Home

млг - Department of Mathematics, University of Utah

1. Effect of the Phase Condition Time Integration with Euler s Method o AUTO Demos BVP 11 1 11 2 11 3 11 4 11 5 11 6 11 7 11 8 exp Bratu s Equation int Boundary and Integral Constraints o e A Nonlinear ODE Eigenvalue Problem A Linear ODE Eigenvalue ProbleM bvp lin non kar spb ezp A Non Autonomous BVP The Von Karman Swirling Flows e A Singularly Perturbed BVP Complex Bifurcation in a BVP 45 46 46 46 46 49 49 50 50 50 53 53 54 99 56 56 97 58 99 60 61 64 65 66 67 68 69 70 72 73 74 12 AUTO Demos Parabolic PDEs 83 12 1 pdl Stationary States 1D Problem lies see 6h od be ee a 84 12 2 pd2 Stationary States 2D Problem ces oe AAA e NE AA e 85 12 3 wan Periodi Waves cenre toaca ta a ee ee ee 86 12 4 bre Chebyshev Collocation in Space o eee 87 12 5 brf Finite Differences in Space eta td Se we ele e ak 88 12 6 bru Euler Time Integration the Brusselator 89 13 AUTO Demos Optimization 90 13 1 opt A Model Algebraic Optimization Problem 91 13 2 ops Optimization of Periodic Solutions os E a SS 92 13 3 obv Optimization for a BVP aa a ae Bae hia ee a 96 14 AUTO Demos Connecting orbits 98 14 1 fsh A Saddle
2. Figure 19 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 19 4 Three dimensional blow up of the solution curves t at labels 1 dotted and 2 solid line from Figure 3 8 132 0 930 1 030 1 010 0 990 0 970 0 950 0 930 1 020 1 000 0 980 0 960 0 940 x Figure 19 5 Computed homoclinic orbits approaching the BT point Note that these solutions were obtained by choosing a smaller step DS and more output smaller NPR in r kpr 4 A blow up of the region close to the origin of this figure is shown in Figure 19 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 ats PAR 10 diga 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 19 5 Note how the computed homoclinic orbits approaching the BT point have their endpoints well away from
3. O 080 0 090 Figure 8 1 The bifurcation diagram of demo ab 0 100 0 110 51 0 120 0 130 0 140 10 Figure 8 2 The phase plot of solutions 6 7 and 10 in demo ab COMMAND ACTION cp r ab 3 r ab changes from r ab 1 IRS NICP ICP ISW DSMAX r ab compute a locus of folds Osv 2p save output files as p 2p q 2p d 2p cp r ab 4 r ab changes from r ab 3 DS sign r ab compute the locus of folds in reverse direction ap 2p append the output files to p 2p q 2p d 2p cp r ab 5 r ab changes from r ab 4 IRS Or ab compute a locus of Hopf points ap 2p append the output files to p 2p q 2p d 2p Table 8 9 Commands for Runs 3 4 and 5 of demo ab AUTO COMMAND ACTION lb 2p run the relabeling program on p 2p and q 2p Table 8 10 Command to run the relabeling program on p 2p and q 2p 52 RELABELING COMMAND ACTION l list the labeled solutions in q 2p r relabel the solutions l list the new solution labeling w rewrite p 2p and q 2p Table 8 11 Relabeling commands for the files p 2p and q 2p 8 10 Plotting the 2 Parameter Diagram To run PLAUT on the files p 2p and q 2p enter the command listed in Table 8 12 The PLAUT commands for plotting the two parameter diagram are then as given in Table 8 13 The saved plot is shown in Figure 8 3 AUTO COMMAND ACTION run PLAUT to graph the contents of p
4. Table 13 5 Commands for running demo ops 95 13 3 obv Optimization for a BVP This demo illustrates use of the method of successive continuation for a boundary value opti mization problem A detailed description of the basic method as well as a discussion of the specific application considered here is given in Doedel Keller amp Kern vez 1991b The required extended system is fully programmed here in the user supplied subroutines in obv f For the case of periodic solutions the optimality system can be generated automatically see the demo ops Consider the system u t ua t ub t Azer urA2rs where p u1 Az Az ui Agu Azuf with boundary conditions u1 0 0 u 1 0 The objective functional is u f u t ale d E The successive continuation equations are given by u t u t ub t D wilt APA p wo t 2y u t 1 walt wr t where 9 Pu 2P 24941 Argus Ou with u 0 0 w1 0 61 9 wa 0 0 u 1 0 w1 1 62 0 we 1 0 w OO eee DE dz RO on de 0 Amat Em de 0 Io Aye 4128 21 wo t Ey Ta dt 0 Gis M1 er u1 42 43 21 1 wo t SA 73 dt 0 In the first run the free equation parameter is A All adjoint variables are zero 13 3 13 4 13 5 13 6 13 7 Three extrema of the objective function are located These correspond to branch points and in the second run branch s
5. 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 Lorenz J 1982 Nonlinear boundary value problems with turning points and properties of difference schemes in W Eckhaus amp E M de Jager eds Singular Perturbation Theory and Applications Springer Verlag Rodr guez Luis A J 1991 Bifurcaciones multiparam tricas en osciladores aut nomos PhD thesis Department of Applied Mathematics University of Seville Spain Rucklidge A amp Mathews P 1995 Analysis of the shearing instability in nonlinear convection and magnetoconvection Submitted to Nonlinearity Russell R D amp Christiansen J 1978 Adaptive mesh selection strategies for solving boundary value problems SIAM J Numer Anal 15 59 8
6. 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 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 a 2 parameter PAR 1 1 35335E 01 1 09381E 08 the locus of PAR 1 11 1 31939E 03 Appended to 2p a 2 parameter PAR 1 8 80940E 05 L2 NORM 3 98712E 00 3 14630E 00 2 21917E 00 1 69684E 00 1 60388E 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 O O O O OO locus of folds L2 NORM 2 06012E 00 2 13650E 01 folds in reverse direction L2 NORM U 1 4 99653E 01 9 53147E 01 U 1 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 OOO ON U 2 1 99861E 00 2 13437E 01 U 2 9 96432E 01 3 58651E 03 9 96426E 01 locus of Hopf points L2 NORM 1 17440E 01 48 U 1 9 14609E 01 U 2 1 17083E 01 PERIOD 2 721E 00 6 147E 00 1 399E 01 9 956E 01 1 867E 03 PAR 3 2 499F 00 3 748E 01 PAR 3 1 050E 00 PAR 3 9 362E 02 8 5 Executing Selected Runs Aut
7. 2 0 2 o fo o 1 f t Case 2 2 1 fo 1 fo Jo 1 ja Table 14 3 Problem coefficients in demo stw With w x t u x ct z x ct one obtains the reduced system ur 2 Ua ub z cus B u u2 C u1 A u1 14 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 uy 2 A ui ue uh Z cuz B u u2 C u1 14 6 Sharp traveling waves then correspond to heteroclinic connections in this transformed system 101 Finally we map 0 7 gt 0 1 by the transformation 2 T With this scaling of the independent variable the reduced system becomes ui E TA u ua ub T leu Blu1 uz C u 14 7 For Case 1 this equation has a known exact solution namely 1 _l a CO TT This solution has wave speed c 1 In the limit as T gt its phase plane trajectory connects the stationary points 1 0 and 0 3 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 COMMAND ACTION mkdir stw create an empty work directory cd stw change directory dm stw copy the demo files to the work di
8. make fourth we obtain the output BR PT TY LAB PAR 1 L2 NORM MAX U 1 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 151 gt 0 25 0 50 0 75 1 00 0 00 0 20 0 40 0 60 0 80 1 00 0 10 0 30 0 50 0 70 0 90 x T Figure 22 3 Two R2 reversible homoclinic orbits at P 1 6 corresponding to labels 1 smaller amplitude and 5 larger amplitude 0 00 0 20 0 40 0 60 0 80 1 00 0 10 0 30 0 50 0 7 0 0 90 x T Figure 22 4 An Ro reversible homoclinic orbit at label 8 152 22 4 Detailed AUTO Commands COMMAND ACTION mkdir rev cd rev dm rev cp rev f 1 rev f cp rev dat 1 rev dat cp r rev 1 r rev cp s rev s rev fc rev h rev dat Osv 1 Cp r rev 2 r rev Cp S rev 2 s rev h rev 1 ap 1 cp rev f 3 rev f cp rev dat 3 rev dat cp r rev 3 r rev Cp s rev 3 S rev fc rev h rev dat svu 3 cp r rev 4 r rev Cp 8 rev 4 S rev h rev 3 ap 3 create an empty work directory change directory copy the demo files to the work directory get equations file to rev f get the starting data to rev dat get the AUTO constants file get the HomCont constants file use the starting data in rev dat to crea
9. one need only define the vector field and the objective functional as in done in the file ops f For reference purpose it is convenient here to write down the full extended system in its general form u t Tf u t d TER period u f ER AER w t T fu u t A w t suple ygulu t A wE ER k y ER u 1 u 0 0 w 1 w 0 0 Jo ut u t dt 0 13 2 Jo o g u t A dt 0 Jo w t w t K 7 a dt 0 a ER Jo F u t A wE ygr u t A ro dt 0 mER So Tf ult A w t yg u t A 7 dt 0 TER t 1 n Above up is a reference solution namely the previous solution along a solution branch 92 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 A3 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 k 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
10. 0 1 and es 1 Koper 1995 To copy across the demo kpr and compile we type dm kpr 19 2 The Primary Branch of Homoclinics First we locate a homoclinic orbit using the homotopy method The file kpr f already con tains approximate parameter values for a homoclinic orbit namely A PAR 1 1 851185 k PAR 2 0 15 The files r kpr 1 and s kpr 1 specify the appropriate constants for con tinuation 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 ovis 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 p 1 q 1 d 1 Upon plotting the data at label 2 see Figure 19 1 it can be noted that although the right hand projection boundary condition is satisfied the solution is still quite away from the equilibrium 129 1 000 1 020 1 000 0 980 0 960 0 940 1 010 0 990 0 970 0 950 x Figure 19 1 Projection on the x y plane of solutions of the boundary value problem with 2T 19 08778 1 000 1 020 1 000
11. 145 Detailed AUTO Commands COMMAND ACTION mkdir she cd she dm she cp r she 1 r she cp s she 1 s she fc she h she dat sv 1 cp r she 2 r she cp s she 2 s she h she dat sv 2 cp r she 3 r she cp s she 3 s she Oh she 2 ap 2 create an empty work directory change directory copy the demo files to the work directory get the AUTO constants file get the HomCont constants file use the starting data in she dat to create q dat continue heteroclinic orbit restart from q dat save output files as p 1 q 1 d 1 get the AUTO constants file get the HomCont constants file repeat with IEQUIB 1 save output files as p 2 q 2 d 2 get the AUTO constants file get the HomCont constants file continue in reverse direction restart from q 2 append output files to p 2 q 2 d 2 Table 21 1 Detailed AUTO Commands for running demo she 146 A Pea x a A lt a E p x A we va Se oe Figure 21 1 Projections into x y z space of the family of heteroclinic orbits 147 Chapter 22 HomCont Demo rev 22 1 A Reversible System The fourth order differential equation u Py u u 0 arises in a number of contexts e g as the travelling wave equation for a nonlinear Schr dinger equation with fourth order dissipation Buryak amp Akhmediev 1995 and as a model of a strut on a symmetric nonlinear elastic foundation Hunt Bolt amp Thompson 1989 It may be
12. 7 FED 19 1 foo a foos7 va 1 u5 1 0 11 8 ui 1 fees where alfo 7 ZE 42 529 Blf 1 Hli HAP RT 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 f The period T is fixed T 500 The starting solution is u 0 i 1 5 at y 1 fo 0 11 9 COMMAND ACTION mkdir kar create an empty work directory cd kar change directory dm kar copy the demo files to the work directory cp r kar 1 r kar get the constants file r kar computation of the solution branch sv kar save output files as p kar q kar d kar Table 11 6 Commands for running demo kar 80 11 7 spb A Singularly Perturbed BVP This demo illustrates the use of continuation to compute solutions to the singularly perturbed boundary value problem uy u9 u A ujue uj 1 u1 11 10 with boundary conditions u 0 3 2 u 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 COMMAND ACTION mkdir spb crea
13. Repeating computations in the opposite direction along the curve IRS 1 DS 0 01 in r mtn 2 make second one obtains BR PT TY LAB PAR 1 ae 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 Ds K Z 5 1803 0 063855 Note that these data were obtained using a smaller value of NTST than the original computation compare r mtn 1 with r 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 123 18 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 Dp make third starts from D Note that now NUNSTAB 1 IEQUIB 1 has been specified in s mtn 3 Also test functions w9 and 419 have been activated in order to monitor for non hyperbolic equilibria along the homoclinic locus We get the following output BR PT TY LAB 10 20 30 40 EP Be Pe 11 12 13 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 indi
14. cp s san 9 s san get the HomCont constants file h san 6 run AUTO HomCont restart solution read from q 6 ap 6 append output files to p 6 q 6 d 6 Table 16 2 These two sets of AUTO Commands are equivalent 114 Chapter 17 HomCont Demo san 17 1 Sandstede s Model Consider the system Sandstede 1995a t ax by azx 4 f az x 2 32 y be ay 3b2 3ary jfi az 2y 17 1 Z cz ur yzy a6 2 1 1 y as given in the file san f Choosing the constants appearing in 17 1 appropriately allows for computing inclination and orbit flips as well as 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 Sana E aers This solution is specified in the routine STPNT 17 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 86 1 and y u ft 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 09 115 and s
15. 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 7 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 6 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 value of ISW ISW 1 If IRS is the label of a branch point or a period doubling bifurcation then branch switching will be done For period doubling bifurcations it is recommended that NTST be increased For examples see Run 2 and Run 3 of demo lor where branch switching is done at period doubling bifurcations and Run 2 and Run 3 of demo bvp where branch switching is done at a transcritical branch point ISW 2 If IRS is the label of a fold a Hopf bifurcation point 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 al
16. lt 20 2 35 ra Dab ee Dab as 35 6 7 7 Boundary value folds stant eas AA AR AAA 35 6 7 8 Optimization problems 450 it a 35 6 7 9 Internal free parameters eee ER eh O a 36 6 7 10 Parameter overspecification ooa s s e s a ee 36 6 8 Computation Constants ies he ie oe Wale ed A Sh aoe ee 37 Oris SEP sitet E E RN E 37 Do EOP ston Geek newness Stat eh ee re win Ore epee en oe a ee 37 6837 A II ce eee ac oe ee ere ba AO a at 37 03A MABES e a cee ake Ev a cay he A kG A ele ee See Get eee Goede ae Aa 38 Gor SIRO io Btn Be tate Hel america dnt ae ek ek aw eek fh cee ag 38 O10 UPS iby Gat gues sed eet ewe at ee ee ye Pe eS ee OE ye oh 38 6 9 O tp tC trol as eee elt ak ge hk ten Ac ak Aa ee ge aa 40 GU NER so Ate he oo te Hi of ob Pad PLAS each Bde Sank Be Gute B nt Be aoe 40 Oe SLOG ot tears ia a Oe Pt ae EAN Ae Se ee eS 40 AS MPL gale gee Arp ee Si Age A ama a ent le Bete eee i 41 CIA A A ae ol te age tae O Ye op ae Yet ea oy dy ag 41 Notes on Using AUTO 42 7 1 Restrictions on the Use of PAR rs A E A A IS OS 42 ae o e A O be Be 42 To Correctness Or Reus lt A E A eh ON eo ed 43 7 4 Bifurcation Points and Folds os ek a oe ee ee eee ee cee So 43 2 Ploqtier Multipliers i e O A AAA Ae A AA 43 7 6 Memory Requirements led yt A A y A Be eB 44 8 AUTO Demos Tutorial 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 8 9 8 10 8 11 8 12 8 13 Introduction ab Copying the Demo Files E
17. q brc d brc cp r brc 3 r bre constants changed IRS ISW r bre compute a solution branch from a secondary periodic bifurcation ap bre append the output files to p brc q brc d bre Table 12 4 Commands for running demo brc 87 12 5 brf Finite Differences in Space This demo illustrates the computation of stationary solutions and periodic solutions to systems of parabolic PDEs in one space variable A fourth order accurate finite difference approximation is used to approximate the second order space derivatives This reduces the PDE to an autonomous ODE of fixed dimension which AUTO is capable of treating The spatial mesh is uniform the number of mesh intervals as well as the number of equations in the PDE system can be set by the user in the file brf inc As an illustrative application we consider the Brusselator Holodniok Knedlik amp Kub ek 1987 u Dy Lugo u v B 1 ut A v D y L Wgy 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 12 4 COMMAND ACTION mkdir brf create an empty work directory cd brf change directory dm brf copy the demo files to the work directory cp r brf 1 r brf get the first constants file Or brf compute
18. t We restart from the data saved at LAB 8 and LAB 13 in q 7 and q 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 TEQUIB 2 etc or by appending Y 0 to the continuation of a saddle homoclinic orbit with TEQUIB 1 The first approach is used in the example mtn for contrast we shall adopt the second approach here 136 make twelfth make thirteenth The projection onto the e k plane of all four of these codimension two curves is given in Figure 19 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 almost overlaid but that as in Figure 19 6 the orbits look quite distinct in phase space 19 5 Detailed AUTO Commands COMMAND ACTION mkdir kpr create an empty work directory cd kpr change directory dm kpr copy the demo files to the work directory cp r kpr 1 r kpr get the AUTO constants file cp s kpr 1 s kpr get the HomCont constants file h kpr continuation in the time length parameter PAR 11 sv 1 save output files as p 1 q 1 d 1 cp r kpr 2 r kpr get the AUTO constants file cp s kpr 2 s kpr get the HomCont constants file Oh kpr 1 locate the homoclinic orbit restart from q 1 sv 2 save output files as p 2 q 2 d 2 cp r kpr 3 r kpr get the AUTO c
19. 0 980 0 960 0 940 1 010 0 990 0 970 0 950 x Figure 19 2 Projection on the x y plane of solutions of the boundary value problem with 2T 60 0 130 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 me 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 19 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 ins 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 r 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 parameter continuation The fourth run make fourt
20. 1 NDIM 1 0 1 0 The output 149 0 00 0 20 0 40 0 60 0 80 1 00 0 10 0 30 0 50 0 70 0 2 9 0 x T Figure 22 1 R Reversible homoclinic solutions on the half interval T 0 1 where T 39 0448429 for P approaching 2 solutions with labels 1 5 respectively have decreasing amplitude 0 00 0 20 0 40 0 60 0 80 1 00 0 10 0 30 0 50 0 70 0 90 x T Figure 22 2 R reversible homoclinic orbits with oscillatory decay as x gt oo corresponding to label 6 and monotone decay at label 10 150 BR PT TY LAB PAR 1 L2 NORM MAX U 1 1 15 UZ 2 1 700000E 00 3 836401E 01 4 890015E 01 1 16 LP 3 1 711574E 00 3 922135E 01 5 442385E 01 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 r 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 22 3 The computation ends in a no convergence point The solution here is depicted in Figure 22 4 The lack of convergence is due to the large peak and trough of the solution rapidly moving to the left as P gt 2 cf Champneys amp Spence 1993 Continuing from the initial solution in the other parameter direction
