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Schilder, F. (2004). Torcont v1 (2003) user manual.

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1. WH A Create the parameter file expected by torcont with settings problem name pnet run 1 continuation parameter epsilon mesh 30x100 LFil and Reserve see also ex kawa 100 200 use a Krylov subspace of at most Restart dimensions Restart 50 max number of iterations of gmres ItMX 350 dropping tolerance for ilu 0 005 max value of rel residual of gmres TOL 1 0e 8 don t print debugging information of iterative solver and ilu LogFile NULL continuation interval 1 5 7 0 max number of continuation steps 150 print every npr steps npr 5 and run torcont create tc run pnet 1 continuer param epsilon discretisation_pointsl 30 discretisation points2 100 linear solver LFil 100 linear solver Reserve 200 linear solver Restart 50 linear solver ItMX 350 linear solver DropTOL 0 005 linear solver TOL 1 0e 8 linear solver LogFile NULL clog continuer param interval 1 5 7 0 continuer ItMX 150 npr 5 torcont pnet 1 STEP PAR LIxiI I TOL Period T2 0 3 000e 00 8 6358e 01 5 7013e 03 6 9584e 00 5 3 355e 00 8 8125e 01 4 0472e 03 7 1356e 00 oF 10 3 934e 00 9 1147e 01 2 7886e 03 7 4471e 00 d 15 4 531e 00 9 4411e 01 2 3816e 03 7 7914e 00 ix 20 4 932e 00 9 6678e 01 2 5076e 03 8 0341e 00 a 25 5 478e 00 9 9835e 01 1 2610e 02 8 3755e 00 25 30
2. currentdir export fschild examples lang mkdir data currentdir Warning the name changecoords has been redefined Error in mkdir directory exists and is not empty export fschild examples lang kh Po ZG ue V V V V N Definition of the System and Creation of the Shared Object The constants and parameters are defined as a list of name value pairs The system is defined as a function taking a list and a number as arguments and returning a list of expressions gt Constants Params epsilon 0 rho 0 55 omega 3 5 LANG x t gt gt gt x 3 0 7 x 1 omega x 2 omega x 1 x 3 0 7 x 2 0 6 x 3 1 3 x 3 3 x 1 2 x 2 72 1 rho x 3 epsilon x 3 xi 3 L 1 Create a shared object by calling codegen and then compiling and linking The compiler options used are for Solaris You may need different options create ode lang LANG Constants Params compiler gcc fPIC linker gcc fPIC shared codegen lt lang ode gt lang c OK geo PIC o lang o lang c OK L gcc fPIC shared o lang so lang o OK Definition of the Start Solutions Define initial approximations to the quasiperiodic solution This function is a guess This torus function s must always be 2 Pi periodic in each argument gt isol t th gt 0 9 0 3 cos th cos t 0 9 0 3 cos th sin t 0 7 0 5
3. Create the parameter file expected by pofind with settings problem name kawa run 1 initial solution file kawa pol dat and then compute the initial periodic orbit gt create pof run kawa 1 isol kawa pol dat pofind kawa 1 Iterat D mpfung Normen Rechenzeit I SI gamma Ixi I gamma d F x DF x solve 0 0 0 0000e 00 7 3236e 00 2 2614e 01 0 0000e 00 0 0 0 1 1 1 0000e 00 7 7135e 00 2 8230e 02 5 8477e 01 0 0 0 0 0 0 2 1 1 0000e 00 7 6929e 00 7 9791e 04 1 2612e 01 0 0 0 0 0 0 3 1 1 0000e 00 7 6918e 00 6 6770e 06 4 8005e 03 0 0 0 0 0 0 4 1 1 0000e 00 7 6918e 00 2 3129e 07 1 9336e 05 0 0 0 0 0 0 solution written to file data kawa 1 po0 dat L checking for memory leaks no leaks Create the parameter file expected by pocont with settings problem name kawa run 1 continuation parameter k1 continuation interval 0 04 0 15 print every npr steps npr 1 and then continue the periodic solution The solution written to data kawa 1 po0 dat by pofind is used as initial solution gt create poc run kawa 1 continuer param kl continuer param interval 0 04 0 15 npr 1 pocont kawa 1 STEP PAR x data file 0 9 000e 02 1 7199e 00 data kawa 1 po0 dat 1 9 438e 02 1 7195e 00 data kawa 1 pol dat 2 1 027e 01 1 7186e 00 data kawa 1 po2 dat 3 1 187e 01 1 7167e 00 data kawa 1 po3 dat 4 1 361e 01 1 7144e 00 data kawa 1 po4 dat 5 1
4. grl surfdata data4 display grl The following demonstrates how to obtain cross sections corresponding to invariance curves of local stroboscopic maps and how to draw these together with the torus Note that we create cross sections of both the full blue and the dissected coral torus gt data5 sectionl data2 10 data6 sectionl data2 4 data7 2section2 data2 36 data8 section2 data2 31 data5 sectionl data4 1 data6 sectionl data4 15 data7 section2 data4 1 data8 section2 data4 36 gr2 spacecurve data5 thickness 3 color blue gr3 spacecurve data6 thickness 3 color coral gr4 spacecurve data7 thickness 3 color blue gr5 spacecurve data8 thickness 3 color coral gt drsplavilgtly gr2 959 qui gr5 Bifurcation Diagram At last we draw the bifurcation diagram The bifurcation diagrams contain the following columns in each row fpcont PAR x components of fixed point pocont non autonomous PAR x pocont autonomous PAR x T torcont non autonomous PAR x ERR T2 torcont autonomous PAR x ERR T1 T2 where means PAR value of parameter x normalised L2 norm of x T period periodic orbit ERR estimated error of a quasiperiodic solution T1 first basic period quasiperiodic orbit T2 second basic period quasiperiodic orbit The data structure of a bifurc
5. vdp_qpol dat mesh 40x40 LFil and Reserve see also ex kawa 250 300 use a Krylov subspace of at most Restart dimensions Restart 50 max number of iterations of gmres IEMX 350 dropping tolerance for ilu 0 01 max value of rel residual of gmres TOL 1 0e 7 don t print debugging information of iterative solver and ilu LogFile NULL and run torfind gt create tf run vdp 1 isol vdp_qpol dat discretisation_pointsl 40 discret isation points2 40 inear_solver LFil 250 linear_solver Reserve 300 linear_solver Restart 50 linear_solver ItMX 350 linear_solver DropTOL 0 01 linear_solver TOL 1 0e 7 linear_solver LogFile NULL clog torfind vdp 1 Iterat D mpfung Norme Rechenzeit I SI gamma MESM I gamma d F x DF x 0 0 0 0000e 00 1 1593e 02 2 5166e 01 0 0000e 00 0 0 1 1 1 0000e 00 1 1626e 02 3 1900e 01 9 4264e 00 0 3 4 4 2 1 1 0000e 00 1 1612e 02 2 1970e 04 1 6002e 01 0 6 8 6 3 1 1 0000e 00 1 1612e 02 4 7285e 08 4 8280e 04 0 12 8 4 1 1 0000e 00 1 1612e 02 2 2302e 10 5 3201e 05 1 3 17 0 period T1 6 3150750482154807131e 00 period T2 5 7599935747348069981e 00 solution written to file data vdp 1 qpo0 dat checking for memory leaks no leaks Create the parameter file expected by torcont with settings problem name vdp run 1 continuation parameter beta b mesh 40x40 LFi
