 00 1 933188E 07 Output beyond both of these codim 3 points is spurious and both computations end in an MX point no convergence To continue the
112. blem parameter The equation is Ou Pu an a 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 pvl c In the second run the continuation parameter is pj 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 T p u 1 u AUTO COMMAND ACTION mkdir pd1 create an empty work directory cd pdl change directory demo pd1 copy the demo files to the work directory run c pdi1 1 time integration towards stationary state sv 1 save output files as b 1 s 1 d 1 run c pd1 2 s 1 continuation of stationary states read restart data from s 1 constants changed IPS IRS ICP etc sv 2 save output files as b 2 s 2 d 2
113. bles as illustrated in Table 15 1 and Table 15 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 7 4 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 COMMAND ACTION ld san load the problem defition run c san 1 h san 1 get the HomCont constants file and run AUTO HomCont su 6 save output files as b 6 s 6 d 6 Table 15 1 An example of AUTO Commands COMMAND ACTION run c san 9 h san 9 s 6 get the HomCont constants file and run AUTO HomCont restart solution read from s 6 ap 6 append output files to b 6 s 6 d 6 Table 15 2 Another example of AUTO Commands 144 Chapter 16 HomCont Demo san 16 1 Sandstede s Model Consider the system Sandstede 1995a t axt by azx ji az 2 2 31 y ba ay ibs faxry f az 2y 16 1 2 cztpxt yrytaB a 1 2 y as given in the file san c Choosing the constants appearing in 16 1 appropriately allows for computing inclination and orbit flips as well as
114. blocks of code are not defined by some delimiter such as in C but by indentation In Figure 4 7 the commands plot bvp and wait 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 if solution Type name BP is a Python condi 27 copydemo bvp 1d bvp run sv bvp 1d s bvp data sl bvp ch NTST 50 for solution in data if solution Type name BP ch IRS solution Label ch ISW 1 Compute forward run ap bvp Compute back ch DS pr DS run ap bvp plot bvp wait Figure 4 7 This Figure shows a more complex AUTO 2000 CLUI script The source for this script can be found in AUTO_DIR demos python demo2 auto copydemo bvp 1d bvp run Figure 4 8 The first part of the complex AUTO 2000 CLUI script sv bvp 1d s bvp data sl bvp ch NTST 50 Figure 4 9 The second part of the complex AUTO 2000 CLUI script 28 tional It examines the contents of the variable solution which is one of the entries in the array of solutions data and checks to see if the condition solution Type name BP holds For parsed fort 8 files Type name BP corresponds to a
115. cal starting solutions see demos ab and frc For starting from unlabeled numerical data see the fc command Section A 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 q xxx see also the Or and AR commands in Section A Various AUTO constants can be modified when restarting 5 8 6 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 ut f ub p with detection of Hopf bifurcations The sign of PT in fort 7 indicates stability is stable is unstable Demo dd2 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 con tinuation 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 70 for the detection of stationary solutions which can then be entered in the user supplied subroutine s
116. cation 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 2000 CLUI with the Unix command auto and once the AUTO 2000 CLUI is running we use the command from userScript import to import the file userScript py into the AUTO 2000 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 2000 CLUI for example by typing myRun bvp 4 7 Bifurcation Diagram Files Using the commandParseDiagramFile command Section 4 13 18 in the reference the user can parse and read into memory an AUTO 2000 bifurcation diagram file For example the com mand commandParseDiagramFile ab would parse the file b ab if you are using the standard filename translations from Table 4 3 and return an object which encapsulates the bifurcation diagram in an easy to use form The object returned by the commandParseDiagramFile is a list of all of the solutions in the appropriate bifurcation diagram 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 13 prints out the label of the first solutio
117. ce ts ee EY brf Finite Differences in Space ileso a a bru Euler Time Integration the Brusselator 11 1 11 2 11 3 11 4 11 5 11 6 AUTO Demos Optimization 12 1 opt A Model Algebraic Optimization Problem 12 2 ops Optimization of Periodic Solutions errar ee ee 12 3 obv Optimization for a BVP plo alta ad AA AA 13 AUTO Demos Connecting orbits 13 1 fsh A Saddle Node Connection A MAA DR Oe Re 13 2 nag A Saddle Saddle Connection 2 4 4 5 244 24 a 13 3 stw Continuation of Sharp Traveling Waves a 14 AUTO Demos Miscellaneous 14 1 pvl Use of the Subroutine pvls 6 we Yaw Maas be eee de ee eS 14 2 ext Spurious Solutions to BVB dra ai Seek ai dP lle ee as 14 3 tim A Test Problem for Timing AUTO o 9 38 0 soe aa SS als 15 HomCont L5ek Introduction 008 28 3048 ei A nt a A a eee ah als 15 2 HomCont Files and Subroutines 00 000 0 0000 a 15 3 HomCont Constants 2 00 0 0 0 00 000 ee ee 15 3 1 15 3 2 15 3 3 15 3 4 15 3 5 15 3 6 15 3 7 15 3 8 NUNSTAB 20008 de Sy en de ete ee e Re le thes ee ee es a EO SO ee NOTAB 2008 AN eA Gee A eS eh A Ee a TEQUIB Ts gs ob ot os Ay eit ite Pes A eh ta bay aes ae th a e EEWIS E yay ah ee eae en ey oh Jo ay ok ee BL ee A ee feel He e STARTS act Sinus GM as os ti Aw ale rats de Doth dw ties GP ve iad NREV AREV tet 20 he eos od anda A o ae tn y BOG sk BR ee eee oe o
118. ching which is an implementation of Lin s method Lin 1990 Sandstede 1993 C 2001 as described in Oldeman et al 2001 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 16 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 22 1 Branch switching at an inclination flip in Sand stede s model Consider the system Sandstede 1995a t az by ar azz 2 3r y bz ay 3 a bar ay az2y 22 1 2 czetprt3rz a r 1 2 y as given in the file sib c where for simplicity we have set 0 1 and y 3 We study an inclination flip that exists for a 0 375 b 0 625 and c 0 75 This 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
119. configuration that is none of the test functions 4 for i 11 12 13 14 is zero or close to zero see Section 15 6 Finally the values of the NPSI activated test functions are written 15 6 Test Functions Codimension two homoclinic orbits are detected along branches of codim 1 homoclinics by lo cating zeroes of certain test functions w The test functions that are switched on during any continuation are given by the choice of the labels i and are specified by the parameters NPSI 1 IPSI I I 1 NPSI in h xxx Here NPSI gives the number of activated test func tions and IPSI 1 IPSI NPSI give the labels of the test functions numbers between 1 and 16 A zero of each labeled test function defines a certain codimension two homoclinic singular ity specified as follows The notation used for eigenvalues is the same as that in Champneys amp Kuznetsov 1994 or Champneys et al 1996 i 1 Resonant eigenvalues neutral saddle y A4 i 2 Double real leading stable eigenvalues saddle to saddle focus transition 41 fa 141 i 3 Double real leading unstable eigenvalues saddle to saddle focus transition i Ao i 4 Neutral saddle saddle focus or bi focus includes i 1 Re 111 Re A1 1 5 Neutrally divergent saddle focus stable eigenvalues complex Re A1 Re u1 Re u2 i 6 Neutrally divergent saddle focus unstable eigenvalues complex Re 11 Re A1
120. ctories A 0 1 Basic commands Or CR Osv Cap Type r xxx to run 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 Cr xxx yyy to run AUTO with equations file xxx c and restart data file s yyy AUTO constants must be in c xxx Type r xxx yyy zzz to run AUTO with equations file xxx c restart data file s yyy and constants file c zzz The command CR xxx is equivalent to the command Cr xxx above Type CR xxx i to run AUTO with equations file xxx c 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 c constants file c xxx i and restart data file s yyy Type sv xxx to save the output files fort 7 fort 8 fort 9 as b xxx s xxx d xxx respectively Existing files by these names will be deleted 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 196 A 0 2 Plotting commands Cp Type Cp xxx to run the graphics program PLAUT See Chapter B for the graphical inspection of the data files b xxx and s xxx Type p to run the graphics program PLAUT for the graphical inspection of the output files fort 7 and fort 8 Ops Type Cps fig x to convert a saved PLAUT figure fig x from compact PLOT10 format to PostScript format The
121. data dg ab reads in the bifurcation diagram and puts it into the variable data The second command print data 0 prints out all of the data about the first solution in the list The third command print data 0 LAB prints out the label of the first point Query string Meaning 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 branch 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 2000 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 7 Bifurcation to invarient torus ODE TR 8 Normal begin or end EP 9 Abnormal termination MX 9 Table 4 5 This table shows the the various types of points that can be in solution and bifurcation diagram files with their short names and numbers 33 name translations from Table 4 3 and return an object which encapsulate
122. 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 b 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 205 C 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 C 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 c C 2 12 Help button This pulldown menu contains the items AUTO constants for help on AUTO constants and User Manual for viewing the user manual i e this document C 3 Using the GUI AUTO commands are described in Section A and illustrated in the demos In Table C 1 we list the main AUTO commands together with the corresponding GUI button
123. def myRun demo begins the function definition and creates a function named myRun which takes one argument demo The rest of the script is the same except that it has been in dented to indicate that it is part of the function definition and all occurrences of string bvp have been replaced with the variable demo 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 AUTOclui import def myRun demo copydemo demo 1d demo run sv demo 1d s demo data sl demo ch NTST 50 for solution in data if solution Type name BP ch IRS solution Label ch ISW 1 Compute forward run ap demo Compute back ch DS pr DS run ap demo plot demo wait myRun bvp Figure 4 11 This Figure shows a complex AUTO 2000 CLUI script written as a function The source for this script can be found in AUTO_DIR demos python userScript xauto While the script in Figure 4 11 is only slightly different then the one showed in Figure 4 7 it is much more powerful Not only can it be used as a script for running any demo by modifying 30 the last line it can be read back into the interactive mode of the AUTO 2000 CLUI and used to create a new command as in Figure 4 12 First we create a file called userScript py which contains the script from Figure 4 11 with one minor modifi
124. dot ew eae ee he ee ow eS 3 37 Aremments of sStpt acrins aea ir Bi Se WINS Sf SIA amp SI Sale wo Rae 3 4 User Supplied Derivatives Gu see oo el ee a DAG we LES 35 Output Eiles say atoms edad Sobral a ok bre a ee dees Gy ee Se ro a 4 Command Line User Interface 4 1 Typographical Conventions 0 ay ani ye Bone a A AE He 4 2 General Overview 2 8 8 2 tue O sie et ee Ba a eh a First Mee AMIS ll ad ese Gnd do e eee Gone ee ves Ge ee ee Ee ae Ee eS AA CTS oY oh ta et St ae te tn A A A ae ALS Record Example saris S amp B ate Gi a aye A e Anh edt ert Y 4 6 Extending the AUTO 2000 CLUI i eke AR RRR WES 4 7 Bifurcation Diagram Blesa dA ep add sd erate ow DS ets Dt 28 TOMO Pes de ah cline a hte dn tin DS GS EIA at ep ee BLE SPD ae AO Fherattore File 10 sebawa eG BR dado OR da oh De Se BS oe amp os 4 4 10 Two Dimensional Plotting Tool sa be hc ee ON as eS 4 11 Three Dimensional Plotting Tool 3 4 ax 24 3 0 eyes A DAG 4 12 Quick Refer en e ss eth seh Ca GN a OO e AS ay de le es ee et vi 39 4 13 Reference 254 St Sah el oe de ee Ds FI Be Gl 41 4 13 1 commandAppend ees o gt nl rs das Jee ROR e dana 41 413 2 command Cato nabo a si o E e 1 E e det de e ok 41 AL SCOMMIMANG GOs A ian is dt Ad Ses e Sey ett e ld Ss e 42 A Lacio command lea lt a ae a a sl os de 42 4 13 5 commandCopyAndLoadDemo okie da a a As 43 4 13 6 commandCopyDataFiles o e eee ee ee 43 4 13 7 commandCopyDemo ls ie de e de A a
125. e 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 strongly recommended because the Jacobian is used extensively for calculating the linearization at the equilibria and hence for evaluating boundary 137 conditions and certain test functions However note that JAC 1 does not necessarily mean that auto will use the analytically specified Jacobian for continuation 15 3 HomCont Constants An example for the additional file h xxx is listed below 12111 NUNSTAB NSTAB IEQUIB ITWIST ISTART 0 NREV 1 IREV I I 1 NREV 1 NFIXED 1 IFIXED I I 1 NFIXED 13 1 NPSI 1 IPSI 1 I 1 NPSI 9 10 13 The constants specified in h xxx have the following meaning 15 3 1 NUNSTAB Number of unstable eigenvalues of the left hand equilibrium the equilibrium approached by the orbit as t gt 00 15 3 2 NSTAB Number of stable eigenvalues of the right hand equilibrium the equilibrium approached by the orbit as t 00 15 3 3 IEQUIB IEQUIB 0 Homoclinic orbits to hyperbolic equilibria the equilibrium is specified explicitly in pvls and stored in PAR 11 1I I 1 NDIM TEQUIB 1 Homoclinic orbits to hyperbolic equilibria the equilibrium is solved for during continuation Initial values for the equilibrium are stored in PAR 11 I I 1 NDIM in stpnt IEQUIB 2 Homoclinic orbits to a saddle node initial values for the equilibrium ar