21. 2p and q 2p Table 8 12 Command to run PLAUT for files p 2p and q 2p PLAUT COMMAND ACTION d0 set default option ax select axes 15 select real columns 1 p and 5 ps in p 2p bd0 plot the 2 parameter diagram p3 versus p clear the screen cl d2 set other default option bd make a blow up of the current diagram 0 15 0 2 5 enter diagram limits sau save plot fig 3 upon prompt enter a new file name e g fig 3 exit from PLAUT Table 8 13 PLAUT commands for files p 2p and q 2p 8 11 Converting Saved PLAUT Files to PostScript Plots are saved in compact Tektronix PLOT10 format In Table 8 14 it is shown how such files can be converted to PostScript format Note that the latter files are much bigger 53 o 000 0 025 0 050 0 075 0 100 0 125 0 150 Figure 8 3 Loci of folds and Hopf bifurcations for demo ab AUTO COMMAND ACTION Ops fig 1 convert file fig 1 into PostScript file fig 1 ps lpr fig 1 ps system dependent print fig 1 ps on your printer Ops fig 2 convert file fig 2 into PostScript file fig 2 ps lpr fig 2 ps system dependent print fig 2 ps on your printer Ops fig 3 convert file fig 3 into PostScript file fig 3 ps lpr fig 3 ps system dependent print fig 3 ps on your printer Table 8 14 Printing commands for the saved Figures in demo ab 8 12 Using the GUI Demos can also be run using the GUI See Table 5 1 for the correspondence be
22. 30 4 946824E 03 3 288447E 02 3 876291E 02 2 104384E 02 The data are appended to the stored results in p 1 q 1 and d 1 One could now display all data using the AUTO command Qp 1 to reproduce the curve P shown in Figure 18 1 It is worthwhile to compare the homoclinic curves computed above with a curve To const along which the system has a limit cycle of constant large period T 1046 178 which can easily be computed using AUTO or locbif Such a curve is plotted in Figure 18 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 In lla a in the saddle homoclinic case and as a a in the saddle node case 18 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 Dop PAR 3 To achieve this we restart at label 4 corresponding to the codim 2 point D We return to continuation of saddle node homoclinics NUNSTAB 0 IEQUIB 2 but append the defining equation 415 0 to the continuation problem via NFIXED 1 IFIXED 1 15 The new continuation problem is specified in r mtn 6 and s mtn 6 make sixth Notice that we set ILP 1 and choose PAR 3 as the first
23. Bifurcations The commands in Table 8 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 8 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 q ab while the computed solutions were stored in q 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 8 10 Its use is illustrated in Table 8 11 PLAUT COMMAND dl bd bd 08 14 5 4 5 fig 1 dD Ww Q 6 7 10 d 13 d 23 d 2 Q 8 a Table 8 8 C choose one of the default settings make a blow up of current bifurcation diagram save the current plot enter 2D mode for plotting labeled solutions select labeled orbits 6 7 and 10 in q ab default orbit display u versus scaled time select columns 1 and 3 in q ab display the orbits us versus scaled time select columns 2 and 3 in q ab phase plane display uz versus u1 save the current plot exit from 2D mode ommands to be typed in the PLAUT window 10
24. 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 1991b 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 10 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 1 f u p Demo dd2 Find extrema 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 period
25. E Rodr guez Luis A Gamero 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 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 SIAM J Appl Math 50 No 2 460 482 Holodniok M Knedlik P amp Kub ek M 1987 Continuation of periodic solutions in parabolic differential equations in T Kiipper R Seydel amp H Troger eds Bifurcation Analysis Algorithms Applications Vol INSM 79 Birkhauser Basel pp 122 130 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 in P H Rabinowitz ed Applications of Bifurcation Theory Academic Press pp 359 384 155 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
26. Node Connection a 99 14 2 nag A Saddle Saddle Connection o ao a 02 00 0000 100 14 3 stw Continuation of Sharp Traveling Waves oaoa aaa 101 15 AUTO Demos Miscellaneous 103 15 1 pvl Use of the Subroutine PVLS ooa a a eee ee ee 104 15 2 ext Spurious Solutions to BVP aoaaa et ead a e Se ep 105 15 3 tim A Test Problem for Timing AUTO 106 16 HomCont 107 16 1 Tntrod ction Le p ip e a aa E as cm Oa aa 107 16 2 HomCont Files and Subroutines o a a a 107 16 3 Hom Cont Constants a 2 A tal Ge Ee a A A 108 TOO NUNSTAB merreni mea Eye Ba Be Se ae Hele ale Gok St See es E N 108 13 2 NSTAB is ee ti a A ee ee 108 do EQUI B te ee oe Bele els E Da a a al 108 SA IES A A es IA AS AS A 109 O PO TARY boxe e nr td te e od bl hte a ee da 109 16 3 0 NREV 9 TREV Saio Go dew bee e e a ee E 109 16 3 7 NETXED DETXED rahe manari one A a ea ae ee ee snes Y 109 TO NPS Ta MES Dri A Fa et Gee ANA DE BAS en a AA 109 16 4 Restrictions on HomCont Constants e 110 16 5 Restrictions on the Use of PAR 0 e e 110 16 6 Test Functions os A a 111 16 7 Starting Strategies A 112 16 8 Notes on Running HomCont Demos oaoa aa 0 020004 0 113 17 HomCont Demo san 115 tral Sandstede S Model xare oeta A Gna AN Gi Wise a a hes 115 17 2 neha too A A ae Bt ae ee at eh a ee a 115 17 3 Non orientable Resonant Eigenvalues 0 000
27. 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 ul p3 u2 u Uy PiU Ug UUs 10 4 Us U1U2 pous Numerical simulations with a simple initial value solver show the existence of a stable periodic orbit when p 280 po 8 3 ps 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 uz and uz The correponding parameter values are defined in the user supplied subroutine STPNT The AUTO command fc lor then converts the data in lor dat to a labeled AUTO solution with label 1 in a new file q 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 r lor Note that the file q dat should be used for restart only Do not append new output files to q dat as the command fc lor only creates q dat with no corresponding p dat COMMAND ACTION mkdir lor create an empty work directory cd lor change directory dm lor copy the demo files to the work directory cp r lor 1 r lor get the first constants file fc lor convert lor dat to AUTO format in q dat r lor dat compute a sol
28. To to append the output files to other existing data files 5 2 8 Plot button This pull down menu contains the items Plot to run the plotting program PLAUT for the data files p xxx and q xxx where xxx is the active equation name and the item Name to run PLAUT with other data files 5 2 9 Files button This pull down menu contains the item Restart to redefine the restart file Normally when restarting from a previously computed solution the restart data is expected in the file q xxx where xxx is the active equation name Use the Restart button to read the restart data from an other data file in the immediately following run The pull down menu also contains the following items Copy to copy P XXX q xxx d xxx r xxx to p yyy q yyy d yyy Y YYy resp Append to append data files p xxx q xxx d xxx to P yyy q yyy d yyy resp Move to move p xxx q xxx d XXX r xxx to p yyy q yyy d yyy r yyy resp Delete to delete data files p xxx q xxx d xxx Clean to delete all files of the form fort 0 and exe 5 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 26 5 2 11 Misc button This pulldown menu contains the items Tek Window and VT102 Window for opening w
29. a A a 16 Sid s BASIC COMIDAS scs ssr E alae BID oe AE A A 16 3 5 2 Plotting commands e tt e a A O E nas de S 16 3 5 3 File manipulation a A o a ke be 17 93DA DIBSNOSTICE io ngs PEE AE AA A E AA 17 IN A Y 18 A a A A Ye ete rae cae os a ie den Gs 18 3 5 7 HomCont commands o A ee dl ah E eh 19 3 5 8 Copying a A ae foot ee See ae A he pep le Se gee we BSA 19 3 5 9 Pendula animation 2 4 20 428s ame Nh nae Se He eo A HO Re lees 19 3 5 10 Viewing the Manli REA a Ge eo 19 Or Output EN Dek De aS Lote e ac oie Bee A 20 4 The Graphics Program PLAUT 21 4 1 Basic PLAUT Commands he nt Go ati ee E a OR A he el 21 42 erate pions soss tee e eet Gk oe Doe Ge SS ale SE OS OE BES HES 22 43 Other PLAUT Commands 0 0 0 0 0 0 00 2 eee eee 23 14 Ponting PLAUT Piles sae db elk ee be ee O be eA N 23 Graphical User Interface 24 5 1 General Overview 24 5 1 1 The Menu bar 2 20 00 0000 2 24 5 1 2 The Define Constants buttons ooo a a e 24 5 1 3 The Load Constants buttons aoa aoa a a e e e a ee 25 5 1 4 The Stop and Exit buttons eee eee 25 S2 Ehe Ment Bakre sa ta 4 vp be p a a Ge al eat ae Yee 2G 25 9AT gt QUA nous DUO aaa au Pr ee eee a Sg de ey Sn Se a ees 25 02 20 Edit buttons A pia gi ee wet Oe ae EO ee a ee 25 S23 Write BUG OM ie te et ae ete eR ae ees ce a Ee a ee ete ee 25 5 2 4 Define button so ss one Petea daot mu dag a NE a D p i 25 D29 Ru un putton iou e a e e ee ea ach
30. 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 s and Ag i e a two parameter branch of extrema with respect to Az is computed Along this branch the value of the optimality parameter 72 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 T is in fact located and hence an extremum of the objective functional with respect to both Az and Az has been found Note that in general 7 is the value of the functional that vanishes at an extremum with respect to the system parameter A Run 6 In the final run the above found two parameter extremum is continued in three system parameters here A 42 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 o
31. auto 97 test This will execute a selection of demos from auto 97 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 97 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 2 Restrictions on Problem Size There are size restrictions in the file auto 97 include auto h on the following AUTO constants the effective problem dimension NDIM the number of collocation points NCOL the number of mesh intervals NTST the effective number of boundary conditions NBC the effective number of integral conditions NINT the effective number of equation parameters NPAR the number of stored branch points NBIF for algebraic problems and the number of user output points NUZR See Chapter 6 for the significance these constants Their maxima are denoted by the corresponding constant followed by an X For example NDIMX in auto h denotes the maximum value of NDIM If any of these maxima is exceeded in an AUTO run then a message will be printed The exception is the the maximum value of NPAR which if exceeded may lead to unreported errors Upon installation NPARX 36 it should never be decreased below that value see also Section 7 1 Size restr
32. automatically execute all runs of the demo To automatically run a demo in step by step mode type make first make second etc to run each separate computation of the demo At each step the user is encouraged to plot the data saved by using the command p e g p 1 plots the data saved in p 1 and q 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 Note that the sequence of detailed AUTO commands given in these tables can be abbreviated as illustrated in Table 16 1 and Table 16 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 8 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 113 COMMAND ACTION cp r san 1 r san get the AUTO constants file cp s san 1 s san get the HomCont constants file run AUTO HomCont save output files as p 6 q 6 d 6 QH san 1 Osv 6 Table 16 1 These two sets of AUTO Commands are equivalent COMMAND ACTION cp r san 9 r san get the AUTO constants file
33. branch point function in the data file d xxx This function vanishes at a branch point hb Type hb to list the value of the Hopf function in the output file fort 9 This function vanishes at a Hopf bifurcation point Type hb zzz to list the value of the Hopf function in the data file d xxx This function vanishes at a Hopf bifurcation point 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 17 Type Osp zrez 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 it to list the number of Newton iterations per continuation step in fort 9 Type it rrz to list the number of Newton iterations per continuation step in d xxx Ost Type Qst 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 ev Type ev to list the eigenvalues of the Jacobian in fort 9 Algebraic problems Type ev zzz to list the eigenvalues of the Jacobian in d xxx Algebraic problems f1 Type Of to list the Floquet multipliers in the output file fort 9 Differential equations Type f zzz to list the Floquet multipliers in the data file d xxx Differential equations 3 5 5 File edi
34. 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 z v 2 u z D e v z f u z 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 along 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 f
35. 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 Do 0 01 We get among other output BR PT TY LAB PAR 3 L2 NORM PAR 1 PAR 2 1 22 LP 19 1 081212F 02 5 325894F 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 Dp 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 D2 The results of this computation and a similar one starting from D in the opposite direction with DS 0 01 are displayed in Figure 18 3 125 18 5 Detailed AUTO Commands mkdir mtn cd mtn dm mtn cp r mtn 1 r mtn cp s mtn 1 s mtn fe mtn h mtn dat Osv 1 cp r min 2 r mtn cp s mtn 2 s mtn h mtn 1 ap 1 cp r min 3 r mtn cp s mtn 3 s mtn Oh mtn 1 ap 1 cp r min 4 romtn cp s mtn 4 s mtn h mtn 1 Osv 4 cp r min d r mtn cp s mtn 5 s mtn h mtn 1 ap 1 cp r mtn 6 r mtn cp s mtn 6 s mtn Oh mtn 1 sv 6 COMMAND ACTION create an empty work directory change directory copy the demo files to the work directory get the AUTO constants file get the HomCont constants file use the starting data in mtn dat to create q dat continue saddle node homoclinic o
36. criterion for stationary bound states of solitons with radiationless oscillating tails Physical Review E 51 3572 3578 Champneys A amp Kuznetsov Y 1994 Numerical detection and continuation of codimension two homoclinic bifurcations Int J Bifurcation amp Chaos 4 795 822 Champneys A amp Spence A 1993 Hunting for homoclinic orbits in reversible systems a shooting technique Adv Comp Math 1 81 108 Champneys A Kuznetsov Y amp Sandstede B 1996 A numerical toolbox for homoclinic bifurcation analysis 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 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 gt b gt 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 154 Doedel E J amp Kern vez J P 1986b A numerical analysis of wave phenomena i
37. directory cd ezp change directory dm ezp copy the demo files to the work directory cp r ezp 1 r ezp get the first constants file r ezp lst run compute solution branch containing fold sv ezp save output files as p ezp q ezp d ezp cp r ezp 2 r ezp constants changed IRS ISW r ezp 2nd run compute bifurcating complex solution branch ap ezp append output files to p ezp q ezp d ezp cp r ezp 3 r ezp constant changed DS Or ezp 3rd run compute 2nd leg of bifurcating branch ap ezp append output files to p ezp q ezp d ezp Table 11 8 Commands for running demo ezp 82 Chapter 12 AUTO Demos Parabolic PDEs 12 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 problem parameter The equation is Ou du Bio age 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 sc
38. 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 19 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 q 3 make tenth Note that ITWIST 1 in s kpr 10 so that the adjoint is also continued and there is one fixed condition IFIXED 1 13 so that test function 4 3 has been frozen Among the output there is a codimension three point zero of 9 where the neutrally twisted homoclinic orbit collides with the saddle node curve BR PT TY LAB PAR 1 bis J 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 q 3 is continued similarly make eleventh giving among its output another codim 3 saddle node inclination flip point BR PT TY LAB PAR 1 ah 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 non central saddle node homoclinic orbits it is necessary to work on the data without the solution
39. 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 19 for an example however we recommend the homotopy method only for expert users To compute the orientation of a homoclinic orbit in order to detect inclination flip bifur cations it is necessary to compute in tandem a solution to the modified adjoint variational equation by setting ITWIST 1 In order to obtain starting data for such a computation when restarting from a point where just the homoclinic is computed upon increasing ITWIST to 1 AUTO generates trivial data for the adjoint Because the adjoint equations are linear only a single step of Newton s method is required to enable these trivial data to converge to the correct unique bounded solution This can be achieved by making a single continuation step in a trivial parameter i e a parameter that does not appear in the problem Decreasing ITWIST to 0 automatically deletes the data for the adjoint from the continuation problem 16 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 dm zzz Simply typing make or make all will then
40. option is mainly intended for the detecting stationary solutions 39 IPS 17 This option can be used to continue stationary solutions of parabolic systems obtained from an evolution run with IPS 16 For examples see the second runs of demos pdi and pd2 6 9 Output Control 6 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 0 or if NPR gt NMX then no such output is written Note that special solutions such as branch points folds end points etc are always written in fort 8 Furthermore one can specify parameter values where plotting and restart data is to be written see Section 6 9 4 For these reasons and to limit the output volume it is recommended that NPR output be kept to a minimum 6 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 TID 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 dif
41. 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 6 7 10 Parameter overspecification The number of specified parameter indices is allowed to be be greater than the generic number In such case there will be overspecified parameters whose values will appear in the screen and fort 7 output but which are not part of the continuation process A simple example is provided by demo opt where the first three runs have overspecified parameters whose values although constant are printed There is however a more useful application of parameter overspecification In the user supplied 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 onl
42. 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 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 which can be converted to PostScript format with the AUTO commands ps and pr see Section 4 4 21 cl To clear the graphics window l
43. pseudo arclength stepsize after every IADS steps If the Newton Chord iteration converges rapidly then DS will be increased but never beyond DSMAX If a step fails then it will be retried with half the stepsize This will be done repeatedly until the step is successful or until DS reaches DSMIN In the latter case nonconvergence will be signalled The strongly recommended value is IADS 1 which is used in almost all demos 6 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 32 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 Weight 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 6 5 6 NTHU Under certain circumstances one may want to modify the weight accorded to individual