6. LogFile NULL and run torfind create tf run pnet 1 isol pnet qpol dat discretisation_pointsl 30 discretisation points2 100 linear solver LFil 250 linear solver Reserve 250 linear solver Restart 35 linear solver ItMX 350 linear solver DropTOL 0 01 linear solver TOL 1 0e 8 linear solver LogFile NULL clog torfind pnet 1 Iterat D mpfung Normen Rechenzeit I SI gamma biet I gamma d F x DF x solve 0 0 0 0000e 00 4 9607e 01 1 8597e 01 0 0000e 00 0 0 0 1 1 1 0000e 00 4 8752e 01 5 9276e 00 8 6662e 00 0 4 VIS 24 7 2 1 1 0000e 00 4 7434e 01 6 5696e 01 2 9055e 00 0 8 3 6 Fed 3 1 1 0000e 00 4 7311e 01 9 6584e 03 2 9172e 01 ql 5 4 44 5 4 1 1 0000e 00 4 7309e 01 1 8311e 06 3 6620e 03 LD Teak 53 9 5 1 1 0000e 00 4 7309e 01 1 8487e 10 1 1100e 06 1 8 8 9 63 7 period T2 6 9584467149239035422e 00 solution written to file data pnet 1 qpo0 dat L checking for memory leaks no leaks Create s plot of the computed torus in phase space t x 7 x Because t is not plotted ot periodically the torus appears not closed in the plot See the example kawa for the format of the data structure gt datal read_torus_data pnet 1 0 data2 select torus coords datal 1 3 4 surfdata data2 0 6 3 0 44 SSSS8 4 1 SS x 1 __ LYT TU HC OO WMTYORN J SEUI 0 24 uu TT NN ii aN 7
7. we continue the doubled invariant torus emerging from the simple torus by a torus doubling bifurcation in the second basic frequency This run demonstrates how to run torfind and torcont cut out sections of tori extract cross sections corresponding to invariant circles of local stroboscopic maps create 3d plots of computed tori together with space curves Create the parameter file expected by torfind with settings problem name kawa run 3 initial solution file kawa_qpo3 dat owerwrite the value of k1 to 0 0775 mesh 20x80 doubled with respect to th2 LFil and Reserve 100 200 dropping tolerance for ilu 0 01 don t print debugging information LogFile NULL and then compute the initial torus gt create tf run kawa 3 isol kawa qpo3 dat ode k1 0 0775 discretisation_pointsl 20 discretisation_points2 80 linear_solver LFil 100 linear_solver Reserve 200 linear_solver DropTOL 0 01 linear_solver LogFile NULL clog torfind kawa 3 Iterat D mpfung Norme Rechenzeit I SI gamma Ixi I gamma d F x DF x solve 0 0 0 0000e 00 6 9101e 01 2 1539e 00 0 0000e 00 0 0 0 1 1 1 0000e 00 6 9168e 01 2 9498e 02 2 1636e 00 0 3 LI 19527 2 1 1 0000e 00 6 9134e 01 1 6709e 03 5 0092e 01 Q5 2 1735 3 1 1 0000e 00 6 9132e 01 1 1526e 04 6 1077e 02 0 8 3 8 21 22 4 1 1 0000e 00 6 9132e 01 9 5981e 06 2 3312e 03 1 1 S0
8. x 3 x 1 e 1 x 1 2 x 2 x 4 1 d x 3 b x 1 x 3 e 1 x 3 2 x 4 L 1 Create a shared object by calling codegen and then compiling and linking The compiler options used are for Solaris You may need different options create ode vdp VDP Constants Params compiler gcc fPIC gt gt linker gcc fPIC shared codegen lt vdp ode gt vdp c OK gcc lt PIC e o vdp o vdp c 4 OK L gcc fPIC shared o vdp so vdp o OK Definition of the Start Solutions Define initial approximations to the quasiperiodic solution This function is a guess This torus function s must always be 2 Pi periodic in each argument gt isol t th gt 2 sin t 2 cos t 2 sin th 2 19 cos th L Write the initial solution to disk on a 40x40 mesh Parameters problem name vdp run 1 name of the function calculating initial solution values isol TI 6 3 T2 5 8 mesh 40x40 write tss vdp 1 isol 6 3 5 8 40 40 vdp qpol dat Run 1 Continuation of Quasiperiodic Orbits b 0 0 22 This run demonstrates how to run torfind and torcont tune the step size control of the continuer print and interpret debugging information of the continuer create 3D plots and animations Create the parameter file expected by torfind with settings problem name vdp run 1 initial solution file
9. 00 3 0264e 03 1 7952e 00 4 lang 1 qpol dat 10 5 796e 01 1 1676e 00 2 5121e 03 1 7952e 00 4 lang 1 qpo2 dat 15 5 934e 01 1 1602e 00 1 9799e 03 1 7952e 00 4 lang 1 qpo3 dat 20 6 040e 01 1 1546e 00 1 4345e 03 1 7952e 00 4 lang 1 qpo4 dat 25 6 111e 01 1 1507e 00 8 8021e 04 1 7952e 00 4 lang 1 qpo5 dat 30 6 149e 01 1 1486e 00 3 2054e 04 1 7952e 00 4 lang 1 qpo6 dat 31 6 149e 01 1 1486e 00 3 2054e 04 1 7952e 00 4 lang 1 qpo7 dat damped_newton no convergence STEP PAR Llxl TOL Period Tl P file 0 5 500e 01 1 1830e 00 3 3538e 03 1 7952e 00 4 lang 1 qpo0 dat 5 5 361e 01 1 1902e 00 3 6695e 03 1 7952e 00 4 lang 1 qpo8 dat 10 5 128e 01 1 2019e 00 4 1218e 03 1 7952e 00 4 lang 1 qpo9 dat 15 4 873e 01 1 2146e 00 4 5389e 03 1 7952e 00 4 lang 1 qpo10 dat 20 4 599e 01 1 2280e 00 4 9178e 03 1 7952e 00 4 lang 1 qpoll dat 25 4 309e 01 1 2419e 00 5 2599e 03 1 7952e 00 4 lang 1 qpol2 dat 30 4 009e 01 1 2560e 00 5 5733e 03 1 7952e 00 5 lang 1 qpol3 dat 35 3 704e 01 1 2704e 00 5 8772e 03 1 7952e 00 5 eriod T2 3115e 00 2625e 00 2007e 00 1527e 00 1173e 00 0939e 00 0820e 00 0820e 00 eriod T2 3115e 00 3676e 00 4673e 00 5867e 00 728 7e 00 8967e 00 0943e 00 3263e 00 data data data data data data data data data data data data data data data data data data data select torus coords data surfdata data SSW NNI 35 3 704e 01 1 2704e 00 5 8772e 03 lang 1
10. 1301e 02 norm v x 2 2373e 00 rel_abs_diff 1 9625e 02 beta 1 8554e 01 h_facl 1 2139e401 h fac2 1 0772e400 h 1 3867e 01 15 2 013e 01 2 8249e 00 2 8938e 01 5 8351e 00 5 1614e 00 data vdp 1 qpo5 dat continuer step norm x 1 1334e 02 norm v x 2 3340e 00 rel abs diff 2 0412e 02 beta 1 9131e 01 h_facl 1 2247e 01 h_fac2 1 0450e 00 h 1 3766e 01 continuer step norm x 1 1341e 02 norm v x 2 3952e 00 rel_abs_diff 2 0935e 02 beta 1 9711e 01 h_facl 1 1942e 01 h_fac2 1 0145e 00 h 1 3267e 01 continuer step norm x 1 1322e 02 norm v x 2 2104e 00 rel abs diff 1 9352e 02 beta 1 8712e 01 h facl 1 29186401 h fac2 1 0681e400 h 1 3463e 01 18 2 035e 01 2 8302e 00 3 1390e 01 5 8404e 00 5 1589e 00 data vdp 1 qpo6 dat continuer step norm x 1 1278e 02 norm v x 2 1901e 00 rel_abs_diff 1 9248e 02 beta 1 8354e 01 h_facl 1 2988e 01 h_fac2 1 0888e 00 h 1 3926e 01 continuer step norm x 1 1207e 02 norm v x 2 3721e 00 rel abs diff 2 0979e 02 beta 1 9381e 01 h_facl 1 1917e 01 h_fac2 1 0316e 00 h 1 3648e 01 continuer step norm x 1 1112e 02 norm v x 2 3814e 00 rel_abs_diff 2 1240e 02 beta 1 9265e 01 h_facl 1 17706401 h fac2 1 0378e400 h 1 3455e 01 21 1 901e 01 2 7776e 00 2 2214e 00 5 8322e 00 5 1908e 00 data vdp 1 qpo7 dat continuer step norm x 1 1000e 02 norm v x 2 2160e 00 rel abs diff 1 9963e 02 beta