126. e active optimality functionals namely for A2 A3 and T respectively with corresponding variables T2 73 and To respectively These should be set in the first run with IPS 15 and remain unchanged in all subsequent runs Index 3 2 11 22 22 23 31 Variable Az Ao T T2 A2 Ag T Table 12 3 Run 5 file c ops 5 In Run 5 the parameter a which has been replaced by A2 remains fixed and nonzero The variable T monitors the value of the optimality functional associated with A2 The zero of Ta located in this run signals an extremum with respect to Ao Index 3 2 111 22 23 31 Variable Az Ao Ar T A2 As IT Table 12 4 Run 6 file c ops 6 In Run 6 72 which has been replaced by A remains zero Note that To 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 To and 73 or equivalently 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 A4 and its corresponding optimality variable T4 are not used in this demo Also A is used in the last run only and its corresponding optimality variable 7 is never used 124 AUTO COMMAND ACTION mkdir ops create an empty work d
127. e 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 6 2 5 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 branch will be discontinued as soon as the maximum number of iterations ITNW is reached This choice is not recommended Demo tim IADS 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 non convergence will be signalled The strongly recommended value is IADS 1 which is used in almost all demos 64 5 5 5 NTHL By default the pseudo arclength stepsize includes all state variables or state functions and all free parameters Under certain circumstances one may want to modify the weight accorded to individual parameters in the definition of stepsize For this purpose NTHL defines the number of parameters whose weight is to be modified If NTHL 0 then all weights will have default value 1 0 If NTHL gt 0 then one must enter NTHL pairs Parameter Index W
128. e da Es 64 od LADS 253 85 ain do a A 2 A A OR 64 Moro oN THEA cao E rl DP O ef ke AR E AA A ol A 65 5 5 0 NTHU a A ee Re AS 65 Diagrams TES bi ma Sat y ed de ps A 65 56l AA EE AO 65 9 0 2 A A A theme a ake Mees oes et he Se ee amp Oe gk ka ase 65 Orv REL pea yaw gee en og he Pog eh ee ab a Ae a eA ahh 66 OL O a tagcat he Se Se psa ae te gE ae pe Se A te a ee all oh edo a 66 500 O O O aed ost 66 Free Parameters taa ta pie es Gee de ak AE ge Set a 66 Be felis NPCP ie LEPE A A eA ee Bed o ae ashes 66 I2 Eed ponts es Boe ama Pana Aa re Ee a a ES 66 5 7 3 Periodic solutions and rotations 2 0004 66 5 7 4 Folds and Hopf bifurcations eee hee ea a ee e a 67 5 7 5 Folds and period doublings a a e e 67 5 7 6 Boundary value problems oa aaa fea RRs Rae oe 67 5 7 7 Boundary value folds Sy erty 76 a AA 67 5 7 8 Optimization problems eme Ava e aa ae 68 5 7 9 Internal free parameters cad le DAA ES 68 5 7 10 Parameter overspecificationy 4 aaa 6a eke oe RR OE wee 68 Computation Constants us ita 2526 SS ach a Ee A SE eR ee A SS 69 Sila MEP ew OI A ta da ks edd ered e 69 A E A ane A eg 69 A EE 69 DIA MABE a a E ld aan y Ge da A A 70 DG IRS e e ia ee Ds eS 70 Bor EPS olas dlrs ita de oa de gra ky goed 70 5 9 Ottp l Control tasa oe Ba Se ae fe Ae a oa Js Ge ais hese er e A gt INER td Ge ee lL Da A GA e 9972 CRED ss gee pobre o eho hea ge Se eg ae ga a D953 PIPET z
129. e gap is opened and closed in the continuation process This is depicted in Figure 22 5 We are now in a position to continue each of these orbits Column 6 4 006 03 O EE E e 0 008 00 2 008 02 1 008 02 Column 0 Figure 22 5 A plot of e as a function of T during our computation of Shil nikov type two homoclinic orbits Each zero corresponds to a different orbit as a normal homoclinic orbit by setting ISTART 1 and ITWIST 0 We leave this as an exercise to the reader 192 22 3 Branch switching 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 om o i N2 m 15 br ars 7 5 frr 0 22 4 It represents solitary wave solutions r x at r gt 0 as x 00 of the 5th order PDE 2 m 15 wee Of yxa SIT y 2T Tag TT exg 0 where a is the wave speed The ODE corresponds to a Hamiltonian system with Hamiltonian 1 1 1 15 1 H z zot pida 5 42 mn ps4 Ln and 2 m 2 1 QET QST MEF br rr a System 22 4 is also reversible under the transformation Lee t q1 q2 P1 p2 E q q2 p1 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
130. e 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 69 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 or a period doubling or torus bifurcation then a locus of such points will be computed An additional free parameter must be specified for such continuations see also Section 5 7 5 8 4 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 branches will not be computed If MXBF is positive then the bifurcating branches of the first MXBF branch points will be traced out in both directions If MXBF is negative then the bifurcating branches of the first MXBF branch points will be traced out in only one direction 5 8 5 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 subroutine stpnt see also Section 3 3 For representative examples of analyti
131. e 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 T I NDIM 1 2 NDIM right hand equilibrium IEQUIB 2 Heteroclinic orbits to hyperbolic equilibria the equilibria are solved for during continuation Initial values are specified in stpnt and stored in PAR 11 I I 1 NDIM left hand equilibrium PAR 11 I I NDIM 1 2 NDIM right hand equilibrium 138 15 3 4 ITWIST ITWIST 0 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 ITWIST 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 15 7 for more details 15 3 5 ISTART ISTART 1 This option is obsolete in the current version It may be used as a flag that a solution is to be restarted from a previously computed point or from numerical data converted into AUTO format using us In this case IRS gt O ISTART 2 If IRS 0 an explicit solution must be specified in the subroutine stpnt in the usual format ISTART 3 The
132. ee curves each of which is a plot of time versus the value of the first state variable If we want a different plot say the values of the two state variables plotted against each other we use the two remaining widgets in the control panel labeled X and Y For example if change the value of X from t to 0 and the value of Y from 0 to 1 we get a phase plot of the period solutions don t forget to hit the return key when you are done modifying each value This plot is shown in Figure 7 2 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 Further information on the plotting tool can be found in Section 4 10 AUTO COMMAND ACTION plot ab run AUTO to graph the contents of b ab and s ab Table 7 8 Command for plotting the files b ab and s ab 7 8 Following Folds and Hopf Bifurcations The commands in Table 7 9 will execute the remaining runs of demo ab Here as in later demos some of the AUTO constants that have been changed between runs are indicated in the Table AUTO COMMAND ACTION run c ab 3 s ab compute a locus of folds with changes from c ab 1 IRS NICP ICP ISW DSMAX sv 2p save ou
133. eight with each pair on a separate line 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 NTHL 1 with on a separate line the pair 11 0 0 Most demos that compute periodic solutions use this option see for example demo ab 5 5 6 NTHU 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 NTHU defines the number of states whose weight is to be modified If NTHU 0 then all weights will have default value 1 0 If NTHU gt 0 then one must enter NTHU pairs State Index Weight with each pair on a separate line At present none of the demos use this option 5 6 Diagram Limits There are three ways to limit the computation of a branch 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 computa tional window otherwise the computation will stop immediately at the starting point By specifying the maximum number of steps NMX By specifying a negative parameter index in the list associated with the constant NUZR see Section 5 9 4 5 6 1 NMX The maximum number of steps
134. em ut U2 uy 2 uuu 1 u 10 10 with boundary conditions u 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 In 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 run c spb 1 lst run homotopy from A 0 to 1 ev 1 save output files as b 1 s 1 d 1 run c spb 2 s 1 2nd run let e tend to zero restart from s l constants changed IRS ICP 1 NTST DS sv 2 save the output files as b 2 s 2 d 2 run c spb 3 s 2 3rd run continuation in y e 0 001 restart from s 2 Constants changed IRS ICP 1 RLO ITNW EPSL EPSU NUZR sv 3 save the output files as b 3 s 3 d 3 Table 10 7 Commands for running demo spb 111 10 8 ezp Complex Bifurcation in a BVP This demo illustrates the computation of a solution branch to the the complex boundary value problem ee 10 11 with boundary conditions u 0 0 u 1 0 Here uw and uz are all
135. f JAC 1 then derivatives must be given This may be necessary for sensitive problems and is recommended for computations in which AUTO generates an extended system Examples of user supplied derivatives can be found in demos dd2 int plp opt and ops 3 5 Output Files AUTO writes four output files fort 6 A summary of the computation is written in fort 6 which usually corresponds to the window in which AUTO is run Only special labeled solution points are noted namely those listed in Table 3 1 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 branch normal termination MX 9 Abnormal termination no convergence Table 3 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 contain
136. fig 1 ps pr Type pr fig 1 to convert a PLAUT file fig 1 to PostScript format and to print the resulting file fig 1 ps 202 Appendix C Graphical User Interface C 1 General Overview Please note as of July 30 2002 the GUI is being updated so the documentation is this chapter is not being actively maintained The old GUI is provided with this release of AUTO but it is unsupported and may not be included in future releases The AUTO 97 graphical user interface GUI is a tool for creating and editing equations files and constants files see Section 3 1 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 A 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 2000 or in any of its subdirectories Then type auto or its abbreviation a Here we assume that the AUTO aliases have been activated see Section 1 2 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 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 b
137. files as b pen s pen d pen run c pen 2 s pen ap pen a branch of period doubled and out of phase rotations Constants changed IPS NTST ISW NMX append output files tp b pen s pen d pen run c pen 3 s pen ap pen a secondary bifurcating branch without bifurcation detection Constants changed IRS ISP append output files to b pen s pen d pen run c pen 4 s pen ap pen another secondary bifurcating branch without bifurcation detection Constants changed IRS append output files to b pen s pen d pen run c pen 5 s pen svt generate starting data for period doubling continuation Constants changed IRS ICP ICP ISW NMX save output files as b t s t d t run c pen 6 s t sv pd compute a locus of period doubling bi furcations restart from s t Constants changed IRS save output files as b pd s pd d pd Table 9 10 Commands for running demo pen 101 9 11 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 ut a uz h u UY U U2 U3 9 13 Us P u where 1 h x ait 5 ao a1 ja 1 2 1 and where we take 8 14 3 ag 1 7 a 2 7
138. 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 5 7 The following restrictions apply The maximum number of parameters NPARX in auto 2000 src auto_c h has pre defined value NPARX 36 NPARX should not normally be increased and it should never be de creased Any increase of NPARX must be followed by recompilation of AUTO Generally one should only use PAR 1 PAR 9 for equation parameters as AUTO may need the other components internally 6 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 compu tations with ISW 2 However the user has significant 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 problem dependent Generally DSMIN should not be taken too small in
139. fort 9 j7 To use the SGI jot editor to edit the output file fort 7 j8 To use the SGI jot editor to edit the output file fort 8 j9 To use the SGI jot editor to edit the output file fort 9 A 0 6 File maintenance lb Type Clb to run an interactive utility program for listing deleting and relabeling solutions in the output files fort 7 and fort 8 The original files are backed up as fort 7 and fort 8 Type Clb xxx to list delete and relabel solutions in the data files b xxx and s xxx The original files are backed up as b xxx and s xxx Type Clb xxx yyy 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 198 fc 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 5 3 1 and 5 3 2 are used to define the new mesh For examples of using the fc command see demos lor and pen 94to97 Type 94t097 xxx to convert an old AUTO 94 data file s xxx to new AUTO 97 format The original file is backed up as s xxx This conversion is only necessary for files from early versions of AUTO 94 A 0 7 HomCont commands Ch
140. gular Perturbation Theory and Applications Springer Verlag 210 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 B K 2001 Homoclinic branch switching a numerical im plementation of Lin s method http www enm bris ac uk research reports 2001r11 ps gz Applied Nonlinear Mathematics Research Report 2001 11 University of Bristol ac cepted by Int J Bifurcation and Chaos Rodriguez 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 Sandstede B 1995a Constructing dynamical 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 compu
141. h aoi NE TAED IETREDY Cuca Slat eae ai AS OMe a Oa oh Bate ae NESI TRST Fader ate eatin ase a ema a SOAR Ge ae da 15 4 Restrictions on HomCont Constantes 140 15 5 Restrictions on the Use of PAR 20 Gee RE a DAS 141 15 6 Test Functions myni ane o a bea hak a AR Oe Ge AAA 141 15 7 Starting Strategies cn ees a s a a o ok a GR de A 142 15 8 Notes on Running HomCont Demos dete a a a 144 16 HomCont Demo san 145 16 1 Sandstede s Model Ne e e ia sol o 145 16 2 Inclinati n Flip 208 ee ae ks pe a ee te t e ee nr aca 2 145 16 3 Non orientable Resonant Eigenvalues o 0000 eee eee 147 T64 OTDICEl Ds ara o tag eso er AA ANS As e 147 16 5 Detailed AUTO Commands saca as ers Vela Deir ar 149 17 HomCont Demo mtn 152 17 1 A Predator Prey Model with Immigration 0 152 17 2 Continuation of Central Saddle Node Homoclinics 152 17 3 Switching between Saddle Node and Saddle Homoclinic Orbits 2 154 17 4 Three Parameter Continuation res P4 a a ee 155 17 5 Detailed AUTO Commands 5 6 4 6 dr a a ad da 156 18 HomCont Demo kpr 159 18 1 Koper s Extended Van der Pol Model 159 18 2 The Primary Branch of Homoclinics ut o BNE a a eS 159 18 3 More Accuracy and Saddle Node Homoclinic Orbits 163 18 4 Three Parameter Continuation 0 ee 166 18 5 Detailed AUTO Commands raso Oa ae a Se ey we eS Re eS 167 19 HomCo