44. r bup compute the first bifurcating branch in opposite direction ap bup append output files to p bvp q bvp d bvp Table 11 3 Commands for running demo bvp TT 11 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 L2 norm of the first solution component The eigenvalue problem is given by the equations 1 Uy zn us 11 4 26 pix he with boundary conditions u1 0 pz 0 and ui 1 0 We add the integral constraint 1 ur t dt p3 0 0 Then pz is simply the L2 norm of the first solution component In the first two runs ps is fixed while p and ps are free In the third run ps is fixed while p and p are free COMMAND ACTION mkdir lin create an empty work directory cd lin change directory dm lin copy the demo files to the work directory cp r lin 1 r lin get the first constants file r lin lst run compute the trivial solution branch and locate eigenvalues sv lin save output files as p lin q lin d lin cp r lin 2 r lin constants changed IRS ISW DSMAX r lin 2nd run compute a few steps along the bifurcating branch ap lin append output files to p lin q lin d lin cp r lin 3 r lin const
45. s cir 2 We also activate the test functions pertinent to codimension two singular ities which may be encountered along a branch of saddle focus homoclinic orbits viz Ya Ya ws yo and 4109 This must be specified in three ways by choosing NPSI 5 and appropriate IPSI 1 in s cir 2 by adding the corresponding parameter labels to the list of continuation parame ters ICP I in r cir 2 recall that these parameter indices are 20 more than the corresponding w indices and finally adding USZR functions defining zeros of these parameters in r cir 2 Running make second results in BR PT TY LAB PAR 1 PAR 2 PAR 25 PAR 29 1 17 UZ 5 7 256925E 01 4 535645E 01 1 765251E 05 2 888436E 01 1 75 UZ 6 1 014704E 00 9 998966E 03 1 664509E 00 5 035997E 03 1 78 UZ 7 1 026445E 00 2 330391E 05 1 710804E 00 1 165176E 05 1 81 UZ 8 1 038012E 00 1 000144E 02 1 756690E 00 4 964621E 03 1 100 EP 9 1 164160E 00 1 087732E 01 2 230329E 00 5 042736E 02 with results saved in p 2 q 2 d 2 Upon inspection of the output note that label 5 where PAR 25 0 corresponds to a neutrally divergent saddle focus 45 0 Label 7 where PAR 29 0 corresponds to a local bifurcation Y 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 20 1 undergoes a Hopf bifu
46. 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 6 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 computational 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 6 9 4 6 6 1 NMX The maximum number of steps to be taken along any branch 6 6 2 RLO The lower bound on the principal continuation parameter This is the parameter which appears first in the ICP list see Section 6 7 1 6 6 3 RL1 The upper bound on the principal continuation parameter 33 6 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 6 9 3 for choosing another principal solution measure 6 6 5 Al The upper bound on the
47. storing the test functions see Sec tion 16 6 The output is in an identical format to AUTO except that additional information at each computed point is written in fort 9 This information comprises the eigenvalues of the left hand equilibrium the values of each activated test function and if ITWIST 1 whether the saddle homoclinic loop is orientable or not Note that the statement about orientability is only meaningful if the leading eigenvalues are not complex and the homoclinic solution is not in a flip configuration that is none of the test functions y for i 11 12 13 14 is zero or close to zero see Section 16 6 Finally the values of the NPSI activated test functions are written 110 16 6 Test Functions Codimension two homoclinic orbits are detected along branches of codim 1 homoclinics by lo cating zeroes of certain test functions The test functions that are switched on during any continuation are given by the choice of the labels 2 and are specified by the parameters NPSI 1 IPSI 1I I 1 NPSI in s 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 eigenva
48. the stationary solution branch with Hopf bifurcations sv brf save output files as p brf q brf d brf cp r brf 2 r brf constants changed IRS IPS r brf compute a branch of periodic solutions from the first Hopf point ap brf append the output files to p brf q brf d brf cp r brf 3 r brf constants changed IRS ISW r brf compute a solution branch from a secondary periodic bifurcation ap brf append the output files to p brf q brf d brf Table 12 5 Commands for running demo brf 88 12 6 bru Euler Time Integration the Brusselator This demo illustrates the use of Euler s method for time integration of a nonlinear parabolic PDE The example is the Brusselator Holodniok Knedlik amp Kub ek 1987 given by u Dz L uzs u B 1 u A 12 5 Y Dy L eq vv 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 7ax and v x B A 0 7sin ru Note that in the subroutine STPNT the space derivatives of u and v must also be provided see the equations file bru f Euler time integration is only first order accurate so that the time step m
49. via make sixth make seventh make eighth make ninth make tenth with appropriate continuation parameters and user output values set in the corresponding files r san xx All the output is saved to q 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 1 PAR 7 PAR 8 with the test function 1411 for orbit flips with respect to the stable manifold activated 117 make eleventh The output detects an inclination flip by a zero of PAR 31 at PAR 7 0 BR PT TY LAB PAR 7 ae PAR 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 17 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 qe 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 0 118 17 5 Detailed AUTO Commands mkdir san cd san dm san cp r san 1 r san cp s san 1 s san h san sv 1 cp r san 2 r san cp san 2 s san Oh san 1 sv 2 cp r san 3 r san cp s san 3 5 san
50. with c fixed no phase condition Osv 0 save output files as p 0 q 0 d 0 cp r fsh 2 r fsh constants changed IRS ICP NINT DS r fsh 0 continuation in c and T with active phase condition sv fsh save output files as p fsh q fsh d fsh Table 14 1 Commands for running demo fsh 99 14 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 00 lt 2 lt s00 t gt 0 14 3 f w a w 1 w w a Usas 1 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 uj 2 ua z Uy z cu2 z f ui z a where z ct and d dz If a 1 2 and c 0 then there are two analytically known heteroclinic connections one of which is given by 14 4 1 env 2 A 1 Y eve U2 z u z 00 X lt 2 lt X 00 ur 2 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 f and repeat the computations below see also Friedman amp Doedel 1991 COMMAND ACTION mkdir nag create an empty work directory cd nag change directory dm nag copy the demo files t
51. 0 Sandstede B 1995a Constructing dynamical systems possessing homoclinic bifurcation points of codimension two In preparation Sandstede B 1995 Convergence estimates for the numerical approximation of homoclinic solutions In preparation Sandstede B 1995c Numerical computation of homoclinic flip bifurcations In preparation Scheffer M 1995 Personal communication 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 dz Kevrekidis I G 1989 Interactive AUTO A graphical interface for AUTO86 Technical report Department of Chemical Engineering Princeton University 156 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 3 157
52. 0 02 00 008 117 PEA Orbit Elp gt ote ed we 17 5 Detailed AUTO Commands 18 HomCont Demo mtn 18 1 A Predator Prey Model with Immigrati0D o 18 2 Continuation of Central Saddle Node Homoclinics 18 3 Switching between Saddle Node and Saddle Homoclinic Orbits 18 4 Three Parameter Continuation 18 5 Detailed AUTO Commands 19 HomCont Demo kpr 19 1 Koper s Extended Van der Pol Model y ai E A A e a a a NA 19 2 The Primary Branch of Homoclinics ooa 02 02 0220 008 5 19 3 More Accuracy and Saddle Node Homoclinic Orbits 19 4 Three Parameter Continuation 19 5 Detailed AUTO Commands 20 HomCont Demo cir 20 1 Electronic Circuit of Freire et a a a a a 20 2 Detailed AUTO Commands 21 HomCont Demo she 21 1 A Heteroclinic Example 21 2 Detailed AUTO Commands 22 HomCont Demo rev 22 1 A Reversible System 22 2 An R Reversible Homoclinic Solution 0 0 0 0 0000002 ee ae 22 3 An Ro Reversible Homoclinic Solution e 22 4 Detailed AUTO Commands 122 122 122 124 125 126 129 129 129 133 136 137 140 140 143 144 144 146 Preface This 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 19862 Doedel amp Wang 1995 Wang amp Doed
53. 1 cp r tor 3 r tor constants changed IRS ISW NMX Or tor 1 compute a bifurcating branch of periodic solutions restart from q 1 ap 1 append output files to p 1 q 1 d 1 Table 10 10 Commands for running demo tor 69 10 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 r 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 01 ed sind I y 2 01 10 11 p ed sing I 7 d1 2 or in equivalent first order form bY p p Ya Yi ep sind 1 y 02 1 10 12 wh etba sin dg 1 y 01 2 Throughout y 0 175 Initially e 0 1 and J 0 4 Numerical data representing one complete rotation are contained in the file pen dat Each row in pen dat contains five real numbers namely the time variable t da Y and 2 The correponding parameter values are defined in the user supplied subroutine STPNT Actually in this example a scaled time variable t is given in pen dat For this reason the period PAR 11 is also set in STPNT Normally AUTO would automatically set the period according to the data in pen dat The AUTO command fc pen converts the dat
54. 11E 01 which is consistent with the theoretical result that the solution tends uniformly to zero as P gt 0 Note by plotting the data saved in q 1 that only half of the homoclinic orbit is computed up to its point of symmetry See Figure 22 1 The second run continues in the other direction of PAR 1 with the test function Y 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 316007E 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 22 2 22 3 An R Reversible Homoclinic Solution make third Copies the files rev f 3 and rev dat 3 to rev f and rev dat and runs them with the constants stored in r rev 3 and s 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 s rev 1 via NREV 1 IREV I I
55. 6 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 ho motopy method for starting or if the equilibrium contains complex eigenvalues in its lin earization The homotopy method ISTART 3 is not fully implemented for heteroclinic connections TEQUIB lt O saddle node homoclinic orbits IEQUIB 2 or reversible systems IREV 1 Certain test functions are not valid for certain forms of continuation see Section 16 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 16 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 ho moclinic 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 16 7 below PAR 21 PAR 36 These parameters are used for
56. AUTO 97 CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS with HomCont Eusebius J Doedel Alan R Champneys Concordia University University of Bristol Montreal Canada United Kingdom Thomas F Fairgrieve Yuri A Kuznetsov Ryerson Polytechnic University CWI Amsterdam Toronto Canada The Netherlands Bjorn Sandstede Xianjun Wang Weierstrab Institut Concordia University Berlin Germany Montreal Canada March 29 1998 Contents 1 Installing AUTO 8 Toa Misra Otte i eanu ee AR ae Soe E E PS A BE Oe oe 8 1 2 Restrictions on Problem Size aa eas ee O Y eR 9 1 3 Compatibility with Older Versions 2 2 2 4 4 64 2 So ar wea le ee Oe how a 9 2 Overview of Capabilities 10 Qo A ta A eke Saute bee yi ee eee bo cee dad T 10 2 2 Algebraic Systems o id ds lh Ge Sk lage a 10 2 3 Ordinary Differential Equations e e e 11 24 Thera Oe EDS Set rl AR A RR ote is Gee Tene ey nie e AA 12 Quo lt Discretizati n ed ton ap x ett bo o e e a Gh te aoe oh ge a 13 3 How to Run AUTO 14 Bol Wiser Supplied Files 2 grea line Ga ere Ba Ee WO em a ces 14 3 1 1 The equations file xxx f Ge ye ae ey AE we eee 14 3 1 2 The constant HiRes 2 bu be Hagens fees PS ae a ae eet 8 14 3 2 User Supplied Subroutines 2 0 Lia don Gone do Be By Blew Soe eee es 15 3 3 Arsuments of STPS gt Sa ee ee ARS ANA 15 3 4 User Supplied Derivatives 2000200 46 ers mea a a a 16 3 5 Running AUTO using Command Mode ia os
57. EQUIB 1 Homoclinic orbits to hyperbolic equilibria the equilibrium is solved for during continuation Initial values for the equilibrium are stored in PAR 11 1 I 1 NDIM in STPNT TEQUIB 2 Homoclinic orbits to a saddle node initial values for the equilibrium are stored in PAR 11 1 I 1 NDIM in STPNT IEQUIB 1 Heteroclinic orbits to hyperbolic equilibria the equilibria are specified explic itly in PVLS and stored in PAR 11 1 I 1 NDIM left hand equilibrium and PAR 11 1 I NDIM 1 2 NDIM right hand equilibrium TEQUIB 2 Heteroclinic orbits to hyperbolic equilibria the equilibria are solved for during continuation Initial values are specified in STPNT and stored in PAR 11 1 I 1 NDIM left hand equilibrium PAR 11 1 I NDIM 1 2 NDIM right hand equilibrium 108 16 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 16 7 for more details 16 3 5 ISTART ISTART 1 This option is no obsolete in the current version It may be used as a flag
58. HomCont constants file restart and homotopy to PAR 4 1 0 save output files as p 6 q 6 d 6 get the AUTO constants file get the HomCont constants file homotopy to PAR 5 0 0 restart from q 6 append output files to p 6 q 6 d 6 get the AUTO constants file get the HomCont constants file homotopy to PAR 1 0 5 restart from q 6 append output files to p 6 q 6 d 6 get the AUTO constants file get the HomCont constants file homotopy to PAR 2 3 0 restart from q 6 append output files to p 6 q 6 d 6 get the AUTO constants file get the HomCont constants file homotopy to PAR 7 0 25 restart from q 6 append output files to p 6 q 6 d 6 get the AUTO constants file get the HomCont constants file continue in PAR 7 to detect orbit flip restart from q 6 save output files as p 11 q 11 d 11 get the AUTO constants file get the HomCont constants file three parameter continuation of orbit flip restart from q 11 save output files as p 12 q 12 d 12 Table 17 2 Detailed AUTO Commands for running demo san 120 u6 B 0 2265 0 0 2 45 5 0 he 10 0 u5 Figure 17 1 Second versus third component of the solution to the adjoint equation at labels 5 7 and 9 Figure 17 2 Orbits on either side of the orbit flip bifurcation The critical orbit is contained in the x y plane 121 Chapter 18 HomCont Demo mtn 18 1 A Predator Prey
59. MX NPR DS r Irz compute periodic solutions the final orbit is near homoclinic ap lrz append the output files to p 1rz q 1rz d lrz cp r lrz 3 r lrz constants changed IRS Or lrz compute the symmetric periodic solution branch ap lrz append the output files to p 1rz q 1rz d lrz Table 10 1 Commands for running demo Irz 99 10 2 abc The A gt B gt C Reaction This demo illustrates the computation of stationary solutions Hopf bifurcations and periodic solutions in the A gt B gt C reaction Doedel amp Heinemann 1983 uy u t pi l uie uz u pie 1 u psu2 10 2 us U3 p3uz pipae 1 us p2p5ua with po 1 p3 1 55 p4 8 and ps 0 04 The free parameter is p COMMAND ACTION mkdir abe create an empty work directory cd abc change directory dm abc copy the demo files to the work directory cp r abc 1 r abc get the first constants file r abc compute the stationary solution branch with Hopf bifurcations sv abc save output files as p abc q abc d abc cp r abc 2 r abe constants changed IRS IPS NICP ICP r abc compute a branch of periodic solutions from the first Hopf point ap abc append the output files to p abc q abc d abc cp r abc 3 r abe constants changed IRS NMX r abc compute a branch of periodic solutions from the second Hopf point ap abc append the output files to p abc q abc d abc Table 10 2 Commands for runni
60. Model with Immigration Consider the following system of two equations Scheffer 1995 l X AXY X RX 1 a DK K B X 18 1 BR gy A TRAE A BHY The values of all parameters except K Z are set as follows R 0 5 Ay 0 4 B 0 6 Do 0 01 E 0 6 A 1 0 Bo 0 5 D 0 15 The parametric portrait of the system 18 1 on the Z K plane is presented in Figure 18 1 It contains fold t 2 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 18 2 Continuation of Central Saddle Node Homoclinics Local bifurcation analysis shows that at K 6 0 Z 0 06729762 the system has a saddle node equilibrium X Y 5 738626 0 5108401 with one zero and one negative eigenvalue Direct simulations reveal a homoclinic orbit to this saddle node departing and returning along its central direction i e tangent to the null vector Starting from this solution stored in the file mtn dat we continue the saddle node central homoclinic orbit with respect to the parameters K and Z by copying the demo and running it dm mtn make first 122 The file mtn f conta
61. 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 ES arctan Kx which get better as K increases In the numerical calculations below we use K 10 The free parameter is a COMMAND ACTION mkdir chu create an empty work directory cd chu change directory dm chu copy the demo files to the work directory cp r chu 1 r chu get the first constants file r chu lst run stationary solutions Osv chu save output files as p chu q chu d chu cp r chu 2 r chu constants changed IPS IRS ICP ICP r chu 2nd run periodic solutions with detection of period doubling ap chu append the output files to p chu q chu d chu Table 10 12 Commands for running demo chu T2 10 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 t ao ata 10 14 us uw l u This equation has a Hopf bifurcation from the trivial solution at A 0 The bifurcating branch of periodic 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
62. RS ICP 1 NTST NMX DS DSMAX Or fre 0 compute solution branch restart from q 0 sv fre save output files as p frc q frc d frc Table 10 6 Commands for running demo frc 65 10 6 ppp Continuation of Hopf Bifurcations This demo illustrates the continuation of Hopf bifurcations in a 3 dimensional predator prey model Doedel 1984 This curve contain branch points where one locus of Hopf points bifurcates from another locus of Hopf points The equations are ui u1 1 uy pate Uy Pp209 pai Ug psuzuz pi 1 enPov 10 7 Us P3U3 psuzu3 Here p 1 4 p3 1 2 pa 3 ps 3 pe 5 and py is the free parameter In the continuation of Hopf points the parameter pa is also free COMMAND ACTION mkdir ppp create an empty work directory cd ppp change directory dm ppp copy the demo files to the work directory cp r ppp 1 r ppp get the first constants file Or ppp compute stationary solutions detect Hopf bifurcations Osv ppp save output files as p ppp q ppp d ppp cp r ppp 2 r ppp constants changed IPS IRS ICP etc Or ppp compute a branch of periodic solutions Oap ppp append the output files to p ppp q ppp d ppp cp r ppp 3 r ppp constants changed Or ppp compute Hopf bifurcation curves Osv hb save the output files as p hb q hb d hb Table 10 7 Commands for running demo ppp 66 10 7 plp Fold Continuation for Periodic Solutions This demo which corr