11. 2 8238e 00 7 4058e 00 6 0297e 00 vdp 2 qpol18 dat 30 9 676e 02 2 8302e 00 5 9467e 00 6 0252e 00 vdp 2 qpol19 dat 35 9 752e 02 2 8345e 00 1 8941e 00 6 0222e 00 vdp 2 qpo20 dat 40 9 471e 02 2 8383e 00 2 6917e 00 6 0186e 00 vdp 2 qpo21 dat 45 8 979e 02 2 8362e 00 5 1114e 00 6 0156e 00 vdp 2 qpo22 dat 50 8 366e 02 2 8195e 00 3 6437e 00 6 0182e 00 vdp 2 qpo23 dat 55 8 042e 02 2 8008e 00 1 2183e 01 6 0249e 00 vdp 2 qpo24 dat time real 4 28 20 5 user 4 27 33 4 Sys 2 4 command terminated abnormally Torcont was killed here hence the message of time gt 0269e 00 0198e 00 0172e 00 0164e 00 0166e 00 0195e 00 0272e 00 NY CBr BN D dq NY o BN o PB 0488e 00 Period T2 5309e 00 6612e 00 7655e 00 7819e 00 7837e 00 7801e 00 7679e 00 7574e 00 7555e 00 7589e 00 7703e 00 a a a nnn ona won QUO 7797e 00 data data data data data data data data data data data data data data data data data data data data data Here we create two animations of the torus At first we read in all the solutions given in a list nums The for loop creates two sequences of plot structures anl and an2 See example kawa for the format of the torus data structure nums i nums i 2734 y 5276 gt nums 24 14 0 13 anl NULL an2 NULL for i from 1 by 1 to nops nums do printf reading graph d n
12. 5 507e 00 1 0000e 00 1 7242e 02 8 3937e 00 e 35 5 515e 00 1 0005e 00 2 1305e 02 8 3985e 00 i 40 5 520e 00 1 0008e 00 2 3145e 02 8 4016e 00 ee 45 5 524e 00 1 0010e 00 2 4256e 02 8 4042e 00 a 50 5 528e 00 1 0012e 00 2 4240e 02 8 4066e 00 55 5 532e 00 1 0014e 00 2 2975e 02 8 4093e 00 oa 60 5 537e 00 1 0018e 00 2 0943e 02 8 4129e 00 i 65 5 546e 00 1 0023e 00 1 8557e 02 8 4190e 00 70 5 567e 00 1 0036e 00 1 7132e 02 8 4325e 00 E 75 5 602e 00 1 0057e 00 1 7816e 02 8 4552e 00 Mis 80 5 722e 00 1 0128e 00 2 3569e 02 8 5325e 00 idi 85 6 281e 00 1 0463e 00 1 1788e 02 8 9001e 00 n 90 6 745e 00 1 0748e 00 6 4120e 02 9 2135e 00 95 6 866e 00 1 0823e 00 6 5148e 02 9 2947e 00 S 6 941e 00 1 0859e 00 3 6025e 01 9 3314e 00 dat 05 6 975e 00 1 0868e 00 7 7580e 01 9 3359e 00 Sen 7 005e 00 1 0876e 00 4 1035e 01 9 3406e 00 dat STEP PAR IIx TOL Period T2 0 3 000e 00 8 6358e 01 5 7013e 03 6 9584e 00 js 5 2 687e 00 8 4858e 01 1 0881e 02 6 8123e 00 TCU 10 2 385e 00 8 3450e 01 1 9077e 02 6 6801e 00 isti no convergence bifurcation diagramm written to file data pnet 1 bd dat output written to file data pnet checking for memory leaks no leaks real 1 19 53 5 user 1 19 37 3 Sys 14 5 datal read torus data pnet 1 17 data2 select torus coords datal 1 3 surfdata data2 data file data p data p data p data p data p data p data p data p data p data p data p data p data p data p data p data p data p data p
13. 535e 01 1 7121e 00 data kawa 1 po5 dat STEP PAR x data file 0 9 000e 02 1 7199e 00 data kawa 1 po0 dat 1 8 563e 02 1 7204e 00 data kawa 1 po6 dat 2 7 735e 02 1 7212e 00 data kawa 1 po7 dat 3 6 166e 02 1 7225e 00 data kawa 1 po8 dat 4 4 442e 02 1 7236e 00 data kawa 1 po9 dat 5 2 677e 02 1 7244e 00 data kawa 1 pol10 dat bifurcation diagramm written to file data kawa 1l bd dat output written to file data kawa l pocont log checking for memory leaks no leaks real 0 6 user 0 4 sys 0 0 Read the computed data of the periodic orbit into datal Here orbit 0 of run 1 is choosen Then select the columns 2 3 4 from the data which contain the projection onto the x y z subspace The data of a periodic orbit in R n at N mesh points has the following structure note t0 tN tO x1 t0 x2 t0 xn tO tN x1 tN x2 tN xn tN datal read po data kawa 1 0 data select po coords datal 2 3 4 gt spacecurve data thickness 3 color blue axes boxed 1 38 4 1 378 4 1 376 4 1 374 4 1 372 4 1 37 4 1 368 gt N Run 2 Continuation of a Quasiperiodic Orbit Invariant 2 Torus Secondly we continue the invariant torus emerging from the periodic orbit by a Hopf bifurcation This run demonstrates how to run torfind and torcont create 3d plots of computed tori Create the parameter file expected by torfind with settings problem
14. 9658e 05 sin t cos 2 0 th 8 3610e 05 cos t cos 2 0 th 1 8230e 03 sin 3 0 th 4 0360e 05 sin t sin 3 O0 th 3 5066e 05 cos t sin 3 0 th 6 2331e 03 cos 3 0 th 7 7939e 05 sin t cos 3 0 th 6 0643e 05 cos t cos 3 0 th 2 3 175e 03 sin 4 0 th 1 0078e 04 sin t sin 4 0 th 2 0976e 05 cos t sin 4 0 th 9 0188e 03 cos 4 0 th 9 4243e 05 sin t c os 4 0 th 3 5578e 05 cos t cos 4 0 th 1 4481e 03 sin 5 0 t h 2 3374e 04 sin t sin 5 0 th 1 8662e 05 cos t sin 5 0 th 7 2119e 04 cos 5 0 th 8 0794e 07 sin t cos 5 0 th 2 7773 e 05 cos t cos 5 0 th Write discretisations of the initial solutions to unique files This takes some time so do not call this functions if not necessary Note that the first period is always equal to the forcing period For the second period we give initial guesses 60 4 and 118 The computations are done on a 20x40 mesh single torus and a 20x80 mash doubled torus gt Args name run iso T1 T2 N1 N2 write_poss kawa 1 psol 2 Pac 20 write tss kawa 2 tsol Z P 60 4 20 40 write tss kawa 3 dtsol 2 Pi 118 20 80 kawa pol dat kawa qpo2 dat L kawa qpo3 dat gt 3 Run 1 Continuation of a Periodic Orbit At first we continue the periodic orbit from which the invariant torus emerges by a Hopf bifurcation This run demonstrates how to run pofind and pocont create 3D plots of computed periodic orbits