142. hand and right hand end points This can be seen by plotting the solution corresponding to Label 13 using t vs x coordinate 0 as depicted in Figure 22 1 b Hence in order to continue this as a real homoclinic we have to give HomCont special instruc tions 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 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 u PAR 3 PAR 5 to get the desired value c 2 0 rn c sib 4 h sib shift s 3 sv 4 BR PT TY LAB PAR 3 L2 NORM ar PAR 5 3 15 EP 14 2 000000E 00 4 018899F 01 2 661459F 09 The output is saved in the files b 4 s 4 and d 4 Note that PAR 5 remains zero which is exactly what we expect 185 Next we want to add a solution to the adjoint equation to this solution This is achieved by making the change ITWIST 1 saved in h sib twist Also we set ISTART to 1 to tell HomCont that it is should not try to shift the orbit anymore rn c sib 5 h sib twist s 4 sv 5 or alternatively Cec IRS 14 cc ICP 5 8 cc NMX 2 chc ITWIST 1 chc ISTART 1 rn s 4 sv 5 where chc means change HomCont
143. he wave speed This gives the first order system 13 1 u 2 ua z us 2 cu2 Hua 2 ee Its fixed point 0 0 has two positive eigenvalues when c gt 2 The other fixed point 1 0 is a saddle point A branch 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 AUTO COMMAND ACTION mkdir fsh create an empty work directory cd fsh change directory demo fsh copy the demo files to the work directory run c fsh 1 continuation in the period T with c fixed no phase condition sv 0 save output files as b 0 s 0 d 0 run c fsh 2 s 0 continuation in c and T with active phase condition Constants changed IRS ICP NINT DS sv fsh save output files as b fsh s
144. hell or the tcsh shell for other shells you should modify the commands appropriately 12 Chapter 2 Overview of Capabilities 2 1 Summary AUTO can do a limited bifurcation analysis of algebraic systems f u p 0 JE Ju E R 2 1 and of systems of ordinary differential equation ODEs of the form u t F gt f u t p JG ul R 2 2 Here p denotes one or more free parameters It can also do certain stationary solution and wave calculations for the partial differential equation PDE Up Duge f u p C Jul ER 2 3 where D denotes a diagonal matrix of diffusion constants The basic algorithms used in the package as well as related algorithms can be found in Keller 1977 Keller 1986 Doedel Keller amp Kern vez 1991a Doedel Keller amp Kern vez 19910 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 the program can Compute solution branches Demo ab Run 1 Locate branch points and automatically compute bifurcating branches Demo pp2 Run 1 Locate Hopf bifurcation points and continue these in two parameters Demo ab Runs 1 and 5 13 2 3 Locate folds limit points and continue these in two parameters Demo ab Runs 1 3 4 Do each of the above for fixed points of the discrete dynamical system u f u p Demo dd2 Find extrem
145. his 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 aw g by x 2 y y Pr y yla y pa which has the asymptotically stable solution sin 8t y cos Gt We couple this oscillator to the Fitzhugh Nagumo equations 1 Y F v w e w vw dw b rsin 6t 9 6 by replacing sin Gt by x Above F v v v a 1 v and a b e 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 branch with r 0 2 and varying is computed in the second run For detailed results see Alexander Doedel amp Othmer 1990 AUTO COMMAND ACTION mkdir fre create an empty work directory cd frc change directory demo frc copy the demo files to the work directory ld fro load the problem definition run c frc 1 homotopy to r 0 2 sv 0 save output files as b 0 s 0 d 0 run c frc 2 s 0 compute solution branch restart from s 0 Constants changed IRS ICP 1 NTST NMX DS DSMAX sv frc save output files as b frc s frc d frc Table 9 5 Commands for running demo frc 95 9 6 ppp Continuation of Hopf Bifurcations This demo illustrates the continuation of Hopf bifurcations in
146. his is a guide to the software package AUTO for continuation and bifurcation problems in ordinary differential equations Earlier versions of AUTO were described in Doedel 1981 Doedel amp Kern vez 1986a Doedel amp Wang 1995 Wang amp Doedel 1995 For a description of the basic algorithms see Doedel Keller amp Kern vez 1991a Doedel Keller amp Kern vez 19910 This version of AUTO incorporates the HomCont algorithms of Champneys amp Kuznetsov 1994 Champneys Kuznetsov amp Sandstede 1996 for the bifurcation analysis of homoclinic orbits The graphical user interface was written by Wang 1994 The Floquet multiplier algorithms were written by Fairgrieve 1994 Fairgrieve amp Jepson 1991 Acknowledgments The first author is much indebted to H B Keller of the California Institute of Technology for his inspiration 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 and Hans Othmer University of Utah 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 and the pendula animation program An earlier graphica
147. homotopy approach is used for starting see Section 15 7 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 N N 1 2 3 Homoclinic branch switching this description is for reference only and we refer to Chapter 22 to see how this can be used in actual practice and to Oldeman Champneys amp B 2001 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 29 that is furthest from the equilibrium Then follow N 1 shifted copies of the orbit which travel from the point Xo back to the point zp The last part Uy goes from the point o 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 NPARX NDIM NPARX 1 If ITWIST 1 and this was also the case in the preceding run then a copy of the adjoint vector W at xy is stored at the parameters par NPARX NDIM 2 NPARX NDIM 1 and Lin s method can be used to do homoclinic branch switching To be more precise the individual parts u and u 1 are at distances away from each
148. iagramandsolu tionget Parse both bifurcation dia gram and solution commandParseDiagramFile dg diagramget Parse a bifurcation diagram commandParseSolutionFile sl solutionget Parse solution file commandPlotter p2 pl plot 2D plotting of data commandPlotter3D plot3 p3 3D plotting of data commandQueryBranchPoint br bp branchpoint Print the branch point function commandQuery Eigenvalue eigenvalue ev eg Print eigenvalues of Jaco bian algebraic case commandQueryFloquet fl floquet Print the Floquet multipli ers commandQuery Hopf hb hp hopf lp Print the value of the Hopf function 39 commandQuerylterations iterations it Print the number of Newton interations commandQueryLimitpoint Im limitpoint Print the value of the limit point function commandQueryNote nt note Print notes in info file commandQuerySecondaryPeriod sc secondaryperiod Sp Print value of secondary periodic bif fen commandQueryStepsize ss stepsize st Print continuation step sizes commandRun r run rn Run AUTO commandRunnerConfigFort2 changeconstant cc ch Modify continuation con stants commandRunnerLoadName Id load Load files into the AUTO runner commandRunnerPrintFort2 pe pr printconstant Print continuation parame ters
149. ic solutions is vertical and along it the period increases monotonically The branch terminates in a homoclinic orbit containing the saddle point 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 1d phs load the problem definition run c phs 1 detect Hopf bifurcation sv phs save output files as b phs s phs d phs run c phs 2 s phs compute periodic solutions Constants changed IRS IPS NPR ap phs append output files to b phs s phs d phs Table 9 12 Commands for running demo phs 103 9 13 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 ui pxu 1 u1 urua pi 1 e 9 15 Uy U2 p4Uu1u2 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
150. in mtn dat to create s dat continue saddle node homoclinic orbit save output files as b 1 s 1 d 1 run c mtn 2 h mtn 2 s 1 ap 1 continue in opposite direction restart from s 1 append output files to b 1 s 1 d 1 run c mtn 3 h mtn 3 s 1 ap 1 switch to saddle homoclinic orbit restart from s 1 append output files to b 1 s 1 d 1 run c mtn 4 h min 4 s 1 su 47 continue in reverse direction restart from s 1 save output files as b 4 s 4 d 4 run e tmin 5 h mitn 55 T ap 1 other saddle homoclinic orbit branch restart from s 1 append output files to b s 1 d 1 runte mtaco h mtn 6 s 1 su 6 3 parameter non central saddle node homoclinic save output files as b 6 s 6 d 6 Table 17 1 Detailed AUTO Commands for running demo mtn 156 O 0 00 0 07 0 14 0 21 0 28 0 35 Z Figure 17 1 Parametric portrait of the predator prey system 1 5 12 K 9 6 Z l tz 0 06 0 08 0 10 Z Figure 17 2 Approximation by a large period cycle 157 d_0 0 012 Figure 17 3 Projection onto the K Do plane of the three parameter curve of non central saddle node homoclinic orbit 158 Chapter 18 HomCont Demo kpr 18 1 Koper s Extended Van der Pol Model The equation file kpr c contains the equations t e ky 2 32 4 y
151. ines the maximum message length is the system defined maximum string literal length 207 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 Aronson D G 1980 Density dependent reaction diffusion systems in Dynamics and Modelling of Reactive Systems Academic Press pp 161 176 Bai F amp Champneys A 1996 Numerical detection and continuation of saddle node homoclinic 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 with radiationless oscillating tails Physical Review E 51 3572 3578 C Y A 2001 Multipulses of nonlinearly coupled Schr dinger equations JOURNAL of Dif ferential Equations 173 1 92 137 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
152. int from the previous computation label 5 in s 2 make third The output BR PT TY LAB PAR 1 aaa 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 7 428734E 01 3 432139E 05 2 822988E 01 1 26 UZ 12 7 746686E 01 7 746147E 01 5 833163E 01 1 637611E 07 1 28 EP 13 7 746628E 01 7 746453E 01 5Bb 908902E 01 1 426214E 04 contains a neutral saddle focus a Belyakov transition at LAB 10 4 0 a double real leading eigenvalue saddle focus to saddle transition at LAB 11 y gt 0 and a neutral saddle at LAB 12 44 0 Data at several points on the complete branch are plotted in Fig 19 2 If we had continued further by increasing NMX the computation would end at a no convergence error TY MX owing to the homoclinic branch 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 11 170 0 00 0 20 0 40 0 60 0 80 1 00 0 10 0 30 0 50 0 70 0 90 Time Figure 19 1 Solutions of the boundary value problem at labels 6 and 8 either side of the Shil nikov Hopf bifurcation Figure 19 2 Phase portraits of three homoclinic orbits on the branch showing the saddle focus to saddle transition 171 19 2 Detailed AUTO Commands AUTO COMMAND ACTION mkdir
153. ion Point number The number of the point in the given branch Type name A short string which describes the type of the solution see Table 4 5 Type number A number which describes the type of the solution see Table 4 5 P The value of all of the parameters for the solution This is an alias for Parameter parameters The value of all of the parameters for the solution This is an alias for Parameter Table 4 6 This table shows the strings that can be used to query a bifurcation solution object and their meanings 35 data 57 toArray Similarily if one wanted to write out the solution to the file outputfile one would type data 57 writeRawFilename outputfile 4 9 The autorc File Much of the default behavior of the AUTO 2000 CLUI can be controlled by the autorc file The autorc file can exist in either the main AUTO 2000 directory the users home directory or the current directory 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 2000 directory Hence options may be defined on either a per directory per user or global basis 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
154. ion 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 are u Ui Y p elui oa i 1 NDIM 2 with boundary conditions u 0 0 u 1 0 Here e u 3 k 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 AUTO COMMAND ACTION mkdir tim create an empty work directory cd tim change directory demo tim copy the demo files to the work directory runte tim 1 gt Timing run sv tim save output files as b tim s tim d tim Table 14 3 Commands for running demo tim 136 Chapter 15 HomCont 15 1 Introduction HomCont is a collection of subroutines 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 behi
155. ions 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 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 diagnostics file fort 9 6 5 Floquet Multipliers AUTO extracts an approximation to the linearized Poincar map from the Jacobian of the lin earized 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 i
156. irectory cd ops change directory demo ops copy the demo files to the work directory run c ops 1 locate a Hopf bifurcation sv 0 save output files as b 0 s 0 d 0 run c ops 2 s 0 ap 0 compute a branch of periodic solutions restart from s 0 Constants changed IPS IRS NMX NUZR append the output files to b 0 s 0 d 0 run c ops 3 s 0 sv 1 locate a l parameter extremum as a bi furcation restart from s 0 Constants changed IPS IRS ICP save the output files as b 1 s 1 d 1 run c ops 4 s 1 ap 1 switch branches to generate optimality starting data restart from s 1 Constants changed IRS ISP ISW NMX append the output files to b 1 s 1 d 1 run c ops 5 s 1 sv 2 compute 2 parameter branch of 1 parameter extrema restart from s l Constants changed IRS ISW ICP ISW save the output files as b 2 s 2 d 2 run c ops 6 s 2 sv 3 compute 3 parameter branch of 2 parameter extrema restart from s 2 Constants changed IRS ICP EPSL EPSU NUZR save the output files as b 3 s 3 d 3 Table 12 5 Commands for running demo ops 125 12 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