63. Se eS 26 526 Save button 02 graw God Aua wa Be el Se a eas a a GP A 26 Dey Append D tton 4 Wa aa eae eae ee EA a ees 26 duo Plot buttony o rwa esd ae be Ae ROR A eee Se tee a oe 8 26 92 90 Eiles Hutton attic Gu ei Oa eel en ee A T Y 26 52610 Demos buttOne dues de Se Se Se ee dR Sheed Seb loa 26 SALL Mise Hutton o Sea fake hed BA Ges gee hf Ga ge oh be ea Ge OE ae a a gg 27 eal 2 Help putto ie are Seno Sa eee ee a ee de ae Sw bea Pe be 27 Dis Ugn the GUL leat thd oe ae Soe eh gt get Ghee tt 27 pe Customizing the GUL ocs aas eua ad ote a a E h a E eb Beeld A 27 BA Me Printeb tton a A qe T Go tee eee a ee oo 27 BAD GUI Colors a4 nk Baek OR OR Ae Ae E oe A ee ad 27 5A On linehel pe See cann a se Rok mH AN 28 Description of AUTO Constants 29 6 1 The AUTO Constants File 2 000 0 a a 29 6 2 Problem Constants 2 2 0 a 29 622 10 NDIMI aa oh So hun BE ae is See lance fe ys Ala VE in ale Bae a 29 0227 CNBC as o Beenie Seu fois Dec a Sin Oh E OS ale Gy GO otk aD teed wk ad eS 29 Ge 253 NTN fn ac ee Rta A A os 30 ORA DACP oA eB Be atone Ge i Seca ds Sead de aie Gd MN as he ee 30 6 3 Discretization Constants o o er he Ge eH wales aa Ss Dales oe 30 Gs dale ANDES Ts 18202 ei ak A A A Pak aback doh ae oe ae 30 62322 NCUL ss bows eet hie opel th eit ale Ss Bee cee a ee ede d 30 Ord20 ts AD eb A Se teh DOs See ih as i cates aie Se te Bn Ay od Sha 30 A S a 20 3 6 Jes 3S e end ase Ge areal te ac ee de as ee dea kee
64. a in pen dat to a labeled AUTO solution with label 1 in a new file q 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 r pen Note that the file q dat should be used for restart only Do not append new output files to q dat as the command Ofc pen only creates q dat with no corresponding p dat The first run with J 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 70 COMMAND ACTION mkdir pen create an empty work directory cd pen change directory dm pen copy the demo files to the work directory cp r pen 1 r pen get the first constants file r pen dat locate a period doubling bifurcation restart from q dat save output files as p pen q pen d pen cp r pen 2 r pen constants changed IPS NTST ISW NMX r pen a branch of period doubled and out of phase rotations ap pen append output files tp p pen q pen d pen cp r pen 3 r pen constants change
65. ab 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 lb to relabel solutions in fort 7 and fort 8 end To end execution of PLAUT 4 2 Default Options After entering the commands bd0 bd or 2d you will be asked whether you want solution labels grid lines titles or axes labels For quick plotting it is convenient to bypass these selections This can be done by the default commands d0 d1 d2 d3 or d4 below These can be entered as a single command or they can be entered as prefixes in the bd0 and bd commands Thus for example one can enter the command d1bd0 d0 Use solid curves showing solution 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 di 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 so
66. ai 31 GASI AA Ses aT ee AO oa ee Bh ee BE le 31 624 2 ERSU e be ee be es te te ha Be ha Se A 31 624 3 HER SS She cine E ak ee Ae Ge AAS as ads Fee Oh gee aS ee tee es 31 GAA SLIMY ee An he BO A Oe pA 2 ae oo A ete A 31 CAD NENA BO Be at et he en ee ees 31 646 TN E A ie eg A alt ta A a hee oe Nee be It BE SA 31 6 5 Co tinuation Step Sizes sta bac ate ely A deg di dt ee e 32 O Nat ae Glas rel RA RN RR AO 32 Ore DSMIN ama e je ci A se hk ene ene eae eke hens heh ot 32 6 5 3 TBM NN la ara hus a a E E a A E T a EE a te E A a RA E 32 ad TADS Az hi de ink ee we be a a le oe ee ba is TAE E 32 Dedos MHS oo a Gee ere sewer ernest wp ae Bee he ee Sa Se Bee ee Beyer Bo es 32 A E AAA III A IS te te 33 6 6 Diagram MGs fee ee hove ah ea do Es foe Pe EA ARA A 33 CO SNM eis oth E E ode tata cote A AE Od a ty oe 33 61025 CRO a an EE TR Go ee Soe oe ee Se hae 33 620 3 CREL Aun gre a E AS OS 33 GOAN ROP y og go ae ok so tye he oh oa A AR Dat oe A A awe 8 34 Dira AL EN A a ad sacks Be a es Beg Sa Se ey Se oe eS eS 34 Osi Bree Parameters a hara A deh pa kt deta Oh ee ett MRR a ee A am 34 Orle NTC Pins TOPai ge alia Gad a Suter Dona eRe ade ea SEN Eade lah ee ty E 34 6 7 2 Fixed points cose a E Pee ee Bet ae ee e a a i a a 34 6 7 3 Periodic solutions and rotations 0 toe cd Be oe 34 6 7 4 Folds and Hopf bifurcations 20 scsi a Be 35 6 7 5 Folds and period doublings tn a da BEE Bg G AOR es 35 6 7 6 Boundary value problems
67. aled derivative must also be provided see the equations file pv1 f 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 e p u lau COMMAND ACTION mkdir pdl create an empty work directory cd pd1 change directory dm pd1 copy the demo files to the work directory cp r pd1 1 r pd1 get the first constants file Or pdl time integration towards stationary state sv 1 save output files as p 1 q 1 d 1 cp r pd1 2 r pd1 constants changed IPS IRS ICP etc Or pdi 1 continuation of stationary states read restart data from q 1 sv 2 save output files as p 2 q 2 d 2 Table 12 1 Commands for running demo pdl 84 12 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 0u1 0t D 041 03 p u 1 u wuu 15d uz t Deo Ou Ox U2 UU on the space interval 0 L where L PAR 11 1 is fixed throughout as are the diffusion constants D PAR 15 1 and D2 PAR 16 1 The boundary conditions are u 0 u L 0 and u2 0 u2 L 1 for all time In the first run th
68. ameter save output files as p 1 q 1 d 1 constants changed IRS two free equation parameters read restart data from q 1 save output files as p 2 q 2 d 2 constants changed IRS three free equation parameters read restart data from q 2 save output files as p 3 q 3 d 3 constants changed IRS four free equation parameters read restart data from q 3 save output files as p 4 q 4 4 4 Table 13 1 Commands for running demo opt 91 Find the maximum sum of coordinates on the 13 2 ops Optimization of Periodic Solutions This demo illustrates the method of successive continuation for the optimization of periodic solutions For a detailed description of the basic method see Doedel Keller amp Kern vez 19918 The illustrative system of autonomous ODEs taken from Rodriguez Luis 1991 is Y t Aa a 3 z 2 yl A y t x Ag 13 1 a t z2 2 ra with objective functional 1 w f g 2 9 2 Ai Az As A4 de 0 where g x y 2 A1 A2 A3 44 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
69. ands given in for example Table 8 5 can be simplified by using the R com mand For Table 8 5 the equivalent command sequence is given in Table 8 17 COMMAND ACTION OR ab 1 reads AUTO constants from r ab 1 save output files as p ab q ab d ab OR ab 2 reads AUTO constants from r ab 2 append the output files to p ab q ab d ab Table 8 17 Abbreviated AUTO commands 55 Chapter 9 AUTO Demos Fixed points 9 1 enz Stationary Solutions of an Enzyme Model The equations that model a two compartment enzyme system Kern vez 1980 are given by s1 so 1 s2 81 pR s1 9 1 8 so u sa 51 2 pR s2 a where E R s s 1 s8 Hn82 The free parameter is sy Other parameters are fixed This equation is also considered in Doedel Keller amp Kern vez 1991a COMMAND ACTION mkdir enz create an empty work directory cd enz change directory dm enz copy the demo files to the work directory cp r enz 1 r enz get the constants file Or enz compute stationary solution branches sv enz save output files as p enz q enz d enz Table 9 1 Commands for running demo enz 56 9 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 syst
70. ants changed IRS ISW ICP 2 r lin 3rd run compute a two parameter curve of eigenvalues Osv 2p save the output files as p 2p q 2p d 2p Table 11 4 Commands for running demo lin 78 11 5 non A Non Autonomous BVP This demo illustrates the continuation of solutions to the non autonomous boundary value prob lem Uy thd Ga ee 11 5 with boundary conditions u 0 0 uu 1 0 Here x is the independent variable This system is first converted to the following equivalent autonomous system Uy Ua uy peta 11 6 y A with boundary conditions u 0 0 wui 1 0 u3 0 0 For a periodically forced system see demo frc COMMAND ACTION mkdir non create an empty work directory cd non change directory dm non copy the demo files to the work directory cp r non 1 r non get the constants file r non compute the solution branch sv non save output files as p non q non d non Table 11 5 Commands for running demo non 79 11 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 ug Shu lig Tuz uz T 2yu4 u3 20 43 ud 11 7 ta Ts us T 2yue 2u3U4 2u us with left boundary conditions and asymptotic right boundary conditions fo aldo uz 1 uz 1 7 h Y al foo
71. aved in more detail as p 1 q 1 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 s san 2 and IRS 2 NMX 2 and ICP 1 9 saved in r 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 p 2 q 2 and d 2 We can now continue the homoclinic plus adjoint in a 1 PAR 4 PAR 8 by changing the constants stored in r 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 1 I 1 NICP Here PAR 10 is a dummy parameter used in order to make the continuation of the adjoint well posed Theoretically it should be zero if the computation of the adjoint is successful Sandstede 1995a The test functions for detecting resonant bifurcations ISPI 1 1 and inclination flips ISPI 1 13 are also activated Recall that this should be specified in three ways First we add PAR 21 and PAR 33 to the list of continuation parameters in r 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 s san 3 We also add to r san 3 ano
72. bolic PDEs in one space variable using Chebyshev collocation in space More precisely the approximate solution is assumed of the form u x t NH ux t ez x Here uz t corresponds to u x t at the Chebyshev points x _ with respect to the interval 0 1 The polynomials La are the Lagrange interpolating coefficients with respect to points xp yio where o 0 and zn 1 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 Kubi ek 1987 u Dy L Ugo u v B 1 ut A v D L Wgy 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 12 3 COMMAND ACTION mkdir bre create an empty work directory cd bre change directory dm bre copy the demo files to the work directory cp r brc 1 r brc get the first constants file r bre compute the stationary solution branch with Hopf bifurcations sv bre save output files as p brc q brc d brc cp r bre 2 r bre constants changed IRS IPS Or bre compute a branch of periodic solutions from the first Hopf point ap bre append the output files to p brc
73. c 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 BCND and integral constraints in ICND The AUTO constants NBC and NINT must be given correct values IPS 9 This option is used in connection with the HomCont algorithms described in Chapters 16 22 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 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 Starting data can also be specified in STPNT in which case the wave length mu
74. cates 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 410 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 Dz and the whole lower branch of P emanating from it however the solutions between D and D should be considered as spurious therefore we do not save these data The reliable way to compute the lower 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 18 1 1 The program actually computes the saddle saddle heteroclinic orbit bifurcating from the non central saddle node homoclinic at the point D1 see Champneys et al 1996 Fig 2 and continues it to the one emanating from D 124 BR PT TY LAB 10 15 20 16 30 17 40 EP 18 errer PAR 1 4 966429E 00 4 925379E 00 7 092267E 00 1 101819E 01 PAR 2 6 298418E 02 7 961214E 02 1 587114E 01 2 809825E 01 PAR 29 4 382426E 01 3 399102E 01 1 692842E 01 3 482651E 02 PAR
75. ction PSIHO in autlib5 f It is important to remember that in order to specify activated test functions it is required to also add the corresponding label 20 to the list of continuation parameters and a zero of this parameter to the list of user defined output points Having done this the corresponding parameters are output to the screen and zeros are accurately located 16 7 Starting Strategies There are four possible starting procedures for continuation i ii iii iv Data can be read from a previously obtained output point from AUTO e g from contin uation of a periodic orbit up to large period note that the end point of the data stored must be close to the equilibrium These data can be read from fort 8 saved to q xxx by making IRS correspond to the label of the data point in question Data from numerical integration e g computation of a stable periodic orbit or an approx imate homoclinic obtained by shooting can be read in from a data file using the general AUTO utility fc 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 U I J I 1 NDIM where T runs in the interval 0 1 After running fc the restart data is stored in the format of a previously computed solution in q dat When starting from this solution IRS should be set to 1 and the value of ISTART is irrelevant By setting ISTART 2 an e
76. d IRS ISP r pen a secondary bifurcating branch without bifurcation detection ap pen append output files to p pen q pen d pen cp r pen 4 r pen constants changed IRS r pen another secondary bifurcating branch without bifurcation detection ap pen append output files to p pen q pen d pen cp r pen 5 r pen constants changed IRS ICP ICP ISW NMX r pen generate starting data for period doubling continuation sv t save output files as p t q t d t cp r pen 6 r pen constants changed IRS r pen t compute a locus of period doubling bifurcations restart from q t Osv pd save output files as p pd q pd d pd run an animation program to view the solutions in q pen on SGI machines only see also the file auto 97 pendula README Table 10 11 Commands for running demo pen T1 10 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 uy alu h w us U U2 U3 10 13 Us B U2 where i h s az 5 ao a 2 1 r2 1 and where we take 3 14 3 ag 1 7 a 2 7 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
77. d Az as scalar equation parameters as a bifurcation in the third run The parameter Ay and its corresponding optimality variable 74 are not used in this demo Also A is used in the last run only and its corresponding optimality variable 7 is never used 94 COMMAND ACTION mkdir ops cd ops dm ops cp r ops 1 r ops r ops sv 0 Cp r ops 2 r ops Or ops 0 ap 0 Cp r 0ps 3 r Ops Or ops 0 sv 1 Cp r ops 4 Y Ops r ops 1 ap 1 Cp r ops 5 r ops Or ops 1 sv 2 cp r ops 6 r ops Or ops 2 svu 3 create an empty work directory change directory copy the demo files to the work directory get the first constants file locate a Hopf bifurcation save output files as p 0 q 0 d 0 constants changed IPS IRS NMX NUZR compute a branch of periodic solutions restart from q 0 append the output files to p 0 q 0 d 0 constants changed IPS IRS ICP locate a 1 parameter extremum as a bifurcation restart from q 0 save the output files as p 1 q 1 d 1 constants changed IRS ISP ISW NMX switch branches to generate optimality starting data restart from q 1 append the output files to p 1 q 1 d 1 constants changed IRS ISW ICP ISW compute 2 parameter branch of 1 parameter extrema restart from q 1 save the output files as p 2 q 2 d 2 constants changed IRS ICP EPSL EPSU NUZR compute 3 parameter branch of 2 parameter extrema restart from q 2 save the output files as p 3 q 3 d 3
78. ddle node homoclinics restart from q 7 save output files as p 8 q 8 d 8 get the AUTO constants file get the HomCont constants file continue homoclinics from codim 2 point restart from q 8 save output files as p 9 q 9 d 9 get the AUTO constants file get the HomCont constants file 3 parameter curve of inclination flips restart from q 3 save output files as p 10 q 10 d 10 get the AUTO constants file get the HomCont constants file another curve of inclination flips restart from q 3 save output files as p 11 q 11 d 11 get the AUTO constants file get the HomCont constants file continue non central saddle node homoclinics restart from q 7 save output files as p 12 q 12 d 12 get the AUTO constants file get the HomCont constants file continue non central saddle node homoclinics restart from q 8 append output files to p 12 q 12 d 12 Table 19 2 Detailed AUTO Commands for running demo kpr 138 Te 3 5 Ta 9 eps 1 Figure 19 8 Projection onto the PAR 3 PAR 2 plane of the non central saddle node homo clinic orbit curves labeled 1 and 2 and the inclination flip curves labeled 3 and 4 139 Chapter 20 HomCont Demo cir 20 1 Electronic Circuit of Freire et al Consider the following model of a three variable electronic circuit Freire Rodriguez Luis Gamero amp Ponce 1993 t 6 v x By azr bay ay
79. dic 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 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 34 6 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 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 6 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 rotati
80. e amount of diagnostic data can be controlled via the AUTO constant IID see Section 6 9 2 The user has some control over the fort 6 screen and fort 7 output via the AUTO constant IPLT Section 6 9 3 Furthermore the subroutine PVLS can be used to define solution measures which can then be printed by parameter overspecification see Section 6 7 10 For an example see demo pul The AUTO commands sv ap and Odf can be used to manipulate the output files fort 7 fort 8 and fort 9 Furthermore the AUTO command lb can be used to delete and relabel solutions simultaneously in fort 7 and fort 8 For details see Section 3 5 The graphics program PLAUT can be used to graphically inspect the data in fort 7 and fort 8 see Chapter 4 20 Chapter 4 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 p xxx and q xxx To invoke PLAUT use the the p command defined in Section 3 5 The PLAUT window a Tektronix window will appear in which PLAUT commands can be entered For examples of using PLAUT see the tutorial demo ab in particular Sections 8 7 and 8 10 See also demo pp2 in Section 10 3 4 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
81. e 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 u x u2z x Initial data at time zero are u x sin rx L and u 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 f 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 COMMAND ACTION mkdir pd2 create an empty work directory cd pd2 change directory Adm pd2 copy the demo files to the work directory cp r pd2 1 r pd2 get the first constants file Or pd2 time integration towards stationary state sv 1 save output files as p 1 q 1 d 1 cp r pd2 2 r pd2 constants changed IPS IRS ICP etc Or pd2 1 continuation of stationary states read restart data from q 1 sv 2 save output files as p 2 q 2 d 2 Table 12 2 Commands for running demo pd2 85 123 wav Periodic Waves This demo illustrates the computation of various periodic wave solutions to a system of coupled parabolic partial differential eq
82. e for the 24 Define Constants buttons they are grouped by function as in Chapter 6 namely Problem defini tion constants Discretization constants convergence Tolerances continuation Step Size diagram Limits designation of free Parameters constants defining the Computation and constants that specify Output options 5 1 3 The Load Constants buttons The Previous button can be used to load an existing AUTO constants file Such a file is also loaded if it exists by the Equations button on the Menu Bar The Default button can be used to load default values of all AUTO constants Custom editing is normally necessary 5 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 5 2 The Menu Bar 5 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 f The corresponding constants file r xxx is also loaded if it exists The equation name xxx remains active until redefined 5 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 wh