15. Default Values ode lt parname gt lt value gt sets the value of the free parameter lt parname gt to lt value gt overwrites the default value discretisation_points 20 set the number of mesh points linear_solver LFil 10 set the number of values which are reserved for fill in for each line of L and U computed by the ilu preconditioner linear_solver Reserve 20 set the number of additional fill in elements per line if more than LFil elements violate the drop condition of the ilu preconditioner linear solver Restart 15 restart gmres after Restart iterations linear solver ItrMX 150 max number of iterations of gmres this is the global iteration index counting subiterations linear solver DropTOL 0 02 dropping tolerance for the ilu preconditioner linear solver PermTOL 1 criterion for pivotisation should always be set to 1 linear solver TOL 1 0e 4 the stopping criterion for gmres relative residual linear solver LogFile NULL for printing debugging information of the linear solver set LogFile to clog this helps very much to tune the linear solver nonlinear solver ItrMX 10 max number of Newton steps nonlinear solver SubItMX 8 max number of damping steps per Newton step nonlinear solver TOL 1 0e 4 stopping criterion for the Newton iteration continuer param lt parname gt set the primary continuation parameter continuer param interval 1 1 set the parameter interval
16. L and U computed by the ilu preconditioner linear solver Reserve 200 set the number of additional fill in elements per line if more than LFil elements violate the drop condition of the ilu preconditioner linear solver Restart 35 restart gmres after Restart iterations linear solver ItrMX 350 max number of iterations of gmres this is the global iteration index counting subiterations linear solver DropTOL 0 02 dropping tolerance for the ilu preconditioner linear solver PermTOL 1 criterion for pivotisation should always be set to 1 linear solver TOL 1 0e 4 the stopping criterion for gmres relative residual linear solver LogFile 2 NULL for printing debugging information of the linear solver set LogFile to clog this helps very much to tune the linear solver nonlinear solver ItrMX 10 max number of Newton steps nonlinear solver SubItMX 8 max number of damping steps per Newton step nonlinear solver TOL 1 0e 4 stopping criterion for the Newton iteration IN create tc run problem name run continuer param par name options Creates the parameter file expected by pocont Parameters problem name an unique name of your problem as a string e g vdp run number of the run e g 1 2 options a sequence of name value pairs Possible Options and Default Values ode lt parname gt lt value gt sets the value of the free parameter lt parname gt to lt value gt overwrites
17. Reserve see also ex kawa 250 300 use a Krylov subspace of at most Restart dimensions Restart 50 max number of iterations of gmres ItMX 350 dropping tolerance for ilu 0 01 max value of rel residual of gmres TOL 1 0e 7 don t print debugging information of iterative solver and ilu LogFile NULL and run torfind to compute the solution torfind automatically writes the solution to the start solution of run 2 gt Qreate tf run vdp 2 isol data vdp l qpol dat ode b 0 1 discretisation_pointsl 40 discretisation points2 40 linear_solver LFil 250 linear_solver Reserve 300 linear_solver Restart 50 linear_solver ItMX 350 linear_solver DropTOL 0 01 linear_solver TOL 1 0e 7 linear_solver LogFile NULL clog torfind vdp 2 Iterat D mpfung Normen Rechenzeit I SI gamma x I gamma d F x DF x solve 0 0 0 0000e 00 1 1488e 02 1 9697e 00 0 0000e 00 0 0 0 1 1 1 0000e 00 1 1634e 02 6 0309e 01 8 2195e 00 05 3 4 3 134 6 2 1 1 0000e 00 1 1595e 02 1 1565e 01 3 4337e 00 0 26 8 5 228 6 3 1 1 0000e 00 1 1583e 02 2 6079e 03 4 9449e 01 1 0 12 7 322 4 4 1 1 0000e 00 1 1582e 02 7 9621e 06 2 7182e 02 1 3 16 9 418 3 5 1 1 0000e 00 1 1582e 02 1 3183e 08 5 4643e 05 1 6 2120 253715 1 period T1 6 0421905943890745760e 00 period T2 5 5308650816578683873e 00 solution written to file data vdp 2 qpo0 dat estimating
18. course between these two strong resonances there are some more or less weak resonances see therefore also the next plot We read the bifurcation diagram into datal datal then contains two lists of data one for the forward continuation and one for the backward continuation From both lists we extract the parameter and the second basic period By dividing the second period by Pi the first basic period we obtain the inverse rotation number These is plotted blue together with the lines of the 1 2 and the 1 3 resonances red over the parameter epsilon See example kawa for the format of the bifurcation diagram data structure gt datal read bd pnet 1 data2 select bd cols datal 1 4 data3 NULL for i from 1 by 1 to nops data2 1 do data3 data2 1 i l 1 evalf data2 1 i 2 Pi data3 od for i from 1 by 1 to nops data2 2 do data3 data3 data2 2 i 1 evalf data2 2 i 2 Pi od pl plot data3 color blue pl i Bl plot Cl Lb2621s bf2 1 Lbb2 9 Pt edly 0010reored display pl 34 2 84 2 64 2 44 2 24 241 i 2 3 4 5 6 7 A plot of the estimated error of the torus solution The error is plottet over epsilon Areas with larger error values indicate where ilands of weak resonances may reside But numerically this is very hard to verify At strong resonances ou
19. data p data p data p data p data p net net net net net net net net net net net net net net net net net net net net net net net data file data p data p data p l torcont log 4 net net net 1 qpo7 1 qpo0 1 qpol 1 qpo2 1 qpo3 1 qpo4 1 qpo5 1 qpo6 1 qpo8 O a O O Q Q Q a a a 1 qpo9 1 qpol0 1 qpoll 1 gpo12 1 qpol13 1 qpol4 1 qpol5 1 qpol6 1 gpo17 1 qpo18 1 qpol9 1 qpo20 1 qpo21 1 gpo22 1 qpo0 d 1 qpo23 1 qpo24 H ba III 2 III 206A jil Re Here we create an animation of the evolution of the torus under changes of the parameter epsilon At first we read in all solutions given in a list nums The for loop creates a sequence of plot structures gt nums 24 23 0 22 an NULL for i from 1 by 1 to nops nums do printf reading graph d n nums i datal read torus data pnet 1 nums i data2 select torus coords datal 1 3 4 an an surfdata data2 od reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap rea