157. is must be followed by recompilation by typing make in directory auto 2000 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 2 NDIM 1 for algebraic equations 10 and 2 NDIM for ordinary differential equations respectively Similarly for the continuation of Hopf bifurcations the effective dimension is 3 NDIM 2 1 4 Compatibility with Older Versions There are two changes compared to early versions of AUTO 94 The user supplied equations files must contain the subroutine pvls For an example of use of pvls see the demo pvl in Section 14 1 There is also a small change in the q xxx data file If necessary older AUTO 94 files can be converted using the 94to97 command see Section A Data files from AUTO 97 are fully compatible with AUTO 2000 but as AUTO 2000 is written in C user defined function files from AUTO 97 which are generally in Fortran must be rewritten 1 5 Parallel Version AUTO 2000 contains code which allows it to run in on various types of parallel computers Namely it can use either the Pthreads library for running on shared memory multi processors or the MPI message passing library When the configure script is run it will try to find the above two libraries and if it is successful it will include their functionality into AUTO 2000 To force the configure script not to use either of the above librarie
158. it 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 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 13 164 1 00 Uy 25 0 50 0 75 SZN da 1 0 OS 0 0 0 5 1 0 Ls o Figure 18 6 Two non central saddle node homoclinic orbits 1and 3 and 2 a central saddle node homoclinic orbit between these two points 100 200 1s 5 140 D 3 0 0 0 3 1 0 ges 20 Figure 18 7 The big homoclinic orbit approaching a figure of eight 165 make ninth 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 en PAR 2 1 500 EP 24 1 218988E 05 2 181205E 01 2 189666E 00 By plotting phase portraits of orbits approaching this end point see Figure 18 7 we see a canard like like transformation of the big homoclinic or
159. k oe te Ae a Sele does B2 Default Options vq iee 6 ss a ke ders oh a es bee a gk at a Bor Other PLAUT Commands 5 4 4 os wet ee 2 oy be A a BA Printime WE Biles re rd SA he eg he Sele en od on en ese el Sas Graphical User Interface C 1 General Overview C 1 1 The Menu bar C 1 2 The Define Constants buttons ooo e e C 1 3 The Load Constants buttons co a e y E ee EL A C 1 4 The Stop and Exit buttons its al bee s es ar 62 Ehe Menm Bats x a a a a aa e a a a eee tes ld thauations byttons sa bicis we et OE dat BO HAS C 22 Ed buttons occ di ok cs ae ae A oh Sa es C223 Write button as aiie a a ee eee ee ee ed Gig we td fae a T C24 Definesbuttons ws bee Pee A ee eae eh eRe a ea 0 29 R n button d 6 taa ek te era we ele weal Bo waa hw aka E 2 6 Save buttons 6 8 actin hk oe te a amp oh Sale ol Se a a C2 Appendsbulto Sula S48 eI Sk OS A SWAG BGs 9 C28 Plots button ee soc tke whee ele a Da TA sy gh C29 Files buttons 2 030 a p cles e a A a ee ee id C210 Demos button Aani 6 wed de Se he A Sethe Sek ed EN ee ee the a G21 Mise cbuttoriss ara ducts die te he ks ety ee eee ee AS Ud sHelp buttons eho Wei etka ol ak ed A Se da Lar Using slo e ts Fes pital tc tr los acidos et tai a GA Customs the Gr WL s os di e ds E e A ed a th E A CAL PUMA e A Ae A da C42 GUlcolors 0 00 00 00 ot ae e E t AEE a a OEN a E R C 4 3 On line help oot eB ete te ee te ee et ae ee a Re ad 206 Preface T
160. l user interface for AUTO on SGI machines was written by Taylor amp Kevrekidis 1989 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 Work on this updated version was supported by a general research grant from NSERC Canada The development of HomCont has much benefitted from various pieces of 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 and C J Budd University of Bristol Financial support for collaboration was received from the U K Engineering and Physical Science Research Council and the Nuffield Foundation Chapter 1 Installing AUTO 1 1 Typographical Conventions This manual uses the following conventions command This font is used for commands which you can type in PAR This font is used for AUTO parameters filename This font is used for file and directory names variable This font is used for environment variable site This font is used for world wide web and ftp sites function This font is used for function names 1 2 Installation The AUTO files are available via HTTP from http www ama caltech edu redrod auto200
161. label with the first continuation parameter being once again a PAR 1 by changing constants and storing them in c san 5 ac cording to IRS 7 DS 0 05DO NMX 20 ICP 1 4 Running make fifth the output at label 10 BR PT TY LAB PAR 1 PAR 8 PAR 10 PAR 21 1 8 UZ 10 1 304570E 07 3 874816E 12 1 468457E 09 2 609139E 07 indicates that AUTO has detected a zero of PAR 21 implying that a non orientable resonant bifurcation occurred at that point 16 4 Orbit Flip In this section we compute an orbit flip To this end we restart from the original explicit so lution without computing the orientation We begin by separately performing continuation in a A 0 4 a A b f and u 1 in order to reach the parameter values a b a 8 u 0 5 3 1 0 0 25 The sequence of continuations up to the desired parameter values are run via make sixth make seventh make eighth make ninth make tenth with appropriate continuation parameters and user output values set in the corresponding files c san xx All the output is saved to s 6 The final saved point LAB 10 contains a homoclinic solution at the desired parameter values From here we perform continuation in the negative direction of u A PARC7 PAR 8 with the test function 411 for orbit flips with respect to the stable manifold activated 147 make eleventh The output detects an inclination flip by a zero of PAR 31 at PAR 7 0 BR PT TY LAB PAR 7 eeu PAR
162. long such a branch 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 2 4 is approached then the wave tends to a solitary wave solution of 2 3 Demo wav Run 3 Trace out branches 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 branch of approximate 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 computation 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 2 3 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 2 3 are adaptive in space and in time 15 Discretization in time is not very accurate only implicit Euler Indeed time integration of 2 3 has only been included
163. lt filename templates Existing files with these names will be overwritten Aliases SV Save 44 4 13 9 commandCreateGUI Purpose Show AUTOs graphical user interface Description Type commandCreateGUI to start AUTOs graphical user interface NOTE This command is not implemented yet Aliases gui 4 13 10 commandDeleteDataFiles Purpose Delete data files Description Type commandDeleteDataFiles xxx to delete the data files d xxx b xxx and s xxx if you are using the default filename templates Aliases delete dl 45 4 13 11 commandDeleteFortFiles Purpose Clear the current directory of fort files Description Type commandDeleteFortFiles to clean the current directory This command will delete all files of the form fort Aliases df deletefort 4 13 12 commandDouble Purpose Double a solution Description Type commandDouble to double the solution in fort 7 and fort 8 Type commandDouble xxx to double the solution in b xxx and s xxx if you are using the default filename templates Aliases double db 46 4 13 13 commandInteractiveHelp Purpose Get help on the AUTO commands Description Type help to list all commands with a online help Type help xxx to get help for command xxx Aliases man help 4 13 14 commandLs Purpose List the current directory Description Type comm
164. lues 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 C 2 5 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 C 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 C 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 C 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 the item Name to run PLAUT with other data files C 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
165. lution labels and symbols 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 with grid lines d3 As d1 except for periodic solutions use solid circles if stable and open circles if unstable or if the stability is unknown d4 Use solid curves without labels and symbols If no default option d0 d1 d2 d3 or d4 has been selected or if you want to override a default feature then the the following commands can be used These can be 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 i 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 201 B 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 B 4 Printing PLAUT Files Ops Type ps fig 1 to convert a saved PLAUT file fig 1 to PostScript format in
166. mero E amp Ponce E 1993 A case study for homoclinic 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 3 231 239 209 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 ek 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 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
167. meter curve of inclination flips restart from s 3 save output files as b 10 s 10 d 10 run c kpr 11 h kpr 11 s 3 sv 11 another curve of inclination flips restart from s 3 save output files as b 11 s 11 d 11 run c kpr 12 h kpr 12 s 7 sv 12 continue non central saddle node homoclinics restart from s 7 save output files as b 12 s 12 d 12 run c kpr 13 h kpr 13 s 8 ap 12 continue non central saddle node homoclinics restart from s 8 append output files to b 12 s 12 d 12 Table 18 2 Detailed AUTO Commands for running demo kpr 168 Chapter 19 HomCont Demo cir 19 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 i v x By aga bay x r y Pr B Ny 2 bs y 2 19 1 Z y 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 c and the initial run time constants are stored in c cir 1 and h cir 1 We begin by starting from the data from cir dat for a saddle focus homoclinic orbit at v 0 721309 G 0 6 y 0 r 0 6 A3 0 328578 and B3 0 933578 which was obtained by shooting over the time interval 2T PAR 11 36 13 We wish to follow the branch
168. mns the user is likely to use solution_filename The filename of the solution to plot solution_x The column to plot along the X axis for solutions solution_y The column to plot along the Y axis for solutions symbol font The font to use for marker symbols 37 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 torus_symbol The symbol to use for torus bifurcation points type The type of the plot either solution or bifurcation user_point_symbol The symbol to use for user defined output points xlabel The label for the x axis xmargin The margin between the graph and the right and left edges xticks The number of ticks on the x axis ylabel The label for the y axis ymargin The margin between the graph and the top and bottom edges yticks The number of ticks on the y axis Table 4 7 This table shows the options that can be set for the AUTO 2000 CLUI two dimensional plotting window and their meanings 4 11 Three Dimensional Plotting Tool NOTE the documentation in this section is under developement The AUTO 2000 three dimensional plotting tool can use DataViewer or OpenInventor for rendering three dimensional representations of bifurcation diagrams and solutions and is under active development Neither DataViewer n
169. moclinic at the point D see Champneys et al 1996 Fig 2 and continues it to the one emanating from Da 154 BR PT TY LAB PAR 1 rs PAR 2 PAR 29 PAR 30 1 10 15 4 966429E 00 6 298418E 02 4 382426E 01 4 946824E 03 1 20 16 4 925379E 00 7 961214E 02 3 399102E 01 3 288447E 02 1 30 17 7 092267E 00 1 58 114E 01 1 692842E 01 3 876291E 02 1 40 EP 18 1 101819E 01 2 809825E 01 3 482651E 02 2 104384E 02 The data are appended to the stored results in b 1 s 1 and d 1 One could now display all data using the AUTO command Op 1 to reproduce the curve P shown in Figure 17 1 It is worthwhile to compare the homoclinic curves computed above with a curve Ty 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 17 2 It obviously approximates well the saddle homoclinic loci of P but demonstrates much bigger deviation from the saddle node homoclinic segment D D3 This happens because the period of the limit cycle grows to infinity while approaching both types of homoclinic orbit but with different asymptotics as ln lla a in the saddle homoclinic case and as a in the saddle node case 17 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
170. n Type commandQueryStepsize to list the continuation step size for each continuation step in fort 9 Type commandQueryStepsize xxx to list the continuation step size for each con tinuation step in the info file d xxx Aliases ss stepsize st 59 4 13 31 commandRun Purpose Run AUTO Description Type commandRun options to run AUTO with the given options There are four possible options Long name Short name Description equation e The equations file constants C The AUTO constants file solution s The restart solution file h homcont The Homcont parameter file Options which are not explicitly set retain their previous value For example one may type commandRun e ab c ab 1 to use ab c as the equations file and c ab 1 as the constants file if you are using the default filename templates Type commandRun name load all files with base name This does the same thing as running commandRun e name c name s name h name Aliases r run rn 4 13 32 commandRunnerConfigFort2 Purpose Modify continuation constants Description Type commandRunnerConfigFort2 xxx yyy to change the constant xxx to have value yyy Aliases changeconstant cc ch 56 4 13 33 commandRunnerLoadName Purpose Load files into the AUTO runner Description Type commandRunnerLoadName options to modify AUTO runner There are four
171. n 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 G T E Sta 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 gt o PAR IP i i 1 2 NUNSTAB PAR IP NUNSTAB i w i 1 2 NUNSTAB Note that to avoid interference with the test functions 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 18 for an example however we recommend the homotopy method only for expert users 143 To compute the orie
172. n in a bifurcation diagram The queriable parts of the object are listed in Table 4 4 The individual elements of the array may be accessed in two ways either by index of the solution using the syntax or by label number using the syntax For example assume that the parsed object is contained in a variable data The first solution may be accessed using the command data 0 while the solution with label 57 may be accessed using the command data 57 This class has two 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 and returns a standard Python array Second there is a method called writeRawFilename which will create a standard ASCII file which contains the bifurcation diagram For example we again assume that the parsed object is contained in a variable data If one wanted to have the bifurcation diagram returned as a Python array one would type data toArray Similar ily if one wanted to write out the bifurcation diagram to the file outputfile one would type data writeRawFilename outputfile 4 8 Solution Files Using the commandParseSolutionFile command Section 4 13 19 in the reference the user can parse and read into memory an AUTO 2000 bifurcation solution file For example the command commandParseSolutionFile ab would parse the file b ab if you are using the standard file 31 gt cp AUTO _DI