83. e 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 7 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 is to inspect this multiplier in the diagnostics output file fort 9 If this multiplier becomes inaccurate then the automatic detection of potential secondary periodic bifurcations if ISP 2 is discontinued and a warning is printed in fort 9 It is strongly recommended that the contents of this file be habitually inspected in particular to verify whether solutions labeled as BP or TR cf Table 3 1 have indeed been correctly classified 43 7 6 Memory Requirements Pre defined maximum values of certain AUTO constants are in auto 97 include auto h see also Section 1 2 These maxima affect the run time memory requirements and should not be set to unnecessarily
84. e unnecessary files Also enter the command source HOME auto 97 cmds auto env and add this command to your login or cshrc file The Graphical User Interface GUI requires the X Window system and Motif It may be necessary to enter their correct pathname in the appropriate makefile in auto 97 gui Note that AUTO can be very effectively run in Command Mode i e the GUI is not strictly necessary To compile AUTO without GUI type make cmd in directory auto 97 For timing purposes the file auto 97 src autlib1 f contains references to the function etime If this function is not automatically supplied by your f77 compiler then it can be replaced by an appropriate alternative call or it can be disabled by replacing the two occurrences of the string T etime timaray with T 0 To recompile autlib1 f type C 1 in directory auto 97 src To enable the PostScript conversion command ps make the changes indicted in the README file in directory auto 97 tek2ps and recompile by typing make in that directory Moreover to enable the pr command you may have to enter the correct printer name in auto 97 cmds pr To generate the on line manual type make in auto 97 doc To prepare AUTO for transfer to another machine type make superclean in directory auto 97 before 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
85. een 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 13 134 1 00 2 0 15 1 0 0 5 0 0 0 5 Tep L5 Figure 19 6 Two non central saddle node homoclinic orbits 1 and 3 and 2 a central saddle node homoclinic orbit between these two points 1 00 2 0 1 25 1 0 0 5 0 0 0 5 1 0 Les 2 0 Figure 19 7 The big homoclinic orbit approaching a figure of eight 135 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 r kpr 9 This particular computation ends at BR PT TY LAB PAR 1 L2 NORM Lee PAR 2 1 500 EP 24 1 218988F 05 2 181205E 01 2 189666E 00 By plotting phase portraits of orbits approaching this end point see Figure 19 7 we see a canard like like transformation of the big homoclinic orbit to a pair of homoclinic orbits in a figure of eight configuration That we get a figure of eight is
86. el 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 graphical 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 Ove
87. em are uft piuf 1 uf poutus A 9 2 uktl 1 pju pyutul 9 2 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 dm dd2 copy the demo files to the work directory cp r dd2 1 r dd2 get the first constants file r dd2 lst run fixed point solution branches sv dd2 save output files as p dd2 q dd2 d dd2 cp r dd2 2 r dd2 constants changed IRS ISW r dd2 2nd run a locus of Naimark Sacker bifurcations sv ns save output files as p ns q ns d ns Table 9 2 Commands for running demo dd2 97 Chapter 10 AUTO Demos Periodic solutions 10 1 Irz The Lorenz Equations This demo computes two symmetric homoclinic orbits in the Lorenz equations 1 Uy p3 u2 ur Uy Pili U2 U1U3 10 1 1 Uz U1U2 P2U3 Here p is the free parameter and p 8 3 p 10 The two homoclinic orbits correspond to the final large period orbits on the two periodic solution branches COMMAND ACTION mkdir lrz create an empty work directory cd Irz change directory dm lrz copy the demo files to the work directory cp r lrz 1 r lrz get the first constants file Or Irz compute stationary solutions Osv Irz save output files as p 1rz q 1rz d lrz cp r lrz 2 r lrz constants changed IPS IRS NICP ICP N
88. en NCOL 2 may be appropriate 6 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 TAD gt 0 Adapt the mesh every IAD steps along the branch Most demos use IAD 3 which is the strongly recommended value When computing trivial solutions to a boundary value problem for example when all solution components are constant then the mesh adaption may fail under certain circumstances and overflow may occur In such case try recomputing the solution branch with a fixed mesh IAD 0 Be sure to set IAD back to IAD 3 for computing eventual non trivial bifurcating solution branches 30 6 4 Tolerances 6 4 1 EPSL Relative convergence criterion for equation parameters in the Newton Chord method Most demos use EPSL 10 or EPSL 10 which is the recommended value range 6 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 6 4 3 EPSS Relative arclength convergence criterion for the detection of special solutions Most demos use EPSS 107 or EPSS 10 which is the recommended value range Generally EPSS should be approximately 100 to 1000 times the value of EPSL EPSU 6 4 4 ITMX The maximum number of iterations allowed in the accurate location of special solutions such as bifurcatio
89. esponds 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 that model a one compartment activator inhibitor system Kern vez 1980 are given by so s pR s a a a do a SRG a ats where R s a it Se 0 1 s4 s 5 The free parameter is p In the fold continuation sy is also free COMMAND ACTION mkdir plp cd plp dm plp cp r plp 1 r plp r plp sv plp cp r plp 2 r plp Or plp ap plp cp r plp 3 r plp Or plp sv 2p cp r plp 4 r plp Or plp sv tmp cp r plp 5 r plp Or plp tmp ap 2p cp r plp 6 r plp r plp 2p sv iso create an empty work directory change directory copy the demo files to the work directory get the first constants file lst run compute a stationary solution branch and locate HBs save output files as p plp q plp d plp constants changed IPS IRS NMX compute a branch of periodic solutions and locate a fold append output files to p plp q plp d plp constants changed IPS ICP ISW NMX RL1 Compute a locus of Hopf bifurcation points save output files as p 2p q 2p d 2p constants changed IPS IRS ICP NMX generate starting data for the fold continuation save output files as p tmp q tmp d tmp constants changed IRS NUZR fold continuation restart data from q tm
90. expressed as a system Uy U as 22 1 ug Us UA Puz ul u Note that 22 1 is invariant under two separate reversibilities Ry z uz U2 U3 Us t gt us U2 U3 UA t 22 2 and Ro uz U2 U3 Us t gt us U2 U3 U4 t 22 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 f 1 rev dat 1 and rev f 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 22 2 An R Reversible Homoclinic Solution The first run 148 make first starts by copying the files rev f 1 and rev dat 1 to rev f 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 s rev 1 via NREV 1 IREV I I 1 NDIM 0 1 0 1 Note also from r 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 0071
91. ferential 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 the stepsize DS should be small and one should look at the residuals printed in the fort 9 output file Note that the first residual vector printed in fort 9 may be identically zero as it may correspond to the computation of the starting direction Look at the second residual vector in such case This residual vector has dimension NDIM NBC NINT 1 which accounts for the NDIM differential equations the NBC boundary conditions the NINT user defined integral constraints and the pseudo arclength equation For proper 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 40 6 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 Lo2 norm is printed Most demos use this sett
92. first need to continue to a higher value of PAR 11 141 0 00 0 20 0 40 0 60 0 80 1 00 0 10 0 30 0 50 0 70 0 90 Time Figure 20 1 Solutions of the boundary value problem at labels 6 and 8 either side of the Shil nikov Hopf bifurcation Figure 20 2 Phase portraits of three homoclinic orbits on the branch showing the saddle focus to saddle transition 142 20 2 Detailed AUTO Commands COMMAND ACTION mkdir cir cd cir dm cir cp r ctr 1 r cir cp s cir 1 s cir fc cir h cir dat Osv 1 cp r cir 2 r cir cp s cir 2 s cir h cir 1 sv 2 cp r cir 3 T cir cp s cir 3 s cir h cir 2 ap 2 create an empty work directory change directory copy the demo files to the work directory get the AUTO constants file get the HomCont constants file use the starting data in cir dat to create q dat increase the truncation interval restart from q dat save output files as p 1 q 1 d 1 get the AUTO constants file get the HomCont constants file continue saddle focus homoclinic orbit restart from q 1 save output files as p 2 q 2 d 2 get the AUTO constants file get the HomCont constants file generate adjoint variables restart from q 2 append output files as p 2 q 2 d 2 Table 20 1 Detailed AUTO Commands for running demo cir 143 Chapter 21 HomCont Demo 21 1 she A Heteroclinic Example The following sy
93. fy f D and c Another advantage is the compatibility of output data for restart purposes This allows switching back and forth between evolution calculations and wave computations 2 5 Discretization AUTO discretizes ODE boundary value problems which includes periodic solutions by the method of orthogonal collocation using piecewise polynomials with 2 7 collocation points per mesh interval de Boor amp Swartz 1973 The mesh automatically adapts to the solution to equidistribute the local discretization error Russell amp Christiansen 1978 The number of mesh intervals and the number of collocation points remain constant during any given run although they may be changed at restart points The implementation is AUTO specific In particular the choice of local polynomial basis and the algorithm for solving the linearized collocation systems were specifically designed for use in numerical bifurcation analysis 13 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 5 or independently 3 1 1 The equations file xxx f A source file xxx f containing the Fortran subroutines FUNC STPNT BCND 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 97 demos where e g the file ab ab f defines a two dimens
94. g data for the period doubling continuation Osv tmp save output files as p tmp q tmp d tmp cp r pp3 5 r pp3 constants changed IRS Or pp3 tmp period doubling continuation restart from q tmp Osv 2p save output files as p 2p q 2p d 2p Table 10 9 Commands for running demo pp3 68 10 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 e By azz baly x r de Bx 84 y y 2 3 y 2 10 10 a t y where y 0 6 r 0 6 a3 0 328578 and b3 0 933578 Initially y 0 9 and 8 0 5 COMMAND ACTION mkdir tor create an empty work directory cd tor change directory dm tor copy the demo files to the work directory cp r tor 1 r tor get the first constants file r tor lst run compute a stationary solution branch with Hopf bifurcation sv 1 save output files as p 1 q 1 d 1 cp r tor 2 r tor constants changed IPS IRS Or tor 1 compute a branch of periodic solutions restart from q 1 ap 1 append output files to p 1 q 1 d
95. h continues the homoclinic orbit in PAR 1 and PAR 2 Note that several other parameters appear in the output PAR 10 is a dummy parameter that should be zero when the adjoint is being computed correctly PAR 29 PAR 30 PAR 33 correspond to the test functions 49 410 and Y3 That these test functions were activated is specified in three places in r kpr 4 and s kpr 4 as described in Section 16 6 Note that at the end point of the branch reached when after NMX 50 steps PAR 29 is approx imately zero which corresponds to a zero of w9 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 O PAR 2 PAR 10 Poe 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 p 3 plotting PAR 33 as a function of PAR 1 or PAR 2 Figure 19 3 presents solutions g 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 flip 131
96. h san 2 Osv 3 cp r san 4 r san Cp 8 8an 4 s san h san 1 Osv 4 cp r san d r san cp s san 5 s san h san 3 sv 5 COMMAND ACTION create an empty work directory change directory copy the demo files to the work directory get the AUTO constants file get the HomCont constants file continuation in PAR 1 save output files as p 1 q 1 d 1 get the AUTO constants file get the HomCont constants file generate adjoint variables restart from q 1 save output files as p 2 q 2 d 2 get the AUTO constants file get the HomCont constants file continue homoclinic orbit and adjoint restart from q 2 save output files as p 3 q 3 d 3 get the AUTO constants file get the HomCont constants file no convergence without dummy step restart from q 1 save output files as p 4 q 4 d 4 get the AUTO constants file get the HomCont constants file continue non orientable orbit restart from q 3 save output files as p 5 q 5 d 5 Table 17 1 Detailed AUTO Commands for running demo san 119 COMMAND ACTION cp r san 6 r san cp s san 6 s san h san sv 6 cp r san 7 r san cp s san 7 s san h san 6 ap 6 cp r san 8 r san cp s san 8 s san h san 6 ap 6 cp r san 9 r san cp s san 9 s san h san 6 ap 6 cp r san 10 r san cp s san 10 s san h san 6 ap 6 cp r san 11 r san cp s san 11 s san h san 6 Osv 11 cp r san 12 r san cp s san 12 s san Oh san 11 sv 12 get the AUTO constants file get the
97. hange directory dm int copy the demo files to the work directory cp r int 1 r int get the first constants file r int lst run detection of a fold Osv int save output files as p int q int d int cp r int 2 r int constants changed IRS ISW r int 2nd run generate starting data for a curve of folds sv t save the output files as p t q t d t cp r int 3 r int constants changed IRS Or int t 2nd run compute a curve of folds restart from q t Osv Ip save the output files as p 1p q lp d 1p Table 11 2 Commands for running demo int 76 113 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 Yu Ua 11 3 uy p r u ul ey with boundary conditions u1 0 0 ui 1 0 COMMAND ACTION mkdir bup create an empty work directory cd bup change directory dm bup copy the demo files to the work directory cp r bup 1 r bup get the first constants file r bup compute the trivial solution branch and locate eigenvalues Osv bup save output files as p bvp q bvp d bvp cp r bup 2 r bup constants changed IRS ISW NPR DSMAX r bup compute the first bifurcating branch ap bup append output files to p bvp q bvp d bvp cp r bup 3 r bup constants changed DS
98. iation a Here we assume that the AUTO aliases have been activated see Section 1 1 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 button must be used 5 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 5 2 51 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 whil
99. ic 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 such 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 11 variables Demos exp int Determine folds and branch points along solution
100. ich places editor buffer text at the location of the cursor 5 2 3 Write button This pull down menu contains the item Write to write the loaded files xxx f and r 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 5 2 4 Define button Clicking this button will display the full AUTO constants panel Most of its text fields can be edited but some have restricted input values that can be selected with the right mouse button Some text fields will display a subpanel for entering data To actually apply changes made in the panel click the OK or Apply button at the bottom of the panel 25 5 2 5 Run button Clicking this button will write the constants file r xxx and run AUTO If the equations file has been edited then it should first be rewritten with the Write button 5 2 6 Save button This pull down menu contains the item Save to save the output files fort 7 fort 8 fort 9 aS p XXX q 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 5 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 p xxx q xxx d xxx respectively Here xxx is the active equation name It also contains the item Append
101. ictions can be changed by editing auto h This must be followed by recompilation by typing make in directory auto 97 src It is strongly recommended that NCOLX 4 be used and that the value of NDIMX and NTSTX be chosen as small as possible for the intended application of AUTO Note that in certain cases the effective dimension may be greater than the user dimension For example for the continuation of folds the effective dimension is 2NDIM 1 for algebraic equations and 2NDIM for ordinary differential equations respectively Similarly for the continuation of Hopf bifurcations the effective dimension is 3NDIM 2 1 3 Compatibility with Older Versions There are two changes compared to early versions of AUTO94 The user supplied equations files must contain the subroutine PVLS For an example of use of PVLS see the demo pvl in Section 15 1 There is also a small change in the q xxx data file If necessary older AUTO94 files can be converted using the 09 to97 command see Section 3 5 Chapter 2 Overview of Capabilities 2 1 Summary AUTO can do a limited bifurcation analysis of algebraic systems f u p 0 FC a u R 2 1 and of systems of ordinary differential equation ODEs of the form u t f u t p hul ER 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 Ut Duss f u p JJ 06 E R 2 3 where
102. inate 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 6 7 10 41 Chapter 7 Notes on Using AUTO 7 1 Restrictions on the Use of PAR The array PAR in the user supplied subroutines is available for equation parameters that the user wants to vary at some point in the computations In any particular computation the free parameter s must be designated in ICP see Section 6 7 The following restrictions apply The maximum number of parameters NPARX in auto 97 include auto h has pre defined value NPARX 36 NPARX should not normally be increased and it should never be decreased 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 7 2 Efficiency In AUTO efficiency has at times been sacrificed for generality of
103. indows Emacs and Xedit for editing files and Print for printing the active equations file xxx f 5 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 5 3 Using the GUI AUTO commands are described in Section 3 5 and illustrated in the demos In Table 5 1 we list the main AUTO commands together with the corresponding GUI button Table 5 1 Command Mode GUI correspondences The AUTO command Or zzz yyy is given in the GUI as follows click Files Restart and enter yyy as data Then click Run As noted in Section 3 5 this will run AUTO with the current equations file xxx f and the current constants file r xxx while expecting restart data in q yyy The AUTO command ap zzz yyy is given in the GUI by clicking Files A ppend 5 4 Customizing the GUI 5 4 1 Print button The Misc Print button on the Menu Bar can be customized by editing the file GuiConsts h in directory auto 97 include 5 4 2 GUI colors GUI colors can be customized by creating an X resource file Two model files can be found in directory auto 97 gui namely Xdefaults 1 and Xdefaults 2 To become effective edit one 27 of these if desired and copy it to Xdefaults in your home directory Color names can often be found in the system file usr 1ib X11 rgb txt 5 4 3 On line help The file auto 97 include GuiGlobal h contains on line hel