20. number of values which are reserved for fill in for each line of L and U computed by the ilu preconditioner linear_solver Reserve 20 set the number of additional fill in elements per line if more than LFil elements violate the drop condition of the ilu preconditioner linear_solver Restart 15 restart gmres after Restart iterations linear_solver ItMX 150 max number of iterations of gmres this is the global iteration index counting subiterations linear_solver DropTOL 0 02 dropping tolerance for the ilu preconditioner linear_solver PermTOL 1 criterion for pivotisation should always be set to 1 linear_solver TOL 1 0e 4 the stopping criterion for gmres relative residual linear_solver LogFile NULL for printing debugging information of the linear solver set linear_solver LogFile NULL for printing debugging information of the linear solver set LogFile to clog this helps very much to tune the linear solver nonlinear_solver ItMX 10 max number of Newton steps nonlinear_solver SubItMX 8 max number of damping steps per Newton step nonlinear_solver TOL 1 0e 4 stopping criterion for the Newton iteration create_poc_run problem_name run options Creates the parameter file expected by pocont Parameters problem_name an unique name of your problem as a string e g vdp run number of the run e g 1 2 options a sequence of name value pairs Possible Options and
21. nums do printf reading graph d n nums i data read_torus_data vdp 1 nums i datal select_torus_coords data 2 3 4 anl anl surfdata datal an2 an2 surfdata data2 od reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap L reading graph We create two animations which show the evolution of the torus as the continuation progresses For beta near 0 2035 the torus disappears in a Hopf bifurcation to a periodic orbit The angle coordinate 0 becomes obsolete at this point as the dimension of the manifold data2 select_torus_coords data 1 5 6 PpPPpPpPpPpDDHDH OOO 1 OY O1 4S CO PD ES O drops from 2 torus to 1 periodic orbit Therefore the plot over seems to be still a torus but it is rather the same periodic orbit plottet for each value of 0 There is an interesting additional observation The Hopf point seems also to be a limit point of the torus curve This impression is aided by the second animation which clearely shows different pictures for each of the branches In addition to the continuation process which turns around near this Hopf point Further for beta 0 the torus is not unique Thats the reason for choosing beta 0 001 for the initial solution This all together indicates that there is a closed loop of tori with limit points for beta 0 and at t
22. poc run create tf run create tc run fpfind fpcont pofind pocont torfind torcont odeinfo print odeinfo line scan format read bd select bd cols read po data select po coordas read torus data copy periodic solution copy torus solution select torus coords cutl cut2 sectionl section2 contpack m The Example of C Hayashi T Yoshinaga and H Kawakami This example was given by T Yoshinaga and H Kawakamiand shows a cascade of torus doubling bifurcations which lead to a strange attractor This is a nice test example because no numerical obserbable resonances occour The system is given by the equations 9 g d 1 x 32 x 3 7777 o B cosi 3 1 k 3x 2 z gag with the parameter values k 5e 1 B 22 B 3e 1 and k 4e 1 15 Attention Before starting the computations please set the paths in the read and the currentdir commands correctly Ignore the error message issued by mkdir gt restart with plots with process read export fschild maple contpack m currentdir export fschild examples kawa mkdir data currentdir Warning the name changecoords has been redefined Error in mkdir directory exists and is not empty export fschild examples kawa V V V V 3 Definition of the System and Creation of the Shared Object The constants and parameters are defined as a list of name value pairs The system is defined as a functio
23. the default value discretisation_points1 20 set the number of mesh points at the thl axis discretisation points2 20 set the number of mesh points at the th2 axis linear solver LFil 100 set the number of values which are reserved for fill in for each line of L and U computed by the ilu preconditioner linear solver Reserve 200 set the number of additional fill in elements per line if more than LFil elements violate the drop condition of the ilu preconditioner linear solver Restart 35 restart gmres after Restart iterations linear solver ItrMX 350 max number of iterations of gmres this is the global iteration index counting subiterations linear solver DropTOL 0 02 dropping tolerance for the ilu preconditioner linear solver PermTOL 1 criterion for pivotisation should always be set to 1 linear solver TOL 1 0e 4 the stopping criterion for gmres relative residual linear solver LogFile 2 NULL for printing debugging information of the linear solver set LogFile to clog this helps very much to tune the linear solver nonlinear solver ItrMX 10 max number of Newton steps nonlinear solver SubItMX 8 max number of damping steps per Newton step nonlinear solver TOL 1 0e 4 stopping criterion for the Newton iteration continuer param lt parname gt set the primary continuation parameter continuer param interval 1 1 set the parameter interval continuer ItMX 50 set the maximu
24. torcont vdp 2 STEP PAR 1x TOL Period Tl Period T2 file 0 2 000e 01 2 8953e 00 1 8118e 01 6 0422e 00 5 5309e 00 vdp 2 qpo0 dat 5 3 362e 01 2 9568e 00 5 9696e 02 6 0470e 00 5 2557e 00 vdp 2 qpol dat 10 1 020e 00 3 2137e 00 1 2898e 02 6 0418e 00 4 3208e 00 vdp 2 qpo2 dat 15 2 405e 00 3 6314e 00 1 0497e 02 6 0321e 00 3 3598e 00 vdp 2 qpo3 dat 20 4 229e 00 4 1058e 00 1 0368e 02 6 0277e 00 2 7241e 00 vdp 2 qpo4 dat 25 6 447e 00 4 6153e 00 1 1818e 02 6 0256e 00 2 2886e 00 vdp 2 qpo5 dat data data data data data data data 30 8 518e 00 5 0441e 00 2 1908e 01 6 0256e 00 vdp 2 apo6 dat 95 8 586e 00 5 0556e 00 4 3107e 01 6 0255e 00 vdp 2 qpo7 dat 40 8 611e 00 5 0582e 00 5 9263e 01 6 0234e 00 vdp 2 qpo8 dat 45 8 619e 00 5 0591e 00 6 4788e 01 6 0225e 00 vdp 2 apo9 dat 50 8 617e 00 5 0589e 00 7 2793e 01 6 0227e 00 vdp 2 qpol0 dat 55 8 588e400 5 0559e 00 2 8561e 00 6 0253e 00 vdp 2 qpoll dat 60 8 515e 00 5 0436e 00 2 0275e 01 6 0255e 00 vdp 2 qpo12 dat 65 8 314e 00 5 0038e 00 1 0920e 01 6 0251e 00 vdp 2 qpol13 dat Sparskit PGMRES pgmres no convergence STEP PAR Llxl I TOL Period T1 file 0 2 000e 01 2 8953e 00 1 8118e 01 6 0422e 00 vdp 2 qpo0 dat 5 1 430e 01 2 8685e 00 1 6337e 01 6 0366e 00 vdp 2 qpol4 dat 10 1 005e 01 2 8465e 00 3 8479e 01 6 0293e 00 vdp 2 qpol5 dat 15 9 374e 02 2 8343e 00 2 3998e 02 6 0304e 00 vdp 2 qpol6 dat 20 9 252e 02 2 8267e 00 2 2195e 01 6 0311e 00 vdp 2 qpol7 dat 25 9 326e 02