173. n 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 10 which corresponds to the period and stop when the period is equal to 35 rn c sib 3 s 2 sv 3 BR PT TY LAB PAR 5 L2 NORM Saa PERIOD 3 5 5 2 418809E 03 6 705689E 01 1 089749E 01 3 40 8 1 294950E 02 6 145469E 01 i 1 412970E 01 3 81 EP 13 1 046566E 04 4 018291E 01 o 3 499999E 01 184 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 plane It is now instructive to look at a phase space diagram to see what is going on plot s Selecting solution for Type 5 6 7 8 9 10 11 12 13 for Label 0 for X and 1 for Y we obtain the diagram depicted in Figure 22 1 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 0i 0 00e 00 2 008 01 2 008 01 4 008 01 0 008 0 00e 00 0 00 01 2 006 01 6 006 01 1 006 00 Columns t Figure 22 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
174. nd the methods used is explained in Champneys amp Kuznetsov 1994 Bai amp Champneys 1996 Sandstede 1995b 1995c 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 subroutines are in the file auto 2000 src autlib5 c 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 15 2 HomCont Files and Subroutines In order to run HomCont one must prepare an equations file xxx c where xxx is the name of the example and two constants files c xxx and h xxx The first two of these files are in the standard AUTO format whereas the h xxx file contains constants that are specific to homoclinic continuation The choice IPS 9 in c xxx specifies the problem as being homoclinic continuation in which case h xxx is required The equation file kpr c serves as a sample for new equation files It contains the C subroutines func stpnt pvls bend icnd and fopt The final three are dummy subroutines 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 Note that th
175. ndex of the solution using the syntax or by label number using the syntax For example sssume that the parsed object is contained in a variable data The first solution may be accessed using the command data 0 while the solution with label 57 may be accessed using the command data 57 This class has two 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 solution and re turns a standard Python array Second there is a method called writeRawFilename which will create a standard ASCII file which contains the solution 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 For example we again assume that the parsed object is contained in a variable data If one wanted to have the solution with label 57 returned as a Python array one would type 34 Query string Meaning data An array which contains the AUTO 2000 output Branch number The number of the branch to which the solution belongs The ISW value used to start the calcluation See Sec ISW tion 5 8 3 Label The label of the solution The number of collocation points used to compute the NCOL f solution See Section 5 3 2 NTST The number of mesh intervals used to compute the solu tion See Section 5 3 1 Parameters The value of all of the parameters for the solut
176. nt Demo cir 169 19 1 Electronic Circuit of Freire et Ale Ad os dea pa ed AA 169 19 2 Detailed AUTO Commands o o a a a 172 20 HomCont Demo she 173 20 1 Ay Meterochme EMPERO AA 173 20 2 Detailed AUTO Commands Ds a a 175 21 HomCont Demo rev 177 21 1 A Reversible System a e a a 177 21 2 An R Reversible Homoclinic Solution ooo aa a 0000084 177 21 3 An R Reversible Homoclinic Solution 178 21 4 Detailed AUTO Commands a AA a 182 22 HomCont Demo Homoclinic branch switching 183 22 1 Branch switching at an inclination flip in Sandstede s model 183 22 2 Branch switching for a Shil nikov type homoclinic orbit in the FitzHugh Nagumo E A a eg ay RR NR 190 22 3 Branch switching to a 3 homoclinic orbit in a 5th order Korteweg De Vries model A Running AUTO using Command Mode AQUI Basic commands st e aoaie a a a eg a O a A 0 2 Plotting commands ds o ds E a io a A 0 3 File manipulation meek a ee A SR A da des A 0 4 Diagnostics ck Ges b gt os ly has Ge By ve Ge a a eS Oh A 0 5 SB AN a ee eg ee at A ES a A 0 6 File maintenances 4 24 4 6 446 430 66 84 woes gd we ee a be a A 0 7 HomCont commands side a oes a ey Ger A 0 8 Copying a demo dad Aree ag eG ese at ad A 0 9 Pendula animation et as as e OS A A at A 0 10 Viewing the manual Lee a a A Male A B The Graphics Program PLAUT Beals B sic PLAU T C ommands a8 2 sonte a a w
177. ntation 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 15 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 zxx Simply typing make or make all 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 1 plots the data saved in b 1 and s 1 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 ta
178. ocus of folds in reverse direction PAR 1 PAR 1 8 80940E 05 L2 NORM L2 NORM 1 17440E 01 81 U 1 a 2 parameter locus of Hopf points U 1 9 14609E 01 U 2 11 1 31939E 03 9 96432E 01 3 58651E 03 9 96426E 01 Appended to 2p U 2 1 17083E 01 PERIOD 2 721E 00 6 147E 00 1 399E 01 9 956E 01 1 867E 03 PAR 3 2 499E 00 3 748E 01 PAR 3 1 050E 00 PAR 3 9 362E 02 7 5 Executing Selected Runs Automatically As illustrated by the commands in Table 7 6 one can also execute selected runs of demo ab In general this cannot be done in arbitrary order as any given run may need restart data from a previous run Run 3 only requires the results of Run 1 so that the displayed command sequence is indeed appropriate The screen output of these runs will be identical to that of the corresponding earlier runs except for a change in solution labels in Run 3 In real use there are two mains ways in which the AUTO can be used First one can prepare a constants file for each run In the illustrative runs below the constants files were carefully prepared in advance For example the file c ab 1 contains the AUTO constants for Run 1 c ab 3 contains the AUTO constants for Run 3 etc AUTO COMMAND ACTION ld ab load the problem definition ab run c ab 1 execute the run which uses the constants in c ab 1 sv ab save the results of the run into the files b ab s ab and d ab run c ab 3 s
179. oi Sas gore E E eb Sa ASS NUZIRS Ht Ne Sah OE Sek La ON ee A Be pen Na o ot Res Sate do Notes on Using AUTO 6 1 Restrictions on the Use of PAR Tui ca eh SE a Ep eb See eh a 62 is A A Oe oh ee G 6 3 Correctness of Results a bb ela AE e ta ba ed 6 4 Bifurcation Points and Folds e se ak eS ete Fy Sack ee A eS RRS 6 5 Floguet Multipliers es ea Gea na eta dienes DE me lead 6 6 Memory Requirements 2 4 daa oe Pe es a Pe ow a ee eS Ep AUTO Demos Tutorial FL Mtrod ction Sese ai a es Ae ees Y SE Sie Re Pe ex T2 Abe A Tutorial Demos todo a ee ee he a aa ee 7 3 Copying the Demo Files le do le das ee a A 7 4 Executing all Runs Automatically 9 5 4 4 2 4 4 6 4 be ab a eg a 7 5 Executing Selected Runs Automatically 0 2 22 0000 fio Uso AU EO Commands sie 3108 5 ee and hit an he he are ee us de Sided Rake 7 7 Plotting the Results with AUTO 852k at aa Leeda e ae so 7 8 Following Folds and Hopf Bifurcations 0 7 9 Relabeling Solutions in the Data Files 0 0 0 0 a 00004 7 10 Plotting the 2 Parameter Diagram 2 222 ge fh a 2h ante ee SEA AUTO Demos Fixed points 8 1 enz Stationary Solutions of an Enzyme Model 8 2 dd2 Fixed Points of a Discrete Dynamical System AUTO Demos Periodic solutions 9 1 lrz The Lorenz Equations di A Ne a ee eee ed te tk tS 9 2 abc The A gt B gt C Reaction 0 0 0 0 0 0 0 eee ee 9 3 pp
180. on ab load constants ab 1 run Figure 4 5 The commands from Figure 4 2 and they would appear in a AUTO 2000 CLUI script file The source for this script can be found in AUTO_DIR demos python demol auto 4 5 Second Example In Section 4 3 we showed a very simple AUTO 2000 CLUI script in this Section we will describe a more complex example which introduces several new AUTO 2000 CLUI commands as well as some basic Python constructs for conditionals and looping We will not provide an exhaustive reference for the Python 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 4 7 The script begins with a section extracted into Figure 4 8 which performs a task identical to that shown in Figure 4 2 except that the shorthand discussed in Section 4 3 is used for the 1d command The next section of the script extracted into Figure 4 9 introduces three new AUTO 2000 CLUI commands First sv bvp Section 4 13 6 in the reference saves the results of the AUTO 2000 run into files using the base name bvp and the filename extensions in Table 4 3 For example in this case the bifurcation diagram file fort 7 will be saved as b bvp the solution file fort 8 will be saved as s bvp and the diagnostics file fort 9 will be saved as d bv
181. on vanishes at a branch point Type commandQueryBranchPoint xxx to list the value of the branch point func tion in the info file d xxx Aliases br bp branchpoint 51 4 13 23 commandQuery Eigenvalue Purpose Print eigenvalues of Jacobian algebraic case Description Type commandQueryEigenvalue to list the eigenvalues of the Jacobian in fort 9 Algebraic problems Type commandQueryFigenvalue xxx to list the eigenvalues of the Jacobian in the info file d xxx Aliases eigenvalue ev eg 4 13 24 commandQueryFloquet Purpose Print the Floquet multipliers Description Type commandQueryFloquet to list the Floquet multipliers in the output file fort 9 Differential equations Type commandQueryFloquet xxx to list the Floquet multipliers in the info file d xxx0 Aliases fl floquet 92 4 13 25 commandQuery Hopf Purpose Print the value of the Hopf function Description Type commandQueryHopf to list the value of the Hopf function in the output file fort 9 This function vanishes at a Hopf bifurcation point Type commandQuery Hopf xxx to list the value of the Hopf function in the info file d xxx Aliases hb hp hopf lp 4 13 26 commandQuerylterations Purpose Print the number of Newton interations Description Type commandQuerylterations to list the number of Newton iterations per contin uation
182. onitor the optimality functionals associated with the problem parameters and the period Computations can be started at a solution computed with IPS 2 or IPS 15 For a detailed 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 pdl pd2 and bru Note that the space derivatives of the initial conditions must also be supplied in the user supplied subroutine 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 pdl and pd2 5 9 Output Control 5 9 1 NPR This constant can be used to regularly write fort 8 plotting and restart data IF NPR gt O then such output is written every NPR steps IF NPR O 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 5 9 4 For these reasons and to limit the output volume it is recommended that NPR output be kept to a minimum
183. onstants changed ISW save output files as b tmp s tmp d tmp sv 2p run c pp3 5 s tmp period doubling continuation restart from s tmp Constants changed IRS save output files as b 2p s 2p d 2p Table 9 8 Commands for running demo pp3 98 9 9 9 9 tor Detection of Torus Bifurcations This demo uses a model in Freire Rodr guez 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 a t 8 v By azz b3 y x r de fx 8 yy 2 bs y x 9 10 A y where y 0 6 r 0 6 a3 0 328578 and b3 0 933578 Initially y 0 9 and P 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 ld tor load the problem definition run c tor 1 1st run compute a stationary solution branch with Hopf bifurcation svC 12 save output files as b 1 s 1 d 1 run c tor 2 s 1 compute a branch of periodic solutions restart from s l Constants changed IPS IR
184. or OpenInventor are provided with AUTO 2000 and must be downloaded seperately If you are interested in the three dimensional plotting tool please contact redrod acm org 38 4 12 Quick Reference In this section we have created a table of all of the AUTO 2000 CLUI commands their abbrevia tions 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 Aliases Description commandA ppend ap append Append data files commandCat cat Print the contents of a file commandCd cd Change directories commandClean clean cl Clean the current directory commandCopy AndLoadDemo dm demo Copy a demo into the cur rent directory and load it commandCopy DataFiles copy cp Copy data files commandCopyDemo copydemo Copy a demo into the cur rent directory commandCopyFortFiles SV save Save data files commandCreateGUI gui Show AUTOS graphical user interface commandDeleteDataFiles delete dl Delete data files commandDeleteFortFiles df deletefort Clear the current directory of fort files commandDouble double db Double a solution commandInteractiveHelp man help Get help on the AUTO com mands commandLs ls List the current directory commandMoveFiles move mv Move data files to a new name commandParseConstantsFile cn constantsget Get the current continuation constants commandParseDiagramAndSolutionFile bt d
185. or running demo exp 105 10 2 int Boundary and Integral Constraints This demo illustrates the computation of a solution branch to the equation 10 2 with a non separated boundary condition and an integral constraint u 0 1 p2 0 i u t dt ps 0 The solution branch contains a fold which in the second run is 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 run c int 1 1st run detection of a fold sv int save output files as b int s int d int run c int 2 s int 2nd run generate starting data for a curve of folds Constants changed pace0 2cm sv t save the output files as b t s t d t run c int 3 s t 3rd run compute a curve of folds restart from s t Constants changed IRS vs pace0 2cm sv lp save the output files as b lp s lp d lp Table 10 2 Commands for running demo int 106 10 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 branch The branch of solutions that bifurcates at the first eigenvalue is computed in both directions The equations are ui U2 A 10 3 uy pim ul we with boundary conditions u 0 0 uy 1 0 AUTO C