104. ing For algebraic problems the standard definition of L norm is used For differential equations the L norm is defined as 1 NDIM l 2 Lite dz Note that the interval of integration is 0 1 the standard interval used by AUTO For periodic solutions the independent variable is transformed to range from 0 to 1 before the norm is computed The AUTO constants THL and THU see Section 6 5 5 and Section 6 5 6 affect the definition of the L2 norm If 0 lt IPLT lt NDIM then the maximum of the IPLT th solution component is printed If NDIM lt IPLT lt 0 then the minimum of the IPLT th solution component is printed Demo fsh If NDIM lt IPLT lt 2 NDIM then the integral of the IPLT NDIM th solution component is printed Demos exp lor If 2 NDIM lt IPLT lt 3 NDIM then the L2 norm of the IPLT NDIM th solution component is printed Demo frc Note that for algebraic problems the maximum and the minimum are identical Also for ODEs the maximum and the minimum of a solution component are generally much less accurate than the L2 norm and component integrals Note also that the subroutine PVLS provides a second more general way of defining solution measures see Section 6 7 10 6 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 term
105. ins 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 To PAR 11 1046 178 Since a homoclinic orbit to a saddle node is being followed we have also made the choices TEQUIB 2 NUNSTAB 0 NSTAB 1 in s mtn 1 The two test functions 41 and 4 16 to detect non central saddle node homoclinic orbits are also activated which must be specified in three ways Firstly in s mtn 1 NPSI is set to 2 and the active test functions IPSI 1 I 1 2 are chosen as 15 and 16 This sets up the monitoring of these test functions Secondly in r 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 I I 1 NICP Among the output there is a line BR PT TY LAB PAR 1 ee 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 p 1 q 1 and d 1
106. ional dynamical system and the file exp exp f defines a boundary value problem The simplest way to create a new equations file is to copy an appropriate demo file For a fully documented equations file see auto 97 gui aut f In GUI mode this file can be directly loaded with the GU button Equations New see Section 5 2 3 1 2 The constants file r xxx AUTO constants for xxx f are normally expected in a corresponding file r xxx Specific examples include ab r ab and exp r exp in auto 97 demos See Chapter 6 for the significance of each constant 14 3 2 User Supplied Subroutines The purpose of each of the user supplied subroutines in the file xxx f 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 IRS 0 see Section 6 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 should not be a branch point Demos ab exp frc lor BCND A subroutine BCND 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 pv1 3 3 Arguments of STPNT Note that the arguments of STPNT depend on the solution type When starting from a fixed poin
107. ions file xxx f and a corresponding constants file r xxx see Section 3 1 must be in the current user directory Do not run AUTO in the directory auto 97 or in any of its subdirectories 3 5 1 Basic commands Or Type Or xxx to run AUTO Restart data if needed are expected in q xxx and AUTO constants in r xxx This is the simplest way to run AUTO Type Or zzz yyy to run AUTO with equations file xxx f and restart data fileq yyy AUTO constants must be in r xxx Type Or xxx yyy zzz to run AUTO with equations file xxx f restart data file q yyy and constants file r zzz OR The command OR zzz is equivalent to the command Or xxx above Type OR rrr i to run AUTO with equations file xxx f constants file r xxx i and if needed restart data file q xxx Type OR zzz i yyy to run AUTO with equations file xxx f constants file r xxx i and restart data file q yyy Osv Type sv rrr to save the output files fort 7 fort 8 fort 9 as p xxx q xxx d xxx respectively Existing files by these names will be deleted ap Type ap xxx to append the output files fort 7 fort 8 fort 9 to existing data files P XXX q xxx d XXX resp Type Oap zzz yyy to append p xxx q xxx d xxx to P yyy q yyy d yyy resp 3 5 2 Plotting commands Cp Type p zzz to run the graphics program PLAUT See Chapter 4 for the graphical inspection of the data files p xxx and q xxx 16 Type p to run the graphics program PLAUT fo
108. keeping the peak in the solution in the same location COMMAND ACTION mkdir phs create an empty work directory cd phs change directory dm phs copy the demo files to the work directory cp r phs 1 r phs get the first constants file r phs detect Hopf bifurcation sv phs save output files as p phs q phs d phs cp r phs 2 r phs constants changed IRS IPS NPR r phs compute periodic solutions ap phs append output files to p phs q phs d phs Table 10 13 Commands for running demo phs 73 10 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 1 Si ul poui 1 u1 uru pi 1 e Pau I Uy U2 P441U9 10 15 where all problem parameters have a fixed value The equations are the same as those in demo pp2 The continuation parameter is the independent time variable namely PAR 14 Note that Euler time integration is only first order accurate so that the time step must be sufficiently small to ensure correct results Indeed this option has been added only as a convenience and should generally be used only to locate stationary states Note that the AUTO constants DS DSMIN and DSMAX control the step size in the space consisting of time here PAR 14 and the state vector here u1 uz COMMAND ACTION mkdir ivp create an empty work directory cd ivp change directo
109. l known cases that involve a unique homoclinic orbit at the singular point Homoclinic connections to hyperbolic and non hyperbolic equilibria are allowed as are certain heteroclinic orbits Homoclinic orbits in reversible systems can also be computed The theory behind the methods used is explained in Champneys amp Kuznetsov 1994 Bai amp Champneys 1996 Sandstede 19958 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 97 src autlib5 f Expert users wishing to modify the routines may look there Note also that at present HomCont can be run only in AUTO Command Mode and not with the GUI 16 2 HomCont Files and Subroutines In order to run HomCont one must prepare an equations file xxx f where xxx is the name of the example and two constants files r xxx and s xxx The first two of these files are in the standard AUTO format whereas the s xxx file contains constants that are specific to homoclinic continuation The choice IPS 9 in r xxx specifies the problem as being homoclinic continuation in which case s xxx is required The equation file kpr f serves as a sample for new equation files It contains the Fortran subroutines FUNC STPNT PVLS BCND ICND and FOPT The final three are dummy subr
110. 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 97 include auto h followed by recompilation of the AUTO libraries 44 Chapter 8 AUTO Demos Tutorial 8 1 Introduction The directory auto 97 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 f and one or more constants files r 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 instructed 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 82 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 ul u1 pill u e 8 1 Uy us pip2 1 u1 e p3ue S1 8 3 Copying the Demo Files The commands listed in Table 8 1 will copy the demo files to your
111. lid 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 22 4 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 4 4 Printing PLAUT Files n Ops Type ps fig 1 to convert a saved PLAUT file fig 1 to PostScript format in 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 23 Chapter 5 Graphical User Interface 5 1 General Overview The AUTO97 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 3 5 for corresponding commands To use the GUI for a new equation change to an empty work directory For an existing equations file change to its directory Do not activate the GUI in the directory auto 97 or in any of its subdirectories Then type Qauto or its abbrev
112. lues neutral saddle y Aj i 2 Double real leading stable eigenvalues saddle to saddle focus transition 41 H2 i 3 Double real leading unstable eigenvalues saddle to saddle focus transition At Ao i 4 Neutral saddle saddle focus or bi focus includes i 1 Re 111 Re A1 i 5 Neutrally divergent saddle focus stable eigenvalues complex Re A1 Re 11 Re pi2 i 6 Neutrally divergent saddle focus unstable eigenvalues complex Re 11 Re A1 Re A2 i 7 Three leading eigenvalues stable Re A1 Re u1 Re u2 i 8 Three leading eigenvalues unstable Re 111 Re A1 Re 2 i 9 Local bifurcation zero eigenvalue or Hopf number of stable eigenvalues decreases Re 11 0 i 10 Local bifurcation zero eigenvalue or Hopf number of unstable eigenvalues de creases Re A1 0 i 11 Orbit flip with respect to leading stable direction e g 1D unstable manifold i 12 Orbit flip with respect to leading unstable direction e g 1D stable manifold i 13 Inclination flip with respect to stable manifold e g 1D unstable manifold i 14 Inclination flip with respect to unstable manifold e g 1D stable manifold i 15 Non central homoclinic to saddle node in stable manifold i 16 Non central homoclinic to saddle node in unstable manifold 111 Expert users may wish to add their own test functions by editing the fun
113. machine only see demo pen in Section 10 10 and the file auto 97 pendula README 3 5 10 Viewing the manual mn Use Ghostview to view the PostScript version of this manual 19 3 6 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 Branch pont algebrate system A Userspecified regular output pomt BP 6 Branch point differential equations PD 7 Period doubling bifurcation 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 contains complete graphics and restart data for selected labeled solutions The information per solution is generally much more extensive than that in fort 7 The fort 8 output should normally be kept to a minimum fort 9 Diagnostic messages convergence history eigenvalues and Floquet multipliers are written in fort 9 It is strongly recommended that this output be habitually inspected Th
114. matically 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 6 5 2 DSMIN This is minimum allowable absolute value of the pseudo arclength stepsize DSMIN must be pos itive It is only effective if the pseudo arclength step is adaptive i e if TADS 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 7 2 6 5 3 DSMAX The maximum allowable absolute value of the pseudo arclength stepsize DSMAX must be positive It is only effective if the pseudo arclength step is adaptive i e if IADS gt 0 The choice of DSMAX is highly problem dependent most demos use a value that was found appropriate after some experimentation See also the discussion in Section 7 2 6 5 4 IADS This constant controls the frequency of adaption of the pseudo arclength stepsize TADS 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
115. n a reaction diffusion model in H G Othmer ed Nonlinear Oscillations in Biology and Chemistry Vol 66 Springer Verlag pp 261 273 Doedel E J 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 STAM J Numer Anal 28 No 5 1446 1462 Freire
116. ne 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 93 The free scalar variables specified in the AUTO constants files for Run 3 and Run 4 are shown in Table 13 2 Variable x T r Pa Pal LET Table 13 2 Runs 3 and 4 files r ops 3 and r ops 4 The parameter a which is the norm of the adjoint variables becomes nonzero after branch switching in Run 4 The negative indices 22 23 and 31 set the active optimality functionals namely for A 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 za As 7 Table 13 3 Run 5 file r ops 5 In Run 5 the parameter a which has been replaced by A2 remains fixed and nonzero The variable 7 monitors the value of the optimality functional associated with Aj The zero of 7 located in this run signals an extremum with respect to A2 aja PL Table 13 4 Run 6 file r ops 6 In Run 6 7 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 an
117. ng demo abc 60 10 3 pp2 A 2D Predator Prey Model This demo illustrates a variety of calculations The equations which model a predator prey system with harvesting are ti poui 1 u1 U1U2 pi l ev 10 3 Ug Ug p4U1U2 Here p is the principal continuation parameter p 5 p4 3 and initially p 3 For two parameter computations pa is also free The use of PLAUT is also illustrated The saved plots are shown in Figure 10 1 and Figure 10 2 COMMAND ACTION create an empty work directory change directory copy the demo files to the work directory cp r pp2 1 r pp2 get the first constants file r pp2 lst run stationary solutions Osv pp2 save output files as p pp2 q pp2 d pp2 cp r pp2 2 r pp2 constants changed IRS RL1 Or pp2 2nd run restart at a labeled solution ap pp2 append output files to p pp2 q pp2 d pp2 cp r pp2 3 r pp2 constants changed IRS IPS ILP r pp2 3rd run periodic solutions ap pp2 append output files to p pp2 q pp2 d pp2 cp r pp2 4 r pp2 constants changed IRS NTST r pp2 4th run restart at a labeled periodic solution ap pp2 append output files to p pp2 q pp2 d pp2 cp r pp2 5 r pp2 constants changed IRS IPS ISW ICP r pp2 5th run continuation of folds sv lp save output files as p 1p q 1p d 1p cp r pp2 6 r pp2 constants changed IRS Or pp2 6th run continuation of Hopf bifurcations Osv hb save output files as p hb q hb d hb cp r pp2 7 r pp2 constan
118. ns folds and user output points by Miiller s method with bracketing The recom mended value is ITMX 8 used in most demos 6 4 5 NWIN After NWIN Newton iterations the Jacobian is frozen i e AUTO uses full Newton for the first NWTN iterations and the Chord method for iterations NWIN 1 to ITNW The choice NWIN 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 6 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 ITNW 7 may be used for difficult problems for example demos spb chu plp etc 31 6 5 Continuation Step Size 6 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 auto
119. nted 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 r pv1 1 labeled solutions can also be output at selected values of the overspecified parameters COMMAND ACTION mkdir pul create an empty work directory cd pul change directory dm pul copy the demo files to the work directory cp r pul 1 r pul get the constants file Or pul compute a solution branch Osv pul save output files as p pvl q pvl d pvl Table 15 1 Commands for running demo pvl 104 15 2 ext Spurious Solutions to BVP This demo illustrates the computation of spurious solutions to the boundary value problem ul us 0 ub Mm sin u u u 0 t 0 1 15 2 u1 0 0 uz 1 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 Sec tion 6 3 This example is also considered in Beyn amp Doedel 1981 and Doedel Keller amp Kern vez 19910 COMMAND ACTION mkdir ext create an empty work directory cd ext change directory dm ext copy the demo files to the work directory cp r ext 1 r ext get the first constants file Or ext detect bifurcations from the trivial solution branch Osv ext save output files as p ext q ext d ext cp r ext 2 r ext constants changed IRS ISW NUZR Or ext compute a bifurca
120. o the work directory cp r nag 1 r nag get the first constants file r nag compute part of first branch of heteroclinic orbits sv nag save output files as p nag q nag d nag cp r nag 2 r nag constants changed DS r nag compute first branch in opposite direction ap nag append output files to p nag q nag d nag Table 14 2 Commands for running demo nag 100 14 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 aw aqw B w bo byw bow and C w co e w cow Such equations can have sharp traveling wave fronts as solutions i e solutions of the form w x t u x ct for which there is a zo such that u z 0 for z gt zo u z 4 0 for z lt zo and u z gt 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 We 2wWWee 200 w 1 w to a corresponding sharp traveling wave of 2 we 2w w Wee ww w 1 w This problem is also considered in Doedel Keller amp Kern vez 1991b For these two special cases the functions A B C are defined by the coefficients in Table 14 3 Pf ts ats bo bs be co en es Case 1
121. omatically As illustrated by the commands in Table 8 4 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 Unix COMMAND ACTION make first execute the first run of demo ab make third execute the third run of demo ab Table 8 4 Selected runs of demo ab Of course in real use one must prepare a constants file for each run In the illustrative runs above the constants files were carefully prepared in advance For example the file r ab 1 contains the AUTO constants for Run 1 r ab 3 contains the AUTO constants for Run 3 etc 8 6 Using AUTO Commands Next with the commands in Table 8 5 we execute the first two runs of demo ab again but now using the commands that one would normally use in an actual application We still use the demo constants files that were prepared in advance COMMAND ACTION cp r ab 1 r ab get the first constants file r ab compute a stationary solution branch with folds and Hopf bifurcation Osv ab save output files as p ab q ab d ab cp r ab 2 r ab get the second constants file Or ab compute a branch of periodic solutions from the Hopf point ap ab a
122. only special labeled solution points are printed on the screen More detailed results are saved in the data files p ab q 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 p ab q 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 p 2p q 2p and d 2p One could now use PLAUT 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 8 3 Unix COMMAND ACTION remove data files and temporary files of demo ab Table 8 3 Cleaning the demo ab work directory 47 ab BR Peper Pe first run PT TY LAB 1 EP 33 LP 70 LP 90 HB 92 EP oP WN FE 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
123. ons 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 6 7 6 Boundary value problems The simplest case is that of boundary value problems where NDIM NBC and where NINT 0 Then generically one free problem parameter is required for computing a solution 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 6 7 7 Boundar
124. onstants file cp s kpr 3 s kpr get the HomCont constants file OR kpr 2 generate adjoint variables restart from q 2 sv 3 save output files as p 3 q 3 d 3 cp r kpr 4 r kpr get the AUTO constants file cp s kpr 4 s kpr get the HomCont constants file h kpr 8 continue the homoclinic orbit restart from q 3 ap 8 append output files to p 3 q 3 d 3 cp r kpr 5 r kpr get the AUTO constants file cp s kpr 5 s kpr get the HomCont constants file h kpr 8 continue in reverse direction restart from q 3 ap 8 append output files to p 3 q 3 d 3 cp r kpr 6 r kpr get the AUTO constants file cp s kpr 6 s kpr get the HomCont constants file h kpr 2 increase the period restart from q 2 sv 6 save output files as p 6 q 6 d 6 Table 19 1 Detailed AUTO Commands for running demo kpr 137 COMMAND ACTION cp rT kor 7 kor cp s kpr 7 s kpr h kpr 6 sv 7 cp r kpr 8 r kpr cp s kpr 8 s kpr Oh kpr 7 Osv 8 cp r kpr 9 r kpr cp s kpr 9 s kpr Oh kpr 8 sv 9 cp r kpr 10 r kpr cp s kpr 10 s kpr Oh kpr 3 sv 10 cp r kpr 11 r kpr cp s kpr 11 s kpr Oh kpr 3 sv 11 cp r kpr 12 r kpr cp s kpr 12 s kpr Oh kpr 7 sv 12 cp r kpr 13 r kpr cp s kpr 13 s kpr Oh kpr 8 ap 12 get the AUTO constants file get the HomCont constants file recompute the branch of homoclinic orbits restart from q 6 save output files as p 7 q 7 d 7 get the AUTO constants file get the HomCont constants file continue central sa