25. 0 4 3 10 4 3 0 3 2 10 3 2 0 2 1 10 2 1 0 3 1 10 3 1 color red display pl 3 2 5 0 2 4 6 8 areas of the strong 1 1 and 1 3 resonances are clearly visible gt data2 select bd cols datal 1 3 plot data2 0 10 0 3 color blue plot data2 0 10 0 0 2 color blue A plot of the estimated error of the torus solution The error is plottet over d delta The two 0 24 0 18 7 0 16 4 0 14 4 0 12 4 0 14 0 08 4 0 06 0 04 4 0 02 4
26. 02 1 7276e 00 9 2680e 02 1 1847e 02 data kawa 3 qpo2 a STEP PAR IuIxi TOL Period T2 data file 0 7 750e 02 1 7283e 00 1 3375e 01 1 1720e 02 data kawa 3 qpo0 de 5 7 505e 02 1 7290e 00 1 3710e 01 1 1584e 02 data kawa 3 qpo3 on 10 6 826e 02 1 7313e 00 1 3451e 01 1 1107e 02 data kawa 3 qpo4 e 15 6 219e 02 1 7345e 00 1 9847e 01 1 0459e 02 data kawa 3 qpo5 20 5 877e 02 1 7386e 00 1 1273e 00 9 8277e 01 data kawa 3 qpo6 3 25 5 589e 02 1 7439e 00 6 0845e 01 9 3032e 01 data kawa 3 qpo7 9s 30 5 065e 02 1 7491e 00 3 0263e 01 8 9128e 01 data kawa 3 qpo8 25 35 4 508e 02 1 7520e 00 2 5987e 01 8 6736e 01 data kawa 3 qpo9 Di 39 3 962e 02 1 7535e 00 2 6043e 01 8 4890e 01 data kawa 3 qpo10 a bifurcation diagramm written to file data kawa 3 bd dat output written to file data kawa 3 torcont log checking for memory leaks no leaks real 24 53 4 user 24 51 7 sys 0 5 Read the computed data of the invariant torus into datal Here torus 0 of run 3 is choosen Then select the columns 3 4 5 from the data which contain the projection onto the x y z subspace gt datal read torus _data kawa 3 0 data2 select_torus_coords datal 3 4 5 Different from the previous plots we want to cut out some segments for a better view In this example we cut out parts in both direction to show that this is successively possible We obtain a nice view into the doubled torus gt data3 cut1 data2 Ae 0s data4 cut2 data3 56 70
27. 1 8481e 01 h_facl 1 2523e 01 h_fac2 1 0814e 00 h 1 3823e 01 continuer step norm x 1 0889e402 norm v x 2 3212e 00 rel abs diff 2 1124e 02 beta 1 8871e 01 h_facl 1 1835e 01 h_fac2 1 0593e 00 h 1 3910e 01 continuer step norm x 1 0849e 02 norm v x 2 7439e 00 rel_abs_diff 2 5059e 02 beta 2 8563e 01 h_fac1 9 9763e 00 h_fac2 7 0393e 01 h 9 3019e 00 24 1 751e 01 2 7121e 00 7 5788e 01 5 9136e 00 5 3052e 00 data vdp 1 qpo8 dat continuer step norm x 1 1129e 02 norm v x 6 5002e 00 rel_abs_diff 5 7886e 02 beta 6 0485e 01 h_fac1 4 3188e 00 h_fac2 3 4477e 01 h 4 4184e 00 25 1 539e 01 2 7820e 00 1 5968e 00 5 9347e 00 5 3946e 00 data vdp 1 qpo9 dat continuer initialize h 1 0000e 01 h_max 2 5000e 01 h_min 1 0000e 00 h fac min 5 0000e 01 h_fac_max 2 0000e 00 MaxDiff 2 5000e 01 alpha 2 0000 e 01 gamma 9 5000e 01 STEP PAR 1x1 TOL Period Tl Period T2 data file 0 1 000e 03 2 9027e 00 1 1089e 02 6 3151e 00 5 7600e 00 data vdp 1 qpo0 dat continuer step norm x 1 1536e 02 norm v x 7 7010e 01 rel_abs_diff 6 6182e 03 beta 8 8840e 00 h_facl 3 7774e 01 h_fac2 2 2421e 00 h 1 9000e 01 1 2 167e 02 2 8838e 00 3 4924e 02 6 3899e 00 5 8156e 00 data vdp 1 qpol10 dat bifurcation diagramm written to file data vdp 1 bd dat output written to file data vdp 1 torcont log checking for memory leaks no leaks real 1 33 32 5 user 1 33 29 8 sys 1 0 Interpretation of the debug
28. 24 9 5 1 1 0000e 00 6 9132e 01 3 8010e 07 2 4931e 04 1 3 6 2 28 7 6 1 1 0000e 00 6 9132e 01 4 2433e 08 1 2618e 05 Pd 7 4 32 00 period T2 1 1719963027479205664e 02 solution written to file data kawa 3 qpo0 dat checking for memory leaks no leaks Create the parameter file expected by torcont with settings problem name kawa run 3 continuation parameter k1 owerwrite the value of k1 to 0 0775 mesh 20x80 LFil and Reserve 300 300 max number of iterations of gmres IEMX 700 max value of rel residual of gmres TOL 1 0e 8 continuation interval 0 04 0 08 max continuation step size 1 0 and then continue the invariant torus In the output Note the jump in the tolerance for k 1 0 05877 0 05725 which is due to a number of weak resonances gt create tc run kawa 3 continuer param k1 ode k1 0 0775 discretisation_pointsl 20 discretisation_points2 80 linear_solver LFil 300 linear_solver Reserve 300 linear_solver ItMX 700 linear_solver TOL 1 0e 8 continuer param_interval 0 04 0 08 continuer h_max 1 0 torcont kawa 3 STEP PAR LIxi I TOL Period T2 data file 0 7 750e 02 1 7283e 00 1 3375e 01 1 1720e 02 data kawa 3 qpo0 d 5 7 917e 02 1 7278e400 1 1748e 01 1 1804e 02 data kawa 3 qpol d zs 8 8 006e 02 1 7276e400 9 2680e 02 1 1847e 02 data kawa 3 qpo2 d O a O O Q Q Q a 8 8 006e
29. 30E 03 cos 1 os t cos th 1 47861E 04 5 20006E O01 cos 1 1 23093E 03 sin os t sin th 14 01392E 04 cos 1 os t cos th 1 32976E 00 13 91183E 05 ocos 1 6 18952E 02 sin os t sin th 19 68285E 02 cos 1 os t cos th dtsol C th gt 6 3293e 05 4 7468e 01 sin t 8 n 1 0 th 9 9029e 5 17355E 01 sin t t th 6 34324E 01 sin 1 th 8 63900E 02 sin t sin th 4 t cos 7 99765E 01 sin t t th 6 06000E 02 sin 1 th 3 69231E O1 sin t sin ch 4 th 5 t cos 11 91934E 04 sin t t th 6 28914E 05 sin th 3 91103E 04 sin 02 sin t sin t sin th 9 th 8 t cos th 9 00134E 02 c 43208E 01 c 95267E 01 c 74376E 02 c 69171E 05 c 41343E 05 c 1639e 01 cos t 6 1526e 04 si 1 0 th 8 2113e 03 cos t sin 1 0 th 3 1443e 04 cos 1 0 th 3 9425e 02 sin t cos 1 0 th 8 6842e 03 cos t cos 1 0 th 7 1716e 04 sin 2 0 th 3 0896e 01 sin t sin 2 0 th 5 1457e 01 cos t sin 2 0 th 1 0355e 03 cos 2 0 th 5 9266e 01 sin t cos 2 0 th 1 0655e 01 cos t cos 2 0 th 2 9828e 05 sin 3 0 th 6 5673e 02 sin t sin 3 0 th 7 2169e 02 cos t sin 3 0 th 1 7579e 03 cos 3 0 th 7 1634e 02 sin t cos 3 0 th 7 8122e 02 cos t cos 3 0 th 1 1 177e 03 sin 4 0 th 8 6009e 02 sin t sin 4 0 th 1 6180e 01 cos t sin 4 0 th 1 6492e 03 co