186. 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 NWIN THL EPSU EPSL EPSS also affect efficiency Understanding their significance is therefore useful see Section 5 4 and Section 5 5 Finally it is recommended 75 that initial computations be done with ILP 0 no fold detection and ISP 1 no bifurcation detection for ODEs 6 3 Correctness of Results AUTO computed solutions to 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 branch typically serves as a warning Repeating the computation with increased NTST is then recommended 6 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 branch then AUTO may try to locate one fold after another Generally degenerate bifurcat
187. orm of one of the adjoint variables This is done in the third run Along the resulting branch several two parameter extrema are located by monotoring 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 run c obv 1 locate 1 parameter extrema as branch points sv obv save output files as b obv s obv d obv run c obv 2 s obv compute a few step on the first bifurcating branch Constants changed IRS ISW NMX ev 17 save the output files as b 1 s 1 d 1 run c obv 3 s 1 locate 2 parameter extremum restart from s l Constants changed IRS ISW NMX ICP 3 sv 2 save the output files as b 2 s 2 d 2 run c obv 4 s 2 locate 3 parameter extremum restart from s 2 Constants changed IRS ICP 4 sv 3 save the output files as b 3 s 3 d 3 Table 12 6 Commands for running demo obv 127 Chapter 13 AUTO Demos Connecting orbits 128 13 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 Fw w 1 w We look for solutions of the form w x t u x ct where c is t
188. osite direction Constants changed DS ap nag append output files to b nag s nag d nag Table 13 2 Commands for running demo nag 130 13 3 stw Continuation of Sharp Traveling Waves This demo illustrates the computation of sharp traveling wave front solutions to nonlinear diffusion problems of the form w A w Wee B w w C w with A w a w agqw B w bo b w b3w and C w co tcywt cyw 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 zp 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 2wWre 2w2 w 1 w to a corresponding sharp traveling wave of 2 w 2w w Wee ww w 1 w This problem is also considered in Doedel Keller amp Kern vez 19916 For these two special cases the functions A B C are defined by the coefficients in Table 13 3 a Q2 bo by ba Co C1 Ca Case 1 2 0 2 O JOJO 1 1 Case 2 2 1 0 1 JO 0 1 1 Table 13 3 Problem coefficients in demo stw With w x t u x ct z x ct one obtains the reduced system u 2 U2 ub z
189. other along the Lin vector Psi at the left and right hand end points These gaps e are at parameters par 19 2 i Moreover each part except wy41 ends at at a Poincar section which goes through zo and is perpendicular to To The times T that each part u takes are stored as follows T par 9 Ty par 10 and T par 18 2 i fori 1 N 1 Through a continuation in problem parameters gaps and times T it is possible to switch from a 1 homoclinic to an N homoclinic orbit 139 If ITWIST 0 the adjoint vector is not computed and Lin s method is not used Instead AUTO produces a gap e par 21 at the right hand end point p of uy 1 measuring the distance between the stable manifold of the equilibrium and p This technique can also be used to find 2 homoclinic orbits by 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 15 3 6 NREV IREV If NREV 1 then it is assumed that the system is reversible under the transformation t
190. ous 133 14 1 pvl Use of the Subroutine pvls Consider Bratu s equation a 14 1 uy pie with boundary conditions u 0 0 u 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 pvis c namely po the La norm of u1 p the minimum of uz on the space interval 0 1 pa the 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 1 namely p P2 p3 and p4 The first one in this list p is the true continuation parameter The parameters P2 p3 and p4 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 pa p1 P3 p4 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 1 labeled solutions can also be output at selected values of the overspecified parameters AUTO
191. overspecification In the user supplied subroutine 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 example 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 sub routine pvis namely the L2 norm of the first solution component the minimum of the second component and the left boundary value of the second component These solution measures are assigned to PAR 2 PAR 3 and PAR 4 respectively In the constants file c pvl we have NICP 4 with PAR 1 PAR 4 specified as parameters Thus in this example PAR 2 PAR 4 68 are overspecified Note that PAR 1 must appear first in the ICP list the other parameters cannot be used as true continuation parameters 5 8 Computation Constants 5 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 5 8 2 ISP This constant controls the detection of branch points period doubling bifurcations
192. owed to be complex while the parameter p can only take real values In the real case this is Bratu s equation whose solution branch contains a fold see the demo exp It is known Henderson amp Keller 1990 that a simple quadratic fold gives rise to a pitch 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 branch 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 run c ezp 1 lst run compute solution branch containing fold sv ezp save output files as p ezp s ezp d ezp run c ezp 2 s ezp 2nd run compute bifurcating complex solution branch Constants changed IRS ISW ap ezp append output files to p ezp s ezp d ezp run c ezp 3 s ezp 3rd run compute 2nd leg of bifurcating branch constant changed DS ap ezp append output files to p ezp s ezp d ezp Table 10 8 Commands for running demo ezp 112 Chapter 11 AUTO Demos Parabolic PDEs 113 11 1 pdl Stationary States 1D 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 pro
193. p Next 1d s bvp loads the solution file s bvp into memory so that it can be used by AUTO 2000 for further calculations Up to this point all of the commands presented have had analogs in the command language discussed in Section A and the AUTO 2000 CLUI has been designed in this way to make it easy for users to migrate from the old command language to the AUTO 2000 CLUI The next command namely data sl bvp Section 4 13 19 in the reference is the first command which has no analog in the old command language The command sl1 bvp parses the file s bvp and returns a python object which encapsulates the information contained in the file and presents it to the user in an easy to use format Accordingly the command data sl bvp stores this easy to use representation of the object in the Python variable data 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 Finally ch NTST 50 Section 4 13 32 in the reference changes the NTST value to 50 see Section 5 2 1 To be precise the command ch NTST 50 26 gt cat demoi auto copydemo ab load equation ab load constants ab 1 run gt auto demol auto Initializing Copying demo ab Runner configured Runner configured gcc 0 DPTHREADS 0 I home amavisitors redrod src auto 2000 include c ab c gcc 0 ab o o ab exe home amavisitors redrod s
194. patial mesh is uniform the number of mesh intervals as well as the number of equations in the PDE system can be set by the user in the file brf inc As an illustrative application we consider the Brusselator Holodniok Knedlik amp Kub ek 1987 up Dy L Ugo uu 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 11 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 run c brf 1 compute the stationary solution branch with Hopf bifurcations sv brf save output files as b brf s brf d brf run c brf 2 s brf compute a branch of periodic solutions from the first Hopf point Constants changed IRS IPS apC brf append the output files to b brf s brf d brf run c brf 3 s brf compute a solution branch from a sec ondary periodic bifurcation Constants changed IRS ISW ap brf append the output files to b brf s brf d brf Table 11 5 Commands for running demo brf 118 11 6 bru Euler Time Integration the Brusselator This demo illustrates the use of
195. period according to the data in pen dat The AUTO command us pen converts the data in pen dat to a labeled AUTO solution with label 1 in a new file s 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 Note that the file s dat should be used for restart only Do not append new output files to s dat as the command us pen only creates s dat with no corresponding b dat The first run with as free problem parameter starts from the converted solution with label 1 in pen dat A period doubling bifurcation is located and the period doubled branch is computed in the second run Two branch points are located and the bifurcating branches 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 100 AUTO COMMAND ACTION mkdir pen create an empty work directory cd pen change directory demo pen copy the demo files to the work directory 1d pen load the problem definition us pen convert pen dat to AUTO format in s dat run c pen 1 s dat sv pen locate a period doubling bifurcation restart from s dat save output
196. problems where NDIM NBC and where NINT 0 Then generically one free problem parameter is required for computing a solution branch For example in demo exp we have NDIM NBC 2 NINT 0 Thus NICP 1 Indeed 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 NICP 2 Indeed in this demo two free parameters are designated namely PAR 1 and PAR 3 5 7 7 Boundary value folds To continue a locus of folds for a general boundary value problem with integral constraints set NICP NBC NINT NDIM 2 and specify this number of parameter indices to designate the free parameters 67 5 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 NICP 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 NICP 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
197. rameter continuation The fourth run make fourth 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 W9 W19 and 413 That these test functions were activated is specified in three places in c kpr 4 and h kpr 4 as described in Section 15 6 Note that at the end point of the branch reached when after NMX 50 steps PAR 29 is approximately zero which corresponds to a zero of Yg 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 413 which gives the accurate location of two inclination flip bifurcations BR PT TY LAB PAR 1 ans PAR 2 PAR 10 Fec PAR 33 1 6 UZ 10 1 801662E 00 2 002660E 01 7 255434E 07 1 425714E 04 1 12 UZ 11 1 568756E 00 4 395468E 01 2 156353E 07 4 514073E 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 18 3 presents solutions t of the modified adjoint variational equation for details see Champneys et al 1996 at parameter values on the homoclinic branch before and after the first detected inclination 161
198. rc auto 2000 lib o lpthread L home amavisitors redrod src auto 2000 lib lauto_f2c lm Starting ab done 1 1 EP 1 0 000000E 00 0 000000E 00 0 O000000E 00 0 000000E 00 1 33 LP 2 1 057390E 01 1 484391E 00 3 110230E 01 1 451441E 00 1 70 LP 3 8 893185E 02 3 288241E 00 6 889822E 01 3 215250E 00 1 90 HB 4 1 308998E 01 4 271867E 00 8 950803E 01 4 177042E 00 1 92 EP 5 1 512417E 01 4 369748E 00 9 155894E 01 4 272750E 00 Total Time 8 740E 02 ab done gt Figure 4 6 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 2000 CLUI by typing auto demol auto and the output is shown only modifies the in memory version of the AUTO 2000 constants created by the 1d bvp command The original file c bvp is not modified The next section of the script extracted into Figure 4 10 shows as example of looping and conditionals in an AUTO 2000 CLUI script The first line for solution in data is the Python syntax for loops The data variable was defined in Figure 4 9 to be the parsed ver sion of an AUTO 2000 fort 8 file and accordingly contains a list of the solutions from the fort 8 file The command for solution in data is used to loop over all solutions in the data variable by setting the variable solution to be one of the solutions in data and then calling the rest of the code in the block Python differs from most other computer languages in that
199. re so NICP 1 As a concrete example consider Run 1 of demo ab where NICP 1 with ICP 1 1 Thus in this run PAR 1 is designated as the free parameter 5 7 3 Periodic solutions and rotations The continuation of periodic solutions and rotations generically requires two parameters namely one problem parameter and the period Thus in this case NICP 2 For example in Run 2 of demo ab we have NICP 2 with ICP 1 1 and ICP 2 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 one can set NICP 1 and only specify the index of the free problem parameter as AUTO will automatically addd 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 set NICP 2 and specify two free problem parameters For example in Run 7 of demo pp2 we have NICP 2 with PAR 1 and PAR 2 specified as free 66 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 5 7 4 Folds and Hopf bifurcations The continuation of folds for algebraic problems and the continuation of Hopf bifurcations requires two free problem parameters i e NICP 2 For example to continue a fold in Run 3 of demo ab we have NICP 2 with PAR 1 and PAR 3 specified as free par
200. re creating the tar file This will remove all executable object and other non essential files and thereby reduce the size of the package AUTO can be tested by typing make gt TEST amp in directory auto 2000 test This will execute a selection of demos from auto 2000 demos and write a summary of the computations in the file TEST The contents of TEST can then be compared to other test result files in directory auto 2000 test Note that minor differences are to be expected due to architecture and compiler differences Some EISPACK routines used by AUTO for computing eigenvalues and Floquet multipliers are included in the package Smith Boyle Dongarra Garbow Ikebe Klema amp Moler 1976 1 3 Restrictions on Problem Size There are size restrictions in the file auto 2000 src auto_c h on the following AUTO constants the effective number of equation parameters NPAR and the number of stored branch points NBIF for algebraic problems See Chapter 5 for the significance these constants Their maxima are denoted by the corresponding constant followed by an X For example NPARX in auto c h denotes the maximum value of NPAR If the maxima of NBIF is exceeded in an AUTO run then a message will be printed On the other hand the maximum value of NPAR if exceeded may lead to unreported errors Upon installation NPARX 36 it should never be decreased below that value see also Section 6 1 Size restrictions can be changed by editing auto_c h Th