125. or 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 12 Discretization in time is not very accurate only implicit Euler Indeed time integration of 2 3 has only been included as a convenience and it is not very efficient Demos pal 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 pd1 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 speci
126. or 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 116 ITWIST 1 IRS 2 NMX 50 ICP 1 4 NPUSZR 0 as saved in r san 4 and running make fourth we obtain a no convergence error 17 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 label with the first continuation parameter being once again a PAR 1 by changing constants and storing them in r san 5 according to IRS 7 DS 0 05D0 NMX 20 TEP C1 1 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 17 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 jt 8 1 a b 1 and u 4 in order to reach the parameter values a b a 6 u 0 5 3 1 0 0 25 The sequence of continuations up to the desired parameter values are run
127. outines 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 r xxx is identical in format to other AUTO constants files Note that the values of the constants NBC and NINT are irrelevant as these are set automatically by the choice IPS 9 Also the choice JAC 1 is strongly recommended because the Jacobian is used extensively for calculating the linearization at the equilibria and hence for evaluating boundary conditions 107 and certain test functions However note that JAC 1 does not necessarily mean that auto will use the analytically specified Jacobian for continuation 16 3 HomCont Constants An example for the additional file s xxx is listed below 12111 NUNSTAB NSTAB IEQUIB ITWIST ISTART 0 NREV 1 IREV I I 1 NREV 1 NFIXED 1 IFIXED I l 1 NFIXED 13 1 NPSI 1 IPSI 1 I 1 NPSI 9 10 13 The constants specified in s xxx have the following meaning 16 3 1 NUNSTAB Number of unstable eigenvalues of the left hand equilibrium the equilibrium approached by the orbit as t gt oo 16 3 2 NSTAB Number of stable eigenvalues of the right hand equilibrium the equilibrium approached by the orbit as t 00 16 3 3 TEQUIB TEQUIB 0 Homoclinic orbits to hyperbolic equilibria the equilibrium is specified explicitly in PVLS and stored in PAR 11 I I 1 NDIM T
128. p append output files to p 2p q 2p d 2p constants changed IRS ISW NMX NUZR compute an isola of periodic solutions restart data from q 2p save output files as p iso q iso d iso Table 10 8 Commands for running demo plp 67 10 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 u u 1 u1 parte Uy Potig p4U1U9 psuzuz pi 1 e Ps 10 9 Us P3U3 psuzuz The free parameter is p except in the period doubling continuation where both p and py are free COMMAND ACTION mkdir pps create an empty work directory cd pps change directory dm pp3 copy the demo files to the work directory cp r pp3 1 r pp3 get the first constants file Or pp3 1st run stationary solutions Osv pp3 save output files as p pp3 q pp3 d pp3 cp r pp3 2 r pp3 constants changed IRS IPS NMX Or pp3 compute a branch of periodic solutions ap pp3 append output files to p pp3 q pp3 d pp3 cp r pp3 3 r pp3 constants changed IRS ISW NTST Or pp3 compute the branch bifurcating at the period doubling ap pp3 append output files to p pp3 q pp3 d pp3 cp r pp3 4 r pp3 constants changed ISW Or pp3 generate startin
129. p 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 XNI10240 O in auto 97 gui Makefile by CC cc Wf XNI20480 O On other machines the maximum message length is the system defined maximum string literal length 28 Chapter 6 Description of AUTO Constants 6 1 The AUTO Constants File As described in Section 3 1 if the equations file is xxx f then the constants that define the computation are normally expected in the file r xxx The general format of this file is the same for all AUTO runs For example the file r ab in directory auto 97 demos ab is listed below The tutorial demo ab is described in detail in Chapter 8 2101 NDIM IPS IRS ILP 1 1 NICP ICP I I 1 NICP 504311000 NTST NCOL IAD ISP ISW IPLT NBC NINT 100 0 0 15 0 100 NMX RLO RL1 A0 A1 100 1028530 NPR MXBF IID ITMX ITNW NWIN 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 1I 1 NTHL 11 0 0 NTHU I THU I 1 1 NTHU 0 NUZR I UZR I 1 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 6 2 Problem Constants 6 2 1 NDIM Dimension of the system of equations as specified in the user supplied s
130. ppend the output files to p ab q ab d ab Table 8 5 Commands for Run 1 and Run 2 of demo ab It is instructive to look at the constants files r ab 1 and r ab 2 used in the two runs above The significance of each AUTO constant set in these files can be found in Chapter 6 Note in particular the AUTO constants that were changed between the two runs see Table 8 6 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 8 6 49 Reason Tor Chang 1 2 To compute periodic solutions in Run 2 0 4 To specify the Hopf bifurcation restart label The second run has two free parameters 100 To print output every 30 steps in Run 2 Table 8 6 Differences in AUTO constants between r ab 1 and r ab 2 8 7 Plotting the Results with PLAUT The bifurcation diagram computed in the runs above is stored in the file p ab while each labeled solution is fully stored in q ab To use PLAUT to graphically inspect these data files type the AUTO command given in Table 8 7 The PLAUT window a Tektronix window will appear in which one can enter the PLAUT commands given in Table 8 8 The saved plots are shown in Figure 8 1 and in Figure 8 2 AUTO COMMAND ACTION run PLAUT to graph the contents of p ab and q ab Table 8 7 Command for plotting the files p ab and q ab 8 8 Following Folds and Hopf
131. principal solution measure 6 7 Free Parameters 6 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 6 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 here 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 6 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 perio
132. problem there exists an analytic expression for the two equilibria This is specified in the subroutine PVLS of she f Re running with IEQUIB 1 we obtain the output make second 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 286652F 01 2 062194E 01 3 179939E 02 1 25 6 1 555409F 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 extra parameters PAR 12 21 representing the coordinates of the equilibria are no longer part of the continuation problem Also note that AUTO has chosen to take slightly larger steps along the branch Finally we can continue in the opposite direction along the branch from the original starting point again with IEQUIB 1 make third BR PT TY LAB PAR 3 L2 NORM PAR 1 1 5 8 4 997590E 01 4 060153E 01 1 637322E 01 1 10 9 5 705299E 01 4 551872E 01 2 065264E 01 1 15 10 6 416439E 01 5 031844E 01 2 507829E 01 1 20 11 7 133301E 01 5 500668E 01 2 959336E 01 1 25 12 7 857688E 01 5 958712E 01 3 415492E 01 1 30 13 8 590970E 01 6 406182E 01 3 872997E 01 1 35 EP 14 9 334159E 01 6 843173E 01 4 329270E 01 The results of both computations are presented in Figure 21 1 from which we see that the orbit shrinks to zero as PAR 1 u gt 0
133. 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 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 6 4 and Section 6 5 Finally it is recommended tha
134. q xxx to new AUTO97 format The original file is backed up as q xxx This conversion is only necessary for files from early versions of AUTO94 3 5 7 HomCont commands Ch Use Ch instead of Cr when using HomCont i e when IPS 9 see Chapter 16 Type h xxx to run AUTO HomCont Restart data if needed are expected in q xxx AUTO constants in r xxx and HomCont constants in s xxx Type h zzz yyy to run AUTO HomCont with equations file xxx f and restart data file q yyy AUTO constants must be in r xxx and HomCont constants in s xxx Type h rex yyy zzz to run AUTO HomCont with equations file xxx f restart data file q yyy and constants files r zzz and s zzz H The command H xxx is equivalent to the command Oh zxx above Type OH zzz iin order to run AUTO HomCont with equations file xxx f and constants files r xxx i and s xxx i and if needed restart data file q xxx Type H zzz i yyy to run AUTO HomCont with equations file xxx f constants files r xxx i and s xxx i and restart data file q yyy 3 5 8 Copying a demo dm Type dm zzz to copy all files from auto 97 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 3 5 9 Pendula animation pn Type pn zzz to run the pendula animation program with data file q xxx On SGI
135. r y Bx B y 2 b3 y x 20 1 2 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 f and the initial run time constants are stored in r cir 1 and s cir 1 We begin by starting from the data from cir dat for a saddle focus homoclinic orbit at y 0 721309 6 0 6 y 0 r 0 6 43 0 328578 and Bs 0 933578 which was obtained by shooting over the time interval 27 PAR 11 36 13 We wish to follow the branch in the v plane but first we perform continuation in 7 1 to obtain a better approximation to a homoclinic orbit make first yields the output BR PT TY LAB PERIOD L2 NORM ed 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 3 The new constants defining the continuation are given in 140 r cir 2 and
136. r the graphical inspection of the output files fort 7 and fort 8 Ops Type ps fig x to convert a saved PLAUT figure fig x from compact PLOT10 format to PostScript format The converted file is called fig x ps The original file is left unchanged 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 3 5 3 File manipulation cp Type cp xxx yyy to copy the data files p xxx q xxx d xxx r xxx to P yyy q yyy d yyy Y yyy respectively Omv Type mw xxx yyy to move the data files p xxx q xxx d xxx r xxx to p yyy q yyy d yyy Y yyy respectively df Type Qdf 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 x o and exe d1 Type dl xxx to delete the data files p xxx q xxx d xxx 3 5 4 Diagnostics lp Type lp to list the value of the limit point function in the output file fort 9 This function vanishes at a limit point fold Type lp xxx to list the value of the limit point function in the data file d xxx This function vanishes at a limit point fold Obp Type bp to list the value of the branch point function in the output file fort 9 This function vanishes at a branch point Type bp xxx to list the value of the
137. r 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 Installation The AUTO files auto ps Z auto tar Z and README are available via FTP from directory pub doedel auto at ftp cs concordia ca The README file contains the instructions for print ing this manual Below it is assumed that you are using the Unix shell csh and that the file auto tar Z is in your main directory While in your main directory enter the commands uncompress auto tar Z followed by tar sufo auto tar This will result in a directory auto with one subdirectory auto 97 Type cd auto 97 to change directory to auto 97 Then type make sgi to compile AUTO on Silicon Graphics machines or make solaris on SUN Solaris with ANSI C compiler or make on SUN OS with K amp R C compiler and in principle on other Unix systems Upon compilation type make clean to remov
138. rbit save output files as p 1 q 1 d 1 get the AUTO constants file get the HomCont constants file continue in opposite direction restart from q 1 append output files to p 1 q 1 d 1 get the AUTO constants file get the HomCont constants file switch to saddle homoclinic orbit restart from q 1 append output files to p 1 q 1 d 1 get the AUTO constants file get the HomCont constants file continue in reverse direction restart from q 1 save output files as p 4 q 4 d 4 get the AUTO constants file get the HomCont constants file other saddle homoclinic orbit branch restart from q 1 append output files to p q 1 d 1 get the AUTO constants file get the HomCont constants file 3 parameter non central saddle node homoclinic save output files as p 6 q 6 d 6 Table 18 1 Detailed AUTO Commands for running demo mtn 126 O ii 0 00 0 07 0 14 0 21 0 28 0 35 Z Figure 18 1 Parametric portrait of the predator prey system 15 0 06 0 08 0 10 Z Figure 18 2 Approximation by a large period cycle 127 d_0 0 012 Figure 18 3 Projection onto the K Do plane of the three parameter curve of non central saddle node homoclinic orbit 128 Chapter 19 HomCont Demo kpr 19 1 Koper s Extended Van der Pol Model The equation file kpr f contains the equations t a ky 2 32 y B lye 19 1 Zz ey 2 with e
139. rcation when 8 0 Labels 6 and 8 are the user defined output points the solutions at which are plotted in Fig 20 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 point from the previous computation label 5 in q 2 make third The output BR PT TY LAB PAR 1 PAR 2 PAR 22 PAR 24 1 9 UZ 10 7 204001E 01 5 912315E 01 1 725669E 00 3 295862E 05 1 18 UZ 11 7 590583E 01 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 5 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 y2 0 and a neutral saddle at LAB 12 y4 0 Data at several points on the complete branch are plotted in Fig 20 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
140. rectory cp r stw 1 r stw get the constants file r stw continuation of the sharp traveling wave Osv stw save output files as p stw q stw d stw Table 14 4 Commands for running demo stw 102 Chapter 15 AUTO Demos Miscellaneous 103 15 1 pvl Use of the Subroutine PVLS Consider Bratu s equation E Uy U2 1 u Uy pie Es 15 1 with boundary conditions u 0 0 ui 1 0 As in demo exp a solution curve requires one free parameter here p Note that additional parameters are specified in the user supplied subroutine PVLS in file pvls f namely po the L2 norm of u1 p the minimum of uz on the space interval 0 1 pa the boundary value w2 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 r pv1 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 po p1 P3 pa In general true continuation parameters must appear first in the parameter specification in the AUTO constants file Overspecified parameters will be pri
141. ry dm ivp copy the demo files to the work directory cp r iup 1 r ivp get the constants file Or ivp time integration Osv ivp save output files as p ivp q ivp d ivp Table 10 14 Commands for running demo ivp 74 Chapter 11 AUTO Demos BVP 11 1 exp Bratu s Equation This demo illustrates the computation of a solution branch to the boundary value problem Uy Ua 1 O a 11 1 with boundary conditions u 0 0 1 0 This equation is also considered in Doedel Keller amp Kern vez 1991a COMMAND ACTION mkdir exp create an empty work directory cd exp change directory dm exp copy the demo files to the work directory cp r exp 1 r exp get the first constants file r exp lst run compute solution branch containing fold sv exp save output files as p exp q exp d exp cp r exp 2 r exp constants changed IRS NTST A1 DSMAX r exp 2nd run restart at a labeled solution using increased accuracy ap exp append output files to p exp q exp d exp Table 11 1 Commands for running demo exp 79 11 2 int Boundary and Integral Constraints This demo illustrates the computation of a solution branch to the equation 11 2 with a non separated boundary condition and an integral constraint salen oa HOt te 0 The solution branch contains a fold which in the second run is continued in two equation parameters COMMAND ACTION mkdir int create an empty work directory cd int c
142. s demo illustrates the computation of travelling wave front solutions to the Fisher equation Wt Wee f w o lt 2r lt oo t gt 0 f w wil w 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 14 1 ui 2 u2 2 uz cu2 z f ur z Its fixed point 0 0 has two positive eigenvalues when c gt 2 The other fixed point 1 0 isa 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 14 2 COMMAND ACTION mkdir fsh create an empty work directory cd fsh change directory dm fsh copy the demo files to the work directory cp r fsh 1 r fsh get the first constants file r fsh continuation in the period T
143. s 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 r kpr 7 The files r kpr 9 and s 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 q 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 415 Among the output we find BR PT TY LAB PAR 1 a 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 pe 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 19 6 shows clearly that between the two codimension two points the homoclinic orbit rotates between the two components of the 1D stable manifold i e betw
144. so Section 6 7 37 6 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 6 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 analytical starting solutions see demos ab and frc For starting from unlabeled numerical data see the fc command Section 3 5 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 OR commands in Section 3 5 Various AUTO constants can be modified when restarting 6 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 sol
145. st 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 period PAR 12 the norm of the adjoint variables PAR 14 and PAR 15 are internal optimality variables PAR 21 29 and PAR 31 are used to monitor the optimality functionals associated with the problem parameters and the period 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 pdi 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
146. stem of five equations Rucklidge amp Mathews 1995 e amp de amp UX rLry ZU a 4oxu 4dopz 90 2 4x4 4p2 4 1 0 04 4 0Qu tr 3 1 0 22 40 Cu 4 u 4 21 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 j1 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 f and starting data for the orbit at u 0 5 0 163875 are stored in she dat with a truncation interval of PAR 11 85 07 We begin by computing towards 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 PPR Pe 1 PT TY LAB 5 10 15 20 25 30 EP 7 Oo FF WN NNUU 1 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 1712726E 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 144 Alternatively for this
147. t 42 initial computations be done with ILP 0 no fold detection and ISP 1 no bifurcation detection for ODEs 7 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 7 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 bifurcations cannot be detected Furthermore bifurcations that are close to each other may not be noticed when the pseudo arclength step size is not sufficiently small Hopf bifurcation points may go unnoticed if no clear crossing of the imaginary axis takes 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 th