30. 332e 04 2 qpol dat 2 2 706e 03 2 qpo2 dat 3 6 075e 03 2 qpo3 dat 4 1 074e 02 2 qpo4 dat 5 1 541e 02 2 qpo5 dat 6 2 010e 02 2 qpo6 dat 7 2 483e 02 2 qpo7 dat 8 2 963e 02 2 qpo8 dat 9 3 455e 02 2 qpo9 dat 10 3 966e 02 lang 2 qpo10 dat 11 4 507e 02 lang 2 qpoll dat 12 5 091e 02 lang 2 qpol2 dat 13 5 739e 02 lang 2 qpol3 dat 14 6 479e 02 lang 2 qpol4 dat JS 7 355e 02 lang 2 qpo15 dat 16 8 429e 02 lang 2 qpol6 dat 17 9 570e 02 lang 2 qpol7 dat 18 1 051e 01 lang 2 qpo18 dat lang lang lang lang lang lang lang lang lang lang STEP file PAR 0 0 000e 00 lang 2 qpo0 dat 1 9 332e 04 lang 2 qpol9 dat bifurcation diagramm written to file 20 2 xl 3288e 00 3288e 00 3288e 00 3289e 00 3289e 00 3290e 00 3290e 00 3291e 00 3292e 00 3293e 00 3294e 00 3296e 00 3297e 00 3298e 00 3298e 00 3299e 00 3299e 00 3298e 00 3297e 00 ixil 3288e 00 3288e 00 output written to file checking for memory leaks real Be user I3 10 0 sys 0 8 gt data read torus data lang 2 data select torus coords data surfdata data TOL 0399e 03 T 3369e 03 1 0275e 02 1 6476e 02 1 6758e 02 Tes 7694e 02 T5 9045e 02 1 0820e 02 1 3100e 02 LA 6002e 02 La 9691e 02 1 1442e 01 1 3066e 01 1 4933e 01 La 7229e 01 1 0330e 01 La 4993e 01 1 1286e 01 1 7633e 01 La TOL 0399e 03 1 3369e 03 1 Peri
31. 4 an NULL for i from 1 by 1 to nops nums do printf reading graph d n nums i data read torus data lang 1 nums data select torus coords data 3 4 5 an an surfdata data od reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap reading grap NO O O N Q i O1 OY 1 DD mE mds Hu m Bou usu as meus mu NEPPPPHHHHH O OOO O O1 iS QON I O reading graph 21 reading graph 22 reading graph 23 reading graph 24 Then we display the tori together in a plot with the option insequence true which creates an animation Select the plot and start the animation with the buttons in the toolbar display an insequence true R N Run 2 Continuation of Quasiperiodic Orbits e 0 0 1 This run demonstrates the same as run 1 and in addition how to obtain a start solution for a new run from a previous run at a specified parameter value Only this additional point is commented here Create a start solution for run 2 from a solution of run 1 Parameters problem name lang old run 1 number of solution 18 new run 2 Because torcont does not print solutions at user specified parameter values yet this function may not be of much use at this time See the
32. Ec University of OPEN ACCESS BRISTOL Schilder F 2004 Torcont v1 2003 user manual Link to publication record in Explore Bristol Research PDF document University of Bristol Explore Bristol Research General rights This document is made available in accordance with publisher policies Please cite only the published version using the reference above Full terms of use are available http www bristol ac uk pure about ebr terms html Take down policy Explore Bristol Research is a digital archive and the intention is that deposited content should not be removed However if you believe that this version of the work breaches copyright law please contact open access bristol ac uk and include the following information in your message Your contact details Bibliographic details for the item including a URL An outline of the nature of the complaint On receipt of your message the Open Access Team will immediately investigate your claim make an initial judgement of the validity of the claim and where appropriate withdraw the item in question from public view TORCONT v1 2003 Continuation Software for Quasi Periodic 2 Tori Frank Schilder Bristol Centre for Applied Nonlinear Mathematics Department of Engineering Mathematics University of Bristol Bristol BS8 1TR UK 1 License Copyright C 2003 by Frank Schilder MAPLE and MAPLE V are registered trademarks of Waterloo Maple Inc This software and
33. an these computed by pocont consider using pofind with a start solution as close to the required parameter as possible This is equal to copying and changing the parameter In future versions this detour should become unneseccary Parameters problem name an unique name of your problem as a string e g vdp run the run the orbit was computed in num the number of the orbit within the run see the output of pocont new run the target run for which the copied solution shall become the start solution copy torus solution problem name run num new run Copies a the quasiperiodic solution num of run run to the start solution of run new run Because torcont does not yet output tori at specified parameter values this function may not be of much use If you want to restart from a solution at a different parameter value than these computed by torcont consider using torfind with a start solution as close to the required parameter as possible This is equal to copying and changing the parameter In future versions this detour should become unneseccary Parameters problem name an unique name of your problem as a string e g vdp run the run the torus was computed in num the number of the torus within the run see the output of torcont new run the target run for which the copied solution shall become the start solution data select torus coords data x y z Select the spec
34. ata kawa 2 qpo0 data kawa 2 qpo5 data kawa 2 qpo6 data kawa 2 qpo7 O Aa Q Q Q a Aa Q Q 15 3 828e 02 1 7396e 00 7 9678e 02 4 7345e 01 data kawa 2 qpo7 d at bifurcation diagramm written to file data kawa 2 bd dat output written to file data kawa 2 torcont log checking for memory leaks no leaks real 4 33 4 user 4 32 28 Sys 0 1 Read the computed data of the invariant torus into datal Here torus 0 of run 2 is choosen Then select the columns 3 4 5 from the data which contain the projection onto the X y Z subspace The data of a 2 torus in R n at a NIXN2 mesh has the following structure note thl 0 thl NI th2 0 th2 N2 thi 0 th2 0 xXl thl 0 th2 0 x2 thl 0 o th2 0 xn thl O0 th2 0 J thl N1 th2 O0 x1 thl N1 th2 O x2 th1_N1 th2 0 xn thl N1 th2 O ly thl 0 th2 1 xl thl 0 th2 1 x2 thl 0 th2 1 yy i xn thl 0 th2 l y J thl Nl th2 1 x1 thl N1 th2 1 x2 thl N1 th2 1 xn thl N1 th2 1 ly L Ehl 0 4 th2 N2 xl thLl 0 th2 N2 x2 thl 0 th2 N2 22 xn thl1 0 4 th2 N2 1 5 th1_N1 th2 N2 x1 thl N1 th2 N2 x2 thl N1 th2 N2 xn thl N1 th2 N2 datal read torus data kawa 2 0 data select torus coords datal 3 4 5 surfdata data gt N Run 3 Continuation of the Doubled Quasiperiodic Orbit Invariant 2 Torus Thirdly