201. results in BR PT TY LAB PAR 1 hes PAR 2 sides 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 1 087 32E 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 Y5 0 Label 7 where PAR 29 0 corresponds to a local bifurcation Yg 0 which we note from the eigenvalues stored in d 2 corresponds to a Shil nikov Hopf bifurcation Note that PAR 2 is also approximately zero at label 7 which accords with the analytical observation that the origin of 19 1 undergoes a Hopf bifurcation when 8 0 Labels 6 and 8 are the user defined output points the solutions at which are plotted in Fig 19 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 branch 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 po
202. ructed to carry out a particular task inspect the appropriate constants file In this chapter we describe the tutorial demo ab in detail A brief description of other demos is given in later chapters 72 ab A Tutorial Demo This demo illustrates the computation of stationary solutions Hopf bifurcations and periodic solutions and the computation loci of folds and Hopf bifurcation points The equations that model an A gt B reaction are those from Uppal Ray amp Poore 1974 namely uy u1 pi l u e 7 1 Ug U2 pipe 1 ui e pzuz GD 7 3 Copying the Demo Files The commands listed in Table 7 1 will copy the demo files to your work directory Unix COMMAND ACTION auto start the AUTO2000 Command Line User Interface AUTO COMMAND ACTION cd go to main directory or other directory mkdir ab create an empty work directory Note the P is used to signify a command which is sent to the shell cd ab change to the work directory demo ab copy the demo files to the work directory Table 7 1 Copying the demo ab files At this point you may want to see what files have been copied to the work directory In particular you may want to edit the equations file ab c to see how the equations have been entered in subroutine func and how the starting solution has been set in subroutine stpnt Note that initially p 0 po 14 and p3 2 for which uw uz
203. s 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 w t Ag 2 3 2 z 2 4 y M y t z A 12 1 Z t 2 2 with objective functional 1 w f aa Az Az Az As dt 0 where g x y zZ 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 automat ically i e one need only define the vector field and the objective functional as in done in the file ops c For reference purpose it is convenient here to write down the full extended system in its general form u t T f u t A TER period u f ER A R w t T fu u t w t kuplt You u t A w ER k y ER 12 2 fo wt wlt K 7 a dt 0 a ER hie f u t A w t Y9T u t A To dt 0 To E R fo Thr ult AJ wE 19 ult A 7r dt 0 RER t 1 ny Above uo is a reference sol
204. s tr triple 4 13 37 commandUserData Purpose Covert user supplied data files Description Type commandUserData 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 For examples of using the userData command see demos lor and pen where it has the old name fc Aliases us userdata 59 4 13 38 command Wait Purpose Wait for the user to enter a key Description Type commandWait to have the AUTO interface wait until the user hits any key mainly used in scripts Aliases walt 60 Chapter 5 Description of AUTO Constants 5 1 The AUTO Constants File As described in Section 3 1 if the equations file is xxx c then the constants that define the computation are normally expected in the file c xxx The general format of this file is the same for all AUTO runs For example the file c ab in directory auto 2000 demos ab is listed below The tutorial demo ab is described in detail in Chapter 7 2101 NDIM IPS IRS ILP 1 1 NICP ICP I I 1 NICP 504311000 NTST NCOL IAD ISP ISW IPLT NBC
205. s one may type configure without mpi or configure without pthreads and then type make One may even preclude both by typing configure without mpi without pthreads and then typing 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 2000 may easily be controlled and even run in a serial mode through the use of command line options at run time The command line options are listed in Table 1 1 v Give verbose output m Use the Message Passing Interface library for parallelization Use the Pthreads library for parallelization This option takes one of three arguments conpar parallelizes the condensation of parameters rou tine t setubv parallelizes the Jacobian setup routine both parallelizes both routines In general the recommended option is both m The number of processing units to use currently only used with the t option Table 1 1 Command line options For example to run the AUTO 2000 executable auto exe in serial mode you just type auto exe 11 To run the same command in parallel using the Pthreads library on 4 processors you type auto exe t both 4 If you were to try and run the above command on a machine which did not have the Pthreads library the command would exit with an error and inform you that the Pthreads library is not available Running the
206. s 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 inspected The amount of diagnostic data can be controlled via the AUTO constant IID see Section 5 9 2 The user has some control over the fort 6 screen and fort 7 output via the AUTO constant IPLT Section 5 9 3 Furthermore the subroutine pvls can be used to define solution measures 19 which can then be printed by parameter overspecification see Section 5 7 10 For an example see demo pvl The AUTO commands sv ap and df can be used to manipulate the output files fort 7 fort 8 and fort 9 Furthermore the AUTO command 1b can be used to delete and relabel solutions simultaneously in fort 7 and fort 8 For details see Section A The graphics program PLAUT can be used to graphically inspect the data in fort 7 and fort 8 see Chapter B 20 Chapter 4 Command Line User Interface 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
207. s likely to use 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 color_list A list of colors to use for multiple plots decorations Turn on or off the axis tick marks etc error_symbol The symbol to use for error points foreground The background color of the plot grid Turn on or off the grid hopf_symbol The symbol to use for Hopf bifurcation points index An array of indicies to plot label An array of labels to plot label_defaults A set of labels that the user is likely to use limit_point_symbol The symbol to use for limit points mark_t The t value to marker with a small ball maxx The upper bound for the x axis of the plot maxy The upper bound for the y 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 period_doubling symbol The symbol to use for period doubling bifurcation points runner The runner object from which to get data 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 colu
208. s the bifurcation solution in a easy to use form The object returned by the commandParseSolutionFile 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 14 prints out the label of the first solution in a bifurcation solution The queriable parts of the object are listed in Table 4 6 AUTO gt data s1 Parsed file fort 8 AUTO gt print datal0 Branch number 2 2TISW 1 Label 6 2NCOL O gt NTST O Parameters 0 0 14 0 2 0 0 0 0 0 0 0 Point number 1 Type name EP Type number 9 p72 0 0 14 0 2 0 0 0 0 0 0 0 parameters 0 0 14 0 2 0 0 0 0 0 0 0 AUTO gt print data 0 Label 6 AUTO gt data 0 data 0 ft 0 0 u 0 0 0 0 Figure 4 14 This figure shows an example of parsing a bifurcation solution The first command data dg ab reads in the bifurcation solution and puts it into the variable data The second command print data 0 prints out all of the data about the first solution in the list The third command print data 0 Label prints out the label of the first point The last command prints the value of the solution at the first point of the first solution The individual elements of the array may be accessed in two ways either by the i
209. s 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 3 1 have indeed been correctly classified 76 6 6 Memory Requirements Pre defined maximum values of certain AUTO constants are in auto 2000 src auto_c h see also Section 1 3 These maxima affect the run time memory requirements and should not be set to unnecessarily large values If an application only solves algebraic systems and if NDIM is large then memory requirements can be much reduced by setting each of NTSTX NCOLX NBCX NINTX equal to 1 in auto 2000 src auto c h followed by recompilation of the AUTO libraries TT Chapter 7 AUTO Demos Tutorial 7 1 Introduction The directory auto 2000 demos has a large number of subdirectories for example ab pp2 exp etc each containing all necessary files for certain illustrative calculations Each subdirectory say xxx corresponds to a particular equation and contains one equations file xxx c and one or more constants files c xxx i one for each successive run of the demo To see how the equations have been programmed inspect the equations file To understand in detail how AUTO is inst
210. sed for discretization NTST remains fixed during any particular run but can be changed when restarting Recommended value of NTST As small as possible to maintain convergence Demos exp ab spb For mesh adaption see IAD in Section 5 3 3 5 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 5 3 3 IAD This constant controls the mesh adaption IAD 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 branch Most demos use IAD 3 which is the strongly recommended value 62 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 branch with a fixed mesh IAD 0 Be sure to set IAD back to IAD 3 for computing eventual non trivial bifurcating solution branches 5 4 Tolerances 5 4 1 EPSL Relative convergence criterion for equation parameters in the Newton Chord method Most demos use EPSL 10 or EPSL 1077 which is the recommended
211. should not be a branch point Demos ab exp frc lor bend A subroutine bend that defines the boundary conditions Demo exp kar icnd A subroutine icnd that defines the integral conditions Demos int lin fopt A subroutine fopt that defines the objective functional Demos opt ops pvls A subroutine pvls for defining solution measures Demo pvl 3 3 Arguments of stpnt Note that the arguments of stpnt depend on the solution type When starting from a fixed point or an analytically or numerically known space dependent solution stpnt must have four arguments namely NDIM U PAR T Here T is the inde pendent space variable which takes values in the interval 0 1 T is ignored in the case of fixed points Demos exp and ab Similarly when starting from an analytically known time periodic solution or rotation the arguments of stpnt are NDIM U PAR T where T denotes the independent time variable which takes values in the interval 0 1 In this case one must also specify the period in PAR 11 Demos frc lor pen When using the fc command Section A for conversion of numerical data stpnt must have four arguments namely NDIM U PAR T In this case only the parameter values need to be defined in stpnt Demos lor and pen 18 3 4 User Supplied Derivatives If AUTO constant JAC equals 0 then derivatives need not be specified in func bend icnd and fopt see Section 5 2 4 I
212. step in fort 9 Type commandQuerylterations xxx to list the number of Newton iterations per continuation step in the info file d xxx Aliases iterations it 53 4 13 27 commandQueryLimitpoint Purpose Print the value of the limit point function Description Type commandQueryLimitpoint to list the value of the limit point function in the output file fort 9 This function vanishes at a limit point fold Type commandQueryLimitpoint xxx to list the value of the limit point function in the info file d xxx Aliases Im limitpoint 4 13 28 commandQueryNote Purpose Print notes in info file Description Type commandQueryNote to show any notes in the output file fort 9 Type commandQueryNote xxx to show any notes in the info file d xxx Aliases nt note 54 4 13 29 commandQuerySecondaryPeriod Purpose Print value of secondary periodic bif fen Description Type commandQuerySecondaryPeriod 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 commandQuerySecondaryPeriod xxx to list the value of the secondary periodic bifurcation function in the info file d xxx Aliases sc secondaryperiod sp 4 13 30 commandQueryStepsize Purpose Print continuation step sizes Descriptio
213. sts 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 C 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 C 2 The Menu Bar C 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 c The corresponding constants file c xxx is also loaded if it exists The equation name xxx remains active until redefined C 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 C 2 3 Write button This pull down menu contains the item Write to write the loaded files xxx c and c xxx by the active equation name and the item Write As to write these files by a selected new name which then becomes the active name 204 C 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 va
214. sure the problem definition is loaded run c ab 1 compute a stationary solution branch with folds and Hopf bifurcation sv ab save output files as b ab s ab d ab run c ab 2 s ab compute a branch of periodic solutions from the Hopf point ap ab append the output files to b ab s ab d ab Table 7 6 Commands for Run 1 and Run 2 of demo ab the demo constants files that were prepared in advance and assume you are in the directory into which the ab demo has already been copied It is instructive to look at the constants files c ab 1 and c ab 2 used in the two runs above The significance of each AUTO constant set in these files can be found in Chapter 5 Note in particular the AUTO constants that were changed between the two runs see Table 7 7 Constant Run 1 Run 2 Reason for Change IPS 1 2 To compute periodic solutions in Run 2 IRS 0 4 To specify the Hopf bifurcation restart label NICP 1 2 The second run has two free parameters ICP 1 1 11 To use and print PAR 1 and PAR 11 in Run 2 NMX 100 150 To allow more continuation steps in Run 2 NPR 100 30 To print output every 30 steps in Run 2 Table 7 7 Differences in AUTO constants between c ab 1 and c ab 2 Actually for periodic solutions AUTO automatically adds PAR 11 the period as second parameter However for the period to be printed one must specify the index 11 in the ICP list as shown in Table 7 7