148. t the subroutine STPNT must have three arguments namely NDIM U PAR See demo ab When starting from an analytically or numerically known space dependent solution STPNT must have four arguments namely NDIM U PAR T Here T is the independent space vari able which takes values in the interval 0 1 Demo exp 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 3 5 for conversion of numerical data STPNT must have three arguments namely NDIM U PAR In this case only the parameter values need to be defined in STPNT Demos lor and pen 15 3 4 User Supplied Derivatives If AUTO constant JAC equals 0 then derivatives need not be specified in FUNC BCND ICND and FOPT see Section sec JAC If 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 Running AUTO using Command Mode AUTO can be run with the GUI described in Chapter 5 or with the commands described below The AUTO aliases must have been activated see Section 1 1 and an equat
149. te an empty work directory cd spb change directory dm spb copy the demo files to the work directory cp r spb 1 r spb get the first constants file r spb lst run homotopy from A 0 to 1 sv 1 save output files as p 1 q 1 d 1 cp r spb 2 r spb constants changed IRS ICP 1 NTST DS Or spb 1 2nd run let tend to zero restart from q 1 sv 2 save the output files as p 2 q 2 d 2 cp r spb 3 r spb constants changed IRS ICP 1 RLO ITNW EPSL EPSU NUZR r spb 2 3rd run continuation in y e 0 001 restart from q 2 sv 3 save the output files as p 3 q 3 d 3 Table 11 7 Commands for running demo spb 81 11 8 ezp Complex Bifurcation in a BVP This demo illustrates the computation of a solution branch to the the complex boundary value problem y ug i pill 11 11 with boundary conditions u1 0 0 u1 1 0 Here u and uz are allowed to be complex while the parameter p can only take real values In the real case this is Bratu s equation whose solution 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 COMMAND ACTION mkdir ezp create an empty work
150. te q dat increase PAR 1 save output files as p 1 q 1 d 1 get the AUTO constants file get the HomCont constants file continue in reverse direction restart from q 1 append output files to p 1 q 1 d 1 get equations file with new value of PAR 11 get starting data with different reversibility get the AUTO constants file get the HomCont constants file use the starting data in rev dat to create q dat restart with different reversibility save output files as p 3 q 3 d 3 get the AUTO constants file get the HomCont constants file continue in reverse direction restart from q 3 append output files to p 3 q 3 d 3 Table 22 1 Detailed AUTO Commands for running demo rev 153 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 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
151. that a solution is to be restarted from a previously computed point or from numerical data converted into AUTO format using fc In this case IRS gt 0 ISTART 2 If IRS 0 an explicit solution must be specified in the subroutine STPNT in the usual format ISTART 3 The homotopy approach is used for starting see Section 16 7 for more details Note that this is not available with the choice IEQUIB 2 16 3 6 NREV IREV If NREV 1 then it is assumed that the system is reversible under the transformation t gt t and U i U i for all 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 16 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 16 3 8 NPSI IPSI Number and labels of activated test functions for detecting homoclinic bifurcations see Sec tion 16 6 for a list If a test function is activated then the corresponding parameter IPSI 1 20 must be added to the list of continuation parameters NICP ICP 1 I 1 NICP and zero of this parameter added to the list of user defined output points NUZR I PAR I I 1 NUZR in r xxx 109 1
152. 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 19 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 133 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 r 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 boat 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 Yg and 19 for non central saddle node homoclinic orbits activated make seventh The saddle node point is now detected at BR PT TY LAB PAR 1 ee PAR 2 PAR 29 PAR 30 1 30 UZ 8 1 765003E 01 2 405345E 00 2 743361E 06 2 309317E 01 which is stored in q 7 That PAR 29 19 is zeroed shows that thi
153. ther user zero for detecting when PAR 4 1 0 Running make third reads starting data from q 2 and outputs to the screen BR PT TY LAB PAR 4 oe PAR 8 PAR 10 ee 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 p 3 q 3 and d 3 Note that the artificial parameter 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 7 and 9 at and on either side of the inclination flip are presented in Fig 17 1 The switching of the solution between components of the leading unstable left eigenvect
154. ting 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 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 3 5 6 File maintenance lb Type lb 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 lb xxx to list delete and relabel solutions in the data files p xxx and q xxx The original files are backed up as p xxx and q xxx Type lb xxx yyy to list delete and relabel solutions in the data files p xxx and q xxx The modified files are written as p yyy and q yyy fc Type fc xxx to convert a user supplied data file xxx dat to AUTO format The converted file is called q 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 r xxx must be present as the AUTO constants NTST and NCOL Sections 6 3 1 and 6 3 2 are used to define the new mesh For examples of using the Ofc command see demos lor and pen 18 94t097 Type 94to097 xxx to convert an old AUTO94 data file
155. ting branch containing spurious bifurcations ap ext append output files to p ext q ext d ext Table 15 2 Commands for running demo ext 105 15 3 tim A Test Problem for Timing AUTO This demo is a boundary value problem with variable dimension NDIM It can be used to time the performance of AUTO for various choices of NDIM which must be even NTST and NCOL The equations are u Uj p e u i 1 NDIM 2 with boundary conditions u 0 0 u 1 0 Here 15 3 1 1 1 1 E k p 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 COMMAND ACTION mkdir tum create an empty work directory cd tim change directory dm tim copy the demo files to the work directory cp r tim 1 r tim get the first constants file r tim Timing run Osv tim save output files as p tim q tim d tim Table 15 3 Commands for running demo tin 106 Chapter 16 HomCont 16 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 al
156. ts changed IRS IPS ISP Or pp2 7th run continuation of homoclinic orbits Osv hom save output files as p hom q hom d hom Table 10 3 Commands for running demo pp2 61 AUTO COMMAND ACTION run PLAUT to graph the contents of p pp2 and q pp2 PLAUT COMMAND ACTION d set convenient defaults bd0 plot the default bifurcation diagram L2 norm versus p clear the screen 1 cl az select axes 13 select real columns 1 and 3 in p pp2 bd0 plot the bifurcation diagram max uj versus p cl clear the screen d3 choose other default settings bd get blow up of current bifurcation diagram 0 1 0 25 1 enter diagram limits sav save plot see Figure 10 1 fig 1 upon prompt enter a new file name e g fig 1 clear the screen cl 2d enter 2D mode for plotting labeled solutions 12 13 14 select labeled orbits 12 13 and 14 in q pp2 d default orbit display u versus time 13 select columns 1 and 3 in q pp2 23 select columns 2 and 3 in q pp2 d phase plane display uz versus u1 sav save plot see Figure 10 2 fig 2 upon prompt enter a new file name exit from 2D mode ex exit from PLAUT Table 10 4 Plotting commands for demo pp2 62 0 25 Figure 10 1 The bifurcation diagram of demo pp2 Figure 10 2 The phase plot of solutions 12 13 and 14 in demo pp2 63 10 4 lor Starting an Orbit from
157. tween Command Mode and GUI actions To activate the GUI type the command in Table 8 15 The GUI actions to execute the first two runs of demo ab are given in Table 8 16 In GUI Mode one can copy demo files to the user work directory using the Equations Demo button To load a selected constants file use the Previous button in the LoadConsts area of the GUI window Press the Filter button in the pop up window to update the displayed list of files and then select the appropriate constants file To execute all runs of a selected demo with the GUI click Demos Select select a demo and click the Run button in the pop up window This will actually run the demo in the corresponding subdirectory of auto 97 demos which is only possible if you have write access to this directory Make sure to click the Demos Reset button afterwards Do not otherwise run AUTO in the 54 AUTO COMMAND ACTION Activate the Graphical User Interface Table 8 15 Command to activate the GUI GUI button ACTION Equations Demo Select demo ab then press OK Push Filter select file r ab 1 then press OK This will execute Run 1 of demo ab Save the output files as p ab q ab d ab Select file r ab 2 then press OK This will execute Run 2 of demo ab Append Append Append the output files to p ab q ab d ab Table 8 16 GUI actions for Run 1 and Run 2 of demo ab directory auto 97 or in any of its subdirectories 8 13 Abbreviated AUTO Commands The AUTO comm
158. uations on the spatial interval 0 1 The equations that model an enzyme catalyzed reaction Doedel amp Kern vez 19866 are Ou Ot Ou Ox pi paR u1 u2 p2 u1 uz t BO ug Ax pi paR t us pr ps us 12 2 All equation parameters except p3 are fixed throughout COMMAND ACTION mkdir wav create an empty work directory cd wav change directory dm wav copy the demo files to the work directory cp r wav 1 r wav get the first constants file Or wav lst run stationary solutions of the system without diffusion sv ode save output files as p ode q ode d ode cp r wav 2 r wav constants changed IPS r wav 2nd run detect bifurcations to wave train solutions sv wav save output files as p wav q wav d wav cp r wav 3 r wav constants changed IRS IPS NUZR ILP r wav 3rd run wave train solutions of fixed wave speed ap wav append output files to p wav q wav d wav cp r wav 4 r wav constants changed IRS IPS NMX ICP NUZR r wav 4th run wave train solutions of fixed wave length sv rng save output files as p rng q rng d rng cp r wav d r wav constants changed IPS NMX NPR ICP r wav 5th run time evolution computation Osv tim save output files as p tim q tim d tim Table 12 3 Commands for running demo wav 86 12 4 bre Chebyshev Collocation in Space This demo illustrates the computation of stationary solutions and periodic solutions to systems of para
159. ubroutine FUNC 6 2 2 NBC The number of boundary conditions as specified in the user supplied subroutine BCND Demos exp kar 29 6 2 3 NINT The number of integral conditions as specified in the user supplied subroutine ICND Demos int lin obv 6 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 BCND ICND and FOPT where applicable This may be necessary for sensitive problems It is also recommended for computations in which AUTO generates an extended system for example when ISW 2 For ISW see Section 6 8 3 Demos int dd2 obt plp ops 6 3 Discretization Constants 6 3 1 NTST The number of mesh intervals used for discretization NTST remains fixed during any particular run but can be changed when restarting For mesh adaption see IAD in Section 6 3 3 Recom mended value of NTST As small as possible to maintain convergence Demos exp ab spb 6 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 th
160. ust 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 COMMAND ACTION mkdir bru create an empty work directory cd bru change directory dm bru copy the demo files to the work directory cp r bru 1 r bru get the constants file Or bru time integration Osv bru save output files as p bru q bru d bru Table 12 6 Commands for running demo bru 89 Chapter 13 AUTO Demos Optimization 13 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 19910 COMMAND ACTION mkdir opt cd opt dm opt cp r opt 1 r opt Or opt sv 1 cp r opt 2 r opt Or opt 1 sv 2 cp r opt 3 r opt r opt 2 sv 3 cp r opt 4 r opt r opt 8 Osv 4 create an empty work directory change directory copy the demo files to the work directory get the first constants file one free equation par
161. ution branch restart from q dat sv lor save output files as p lor q lor d lor cp r lor 2 r lor constants changed IRS ISW NTST switch branches at a period doubling detected in the first run append the output files to p lor g lor d lor Table 10 5 Commands for running demo lor 64 10 5 fre A Periodically Forced System This demo illustrates the computation of periodic solutions to a periodically forced system In AUTO this can be done by adding a nonlinear oscillator with the desired periodic forcing as one of the solution components An example of such an oscillator is Y x by s a y Pa y yl Ey ng which has the asymptotically stable solution x sin Gt y cos Gt We couple this oscillator to the Fitzhugh Nagumo equations w Fv w e w v dw b rsin Pt any by replacing sin 3t 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 COMMAND ACTION mkdir fre create an empty work directory cd fre change directory dm fre copy the demo files to the work directory cp r fre 1 r fre get the first constants file r fre homotopy to r 0 2 Osv 0 save output files as p 0 q 0 d 0 cp r frc 2 r fre constants changed I
162. utions of ODEs with detection of Hopf bifurcations The sign of PT the point number in fort 7 is used to indicate stability is stable is unstable Demo ab IPS 1 Fixed points of the discrete dynamical system u D f u 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 continuation in time Note that the time discretization is only first order accurate so that results should be carefully interpreted Indeed this option has been included primarily for the detection of stationary solutions which can then be entered in the user supplied subroutine STPNT 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 ana lytically 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 38 IPS 4 A boundary value problem Boundary conditions must be specified in the user supplied subroutine BCND and integral constraints in ICND The AUTO constants NBC and NINT must be given correct values Demos exp int kar IPS 5 Algebrai
163. witching is done at one of these Along the bifurcating branch the adjoint variables become nonzero while state variables and A remain constant Any such non trivial 96 solution point can be used for continuation in two equation parameters after fixing the L2 norm 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 COMMAND ACTION mkdir obv create an empty work directory cd obv change directory dm obv copy the demo files to the work directory cp r obv 1 r obv get the first constants file Or obv locate 1 parameter extrema as branch points sv obv save output files as p obv q obv d obv cp r obv 2 r obv constants changed IRS ISW NMX Or obv compute a few step on the first bifurcating branch sv 1 save the output files as p 1 q 1 d 1 cp r obv 3 r obv constants changed IRS ISW NMX ICP 3 Or obv 1 locate 2 parameter extremum restart from q 1 sv 2 save the output files as p 2 q 2 d 2 cp r obv 4 r obv constants changed IRS ICP 4 Or obv 2 locate 3 parameter extremum restart from q 2 sv 3 save the output files as p 3 q 3 d 3 Table 13 6 Commands for running demo obv 97 Chapter 14 AUTO Demos Connecting orbits 141 fsh A Saddle Node Connection Thi
164. work directory Unix COMMAND ACTION cd go to your main directory or other directory mkdir ab create an empty work directory cd ab change to the work directory AUTO COMMAND ACTION copy the demo files to the work directory Table 8 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 f 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 wu uz 0 is a stationary solution 8 4 Executing all Runs Automatically To execute all prepared runs of demo ab simply type the command given in Table 8 2 The resulting screen output is 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 46 Unix COMMAND ACTION execute all runs of demo ab Table 8 2 Executing all runs of demo ab different accuracy but simply that arithmetic differences have accumulated from step to step possibly leading to different step size decisions 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 Note also that
165. xe Plotting the 2 Parameter Diagram A Tutorial Demis en ede Bae PO a a ed bea cuting all Runs Automatically cios o A hee ae a Executing Selected Runs Automatically 0 0 0 2 202 208 Using AUTO Commands oe epee eee a ae Se ee EE Se Ee a ee Plotting the Results with PLAUT o Following Folds and Hopf Bifurcations ao aaa Relabeling Solutions in the Data Files a a aaa a a 00008 Converting Saved PLAUT Files to PostScript o Using the GUI Abbreviated AUTO Commands AUTO Demos Fixed points enz Stationary Solutions of an Enzyme Model 2 dd2 Fixed Points of a Discrete Dynamical System 9 1 9 2 10 AUTO Demos Periodic solutions 11 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 lrz abc pp2 lor fre ppp plp pp3 tor 10 10 pen 10 11 chu 10 12 phs 10 13 ivp The Lorenz Equations The A gt B gt C Reaction A 2D Predator Prey Model Starting an Orbit from Numerical Data o o A Periodically Forced System Continuation of Hopf Bifurcations 2 0048 Fold Continuation for Periodic Solutions 2 08048 Period Doubling Continuation o aoo a 0 002000 0000 4 Detection of Torus Bifurcations potes a oes Y Vee ee Rotations of Coupled Pendula A Non Smooth System Chua s Circuit
166. xplicit 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 Section 5 1 2 of Champneys amp Kuznetsov 1994 under the simplification that we do not solve for the adjoint u t here The basic idea is to start with a small solution in the unstable manifold and perform continuation in PAR 11 27 and dummy initial condition parameters in order to satisfy the correct right hand boundary conditions which are defined by zeros of other dummy parameters w More precisely the left hand end point is placed in the tangent space to the unstable manifold of the saddle and is characterized by NUNSTAB coordinates satisfying the condition El ES da dE E where 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 TEQUIB 0 1 and set IP 12 IEQUIB NDIM Then 112 PAR IP PAR IP i i 1 2 NUNSTAB PAR IP NUNSTAB i wi i 1 2 NUNSTAB Note that to avoid interference with the test functions i e PAR 21 PAR 36
167. y one true continuation parameter namely PAR 1 Three solution measures are defined in the subroutine PVLS namely the L2 norm of the first solution component the minimum of the second compo nent 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 r pvl we have NICP 4 with PAR 1 PAR 4 specified as parameters Thus in this example PAR 2 PAR 4 are overspecified Note that PAR 1 must appear first in the ICP list the other parameters cannot be used as true continuation parameters 36 6 8 Computation Constants 6 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 6 8 2 ISP This constant controls the detection of branch points period doubling bifurcations 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
168. y 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 6 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 35 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 3 in the second run The above procedure can be repeated For example folds from the second run can be continued in a third run with three equation parameters specified in addition to PAR 10 Thus 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 6 7 9 Internal free parameters The actual continuation scheme in AUTO may use additional free

млг - Department of Mathematics, University of Utah

Contents

Download Pdf Manuals

Related Search

Related Contents