215. t model a one compartment activator inhibitor system Kern vez 1980 are given by s so s pR s a a a ay a pR s a a where ah R gt 0 s a ieee K The free parameter is p In the fold continuation s is also free AUTO COMMAND ACTION mkdir plp create an empty work directory cd plp change directory demo plp copy the demo files to the work directory 1ld plp load the problem definition run Ce plp 1 lst run compute a stationary solution branch and locate HBs sv plp save output files as b plp s plp d plp run c plp 2 s plp compute a branch of periodic solutions and locate a fold Constants changed IPS IRS NMX ap plp append output files to b plp s plp d plp run c plp 3 s plp Compute a locus of Hopf bifurcation points Constants changed IPS ICP ISW NMX RL1 sv 2p save output files as b 2p s 2p d 2p run c plp 4 s plp generate starting data for the fold contin uation Constants changed IPS IRS ICP NMX sv tmp save output files as b tmp s tmp d tmp run c plp 5 s tmp fold continuation restart data from s tmp Constants changed IRS NUZR ap 2p append output files to b 2p s 2p d 2p run c plp 6 s 2p compute an isola of periodic solutions restart data from s 2p Constants changed IRS ISW NMX NUZR sv iso save output files as b
216. tation of homoclinic flip bifurcations In preparation Scheffer M 1995 Personal communication Smith B Boyle J Dongarra J Garbow B Ikebe Y Klema X amp Moler C 1976 Matrix Eigensystem Routines EISPACK Guide Vol 6 Springer Verlag Taylor M A amp Kevrekidis I G 1989 Interactive AUTO A graphical interface for AUTO86 Technical report Department of Chemical Engineering Princeton University 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 Tech nical report Center for Research on Parallel Computing California Institute of Technology Pasadena CA 91125 CRPC 95 3 211
217. ter e PAR 10 is zero within the allowed tolerance At label 7 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 branch passes through this point can be read from the information in d 3 However in d 3 the line ORIENTABLE 0 2982090775D 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 of the variable indicating the orientation is small compared to its value at the other regular points Data for the adjoint equation at LAB 5 7and 9 at and on either side of the inclination flip are presented in Fig 16 1 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 146 ITWIST 1 IRS 2 NMX 50 ICP 1 4 NPUSZR O as saved in c san 4 and running make fourth we obtain a no convergence error 16 3 Non orientable Resonant Eigenvalues Inspecting the output saved in the third run we observe the existence of a non orientable homo clinic orbit at label 7 corresponding to N 40 We restart at this
218. th respect to A3 is computed Along this branch the value of the optimality parameter T 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 73 is in fact located and hence an extremum of the objective functional with respect to both A 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 A2 and Az 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 branch 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 branch computed with IPS 15 123 The free scalar variables specified in the AUTO constants files for Run 3 and Run 4 are shown in Table 12 2 Index 3 11 12 22 22 23 31 Variable A3 T a Ta A2 As 7 Table 12 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 th
219. tion on the ab demo see Section 7 2 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 4 2 Let us examine more closely what action each of the commands performs First copydemo ab Section 4 13 7 in the reference copies the files in AUTO_DIR demo ab into the work directory Next load equation ab Section 4 13 33 in the reference informs the AUTO 2000 CLUI that the name of the user defined function file is ab c The command load is one of the most commonly used commands in the AUTO 2000 CLUI since it reads and parses the user files which are manipulated by other commands The AUTO 2000 CLUI stores this setting until it is changed by a command such as another load command The idea of storing information is one of the ideas that sets the CLUI apart from the command language described in Section A Next load 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 by AUTO 2000 unless it is loaded in again after the changes are made Finally run Section 4 13 31 in the reference uses the user defined functions loaded by the load equation ab command and the AUTO constants loaded by the load constants ab 1 to run AUTO 2000 Figure 4 2 showed two of the file types that the load comm
220. tpnt Demo ivp IPS 2 Computation of periodic solutions Starting data can be a Hopf bifurcation point Run 2 of demo ab a periodic orbit 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 subroutine bend 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 subroutine 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 subroutine bend 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 15 21 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 subroutine func
221. tput files as b 2p s 2p d 2p run c ab 4 s ab compute the locus of folds in reverse direction with changes from c ab 3 DS sign ap 2p append the output files to b 2p s 2p d 2p run c ab 4 s ab compute a locus of Hopf points with changes from c ab 4 IRS ap 2p append the output files to b 2p s 2p d 2p Table 7 9 Commands for Runs 3 4 and 5 of demo ab 84 Column 1 5 00e 00 1 00e 00 0 00e 00 0 00e 00 2 00e 01 1 00e 01 Column 0 Figure 7 1 The bifurcation diagram of demo ab Columns 1 1 00e 01 0 00e 00 2 00e 01 6 006 01 1 00e 00 4 00e 01 8 00e 01 Columns 0 Figure 7 2 The phase plot of solutions 6 7 and 10 in demo ab 85 7 9 Relabeling Solutions in the Data Files Next we want to plot the two parameter diagram computed in the last three runs However the solution labels in these runs are not distinct This is due to the fact that in each of these three runs the restart solution was read from s ab while the computed solutions were stored in s 2p Consequently these runs were unaware of each other s results which led to non unique labels For relabeling purpose and more generally for file maintenance there is a utility program that can be invoked as indicated in Table 7 10 Its use is illustrated in Table 7 11 AUTO COMMAND ACTION r1 2p run the relabeling program on b 2p and s 2p
222. tra 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 branch Finally we can continue in the opposite direction along the branch from the original starting point again with IEQUIB 1 BR PT TY LAB 5 10 15 20 25 30 a The results of both computations are presented in Figure 20 1 from which we see that the orbit 35 EP 8 9 10 11 12 13 14 OONN OSA PAR 3 997590E 01 705299E 01 416439E 01 133301E 01 857688E 01 590970E 01 334159E 01 shrinks to zero as PAR 1 p 0 aona A p make third L2 NORM 060153E 01 551872E 01 031844E 01 D00668E 01 958712E 01 406182E 01 843173E 01 174 PwWWNNN BE PAR 1 637322E 01 065264E 01 507829E 01 959336E 01 415492E 01 872997E 01 329270E 01 20 2 Detailed AUTO Commands AUTO COMMAND ACTION run c she 1 h she 1 s dat sv 1 mkdir she create an empty work directory cd she change directory demo she copy the demo files to the work directory us she use the starting data in she dat to create s dat continue heteroclinic orbit restart from s dat save output files as b 1 s 1 d 1 run c she 2 h she 2 s dat sv gt 2 repeat with IEQUIB 1 save output files as b 2 s 2 d 2 run c
223. ts 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 18 3 More Accuracy and Saddle Node Homoclinic Orbits 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 node 163 or BT where the convergence to the 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 make sixth we obtain among the output BR PT TY LAB PERIOD L2 NORM Ave PAR 2 1 35 UZ 6 1 000000E 03 1 661910E 00 1 500000E 01 We can now repeat the computation of the branch of saddle homoclinic orbits in PAR 1 and PAR 2 from this point with the test functions Yy and w9 for non central saddle node homoclinic orbits activated make seventh The saddle node point is now detected at BR PT TY LAB PAR 1 a at PAR 2 PAR 29 PAR 30 1 30 UZ 8 1 765003E 01 2 405345E 00 2 743361E 06 2 309317E 01
224. uch extrema in more parameters Demo ops Compute curves of solutions to 2 2 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 additional parameter 14 variables Demos exp int Determine folds and branch points along solution branches to the above boundary value problem Branch switching is possible at branch points Curves of folds can be computed in two parameters Demos bvp int 2 4 Parabolic PDEs For 2 3 the program can Trace out branches 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 branches of the related ODE u 2 v z v z D lc v z f u z p 2 4 where z x ct with the wave speed c specified by the user Demo wav Run 2 Trace out branches of periodic wave solutions to 2 3 that emanate from a Hopf bifurcation point of Equation 2 4 The wave speed c is fixed a
225. use in numerical bifurcation analysis 16 Chapter 3 How to Run AUTO 3 1 User Supplied Files The user must prepare the two files described below This can be done with the GUI described in Chapter 4 or independently 3 1 1 The equations file xxx c A source file xxx c containing the C subroutines func stpnt bend icnd fopt and pvls Here xxx stands for a user selected name If any of these subroutines is irrelevant to the problem then its body need not be completed Examples are in auto 2000 demos where e g the file ab ab c defines a two dimensional dynamical system and the file exp exp c defines a boundary value problem The simplest way to create a new equations file is to copy an appropriate demo file In GUI mode this file can be directly loaded with the GUI button Equations New see Section C 2 3 1 2 The constants file c xxx AUTO constants for xxx c are normally expected in a corresponding file c xxx Specific examples include ab c ab and exp c exp in auto 2000 demos See Chapter 5 for the significance of each constant 17 3 2 User Supplied Subroutines The purpose of each of the user supplied subroutines in the file xxx c is described below func defines the function f u p in 2 1 2 2 or 2 3 stpnt This subroutine is called only if TRS 0 see Section 5 8 5 for IRS which typically is the case for the first run It defines a starting solution u p of 2 1 or 2 2 The starting solution
226. ution namely the previous solution along a solution branch 122 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 hence use the automatically generated extended system Run 1 Locate a Hopf bifurcation The free system parameter is Az Run 2 Compute a branch of periodic solutions from the Hopf bifurcation Run 3 This run retraces part of the periodic solution branch 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 A3 Run 4 Branch switching at the above found branch point yields nonzero values of the adjoint variables Any point on the bifurcating branch 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 branch Run 5 The above found starting solution is continued in two system parameters here Az and Az i e a two parameter branch of extrema wi
227. utton must be used C 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 C 2 203 C 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 the Define Constants buttons they are grouped by function as in Chapter 5 namely Problem definition constants Discretization constants convergence Tolerances continuation Step Size diagram Limits designation of free Parameters constants defining the Computation and constants that specify Output options C 13 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 exi
228. wever 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 y rn c sib 8 h sib hom s 6 ap 6 BR PT TY LAB PAR 4 L2 NORM Les PAR 5 3 7 UZ 24 1 499999E 01 4 944903E 01 3 602482E 03 3 30 EP 25 7 614033E 02 4 987463E 01 2 648395E 06 187 So the 2 homoclinic orbit converges back to the 1 homoclinic orbit at the inclination flip bifur cation The output is appended to b 6 s 6 and d 6 The resulting 2 homoclinic orbits can be seen using plot 6 and is depicted in Figure 22 3 a Columns 0 1 00 00 Figure 22 3 The 2 homoclinic orbit as a is changed a The two different 3 homoclinic orbits b 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 23 and once again stop when PAR 21 0 2 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 NDIM 3 in the parameter file because HomCont handles this internally rn c sib 10 h sib hbs3 s 6 sv 10 BR PT TY LAB PAR 20 a PAR 21 PAR 23 PAR 5 3 10 26 3 458963E 01 7 878940E 07 6 421573E 07 1 062630E 11
229. which are typed to the AUTO 2000 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 2000 CLUI 4 2 General Overview The AUTO 2000 command line user interface CLUI is similar to the command language de scribed in Section A 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 ca pabilities This chapter will provide documentation for the AUTO 2000 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 documentation on the Python language including tutorials and pointers to further documentation 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 2000 or in any of its subdirectories Then type auto If your command search path has been correctly set see
230. y obtained output point from AUTO e g from continu ation 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 142 ii iii iv Data from numerical integration e g computation of a stable periodic orbit or an approx imate homoclinic obtained by shooting can be read in from a data file using the general AUTO utility us see earlier in the manual 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 After running us the restart data is stored in the format of a previously computed solution in s dat When starting from this solution IRS should be set to 1 and the value of ISTART is irrelevant By setting ISTART 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 amp Monteiro 1993 For details of implementation the reader is referred to Sectio

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