35. ation diagram is a list of such rows which are also lists By reading in multiple bifurcation diagrams you get a list of such lists which may have different formats Read in the bifurcation diagrams of the different runs Note that the bifurcation diagrams of different runs contain different numbers of columns and the same data may appear in columns depending on the problem type But the first two columns always contain the parameter and the norm of the solution Therefore we can select the columns 1 and 2 from all these data sets otherwise we would have to do this for each bifurcation diagram separately datal read bd kawa 1 2 3 datal select bd cols datal 1 2 plot datal x 0 04 0 15 color blue 1 75 1 74 1 73 172 0 04 0 06 0 08 0 1 0 12 0 14 The Nonlinear Parametrically forced Network The parametrically forced network is described by the equation 9 Rd m 0 3 p 3 1 Bsin 21 x 0 and is of interest to electrical engineers It is investigated in dependence of epsilon and the remaining parameters are given by B 1 a e B 2 B poe The system is constructed to posess a so called subharmonic response solution with halved frequency frequence divider for epsilon B Here we investigate the system for epsilon 1 5 6 5 Attention Before starting the computations please set the paths in the read and the currentdir commands correctly Ignore the e
36. bserve this phase lock for the 1 1 and the 1 3 resonance where our method breaks down which here is mostly due to the crude discretisation The other resonances are numerically not observable because for epsilon 0 3 and beta 0 1 these phase locking intervals are very narrow For a finer mesh and higher values of epsilon and beta these are observed also We read the bifurcation diagram into datal datal then contains two lists of data one for the forward and one for the backward continuation From both lists we extract the parameter and the two basic periods By dividing both periods by each other we obtain the rotation number These is plotted blue together with the lines of strong resonances red which are possible for rotation numbers in the interval 1 3 over the parameter d delta Weak er resonances are not considered here and also not really observed numerically See example kawa for the format of the bifurcation diagram structure datal read bd vdp 2 data2 NULL for i from 1 by 1 to nops datal 1 do data2 data2 datal 1 i 1 evalf datal 1 i 4 datal 1 i 5 od for i from 1 by 1 to nops datal 2 do data2 datal 2 ill1 evalf data1 2 i 4 d ata1 2 i 5 data2 od pl plot data2 color blue pl pl plot 0 1 1 10 1 1 0 5 4 10 5 4 0 5 3 10 5 3 0 5 2 10 5 2
37. continuer ItMX 50 set the maximum number of continuation steps in both directions continuer MaxDiff 0 25 set the maximum rel abs difference between predicted and corrected solution for step size control continuer Alpha 7 0 set the maximum angle between the tangent vectors of two solution points continuer h0 0 1 set the initial continuation stepsize continuer h max 0 5 set the maximal continuation stepsize continuer h min 0 01 set the minimal continuation stepsize continuer LogFile for printing debugging information of the continuer set LogFile to clog this helps very much to tune the continuer npr 5 print solution every npr steps create tf run problem name run isol file_name options Creates the parameter file expected by torfind Parameters problem name an unique name of your problem as a string e g vdp run number of the run e g 1 2 isol the name of the file containing the initial solution see write tss options a sequence of name value pairs Possible Options and Default Values ode lt parname gt value sets the value of the free parameter lt parname gt to value overwrites the default value discretisation_points1 20 set the number of mesh points at the thl axis discretisation points2 20 set the number of mesh points at the th2 axis linear solver LFil 100 set the number of values which are reserved for fill in for each line of
38. data read torus data vdp 2 datal select torus coords data data2 2select torus coords data anl anl surfdata datal an2 an2 surfdata data2 od reading reading reading reading reading graph graph graph graph graph 24 23 22 21 20 reading graph 19 reading graph 18 reading graph 17 reading graph 16 reading graph 15 reading graph 14 reading graph 0 reading graph 1 reading graph 2 reading graph 3 reading graph 4 reading graph 5 reading graph 6 reading graph 7 reading graph 8 reading graph 9 reading graph 10 reading graph 11 reading graph 12 L reading graph 13 In this animation you can see some tori which appear not smooth This is due to strong resonances which occour at different subintervals of our continuation interval See the section Bifurcation Diagram for further information gt display anl insequence true display an2 insequence true Nt Sy AE NS MN gt N Bifurcation Diagram Within the continuation interval there occour some strong resonances For easy detection we plot the rotation number together with lines of critical values over the parameter d delta At points where the rotation number crosses such a critical line a strong resonance occurs The rotation number then remains constant over some interval This effect is known as a phase lock In our graph we can o
39. ding grap L reading grap 24 23 OO 1 OY O1 iS CO PO ES O Np H HBHBpdBdPmpmpmnnm O00 10Y 01S COND ES O 2DOOOOOO2O2O02O020202202223223220202o2020o0Zo NIN NFR gt In this animation you can see some tori which are not smooth This is due to week and strong resonances and occours visibly araund epsilon 5 524 and epsilon 7 05 The strong resonance at 7 05 can be observed as a phase lock the detected weak resonance may by due to the relatively crude approximation and does not appear so clearly for finer meshes gt display an insequence true SATO VO 2777722 9722 2009 A AL TON ETT ETN HTN 2 MM LANA 77 2 Y 7 LLLA LLLA Z ZZ si HLL WV 4 gt N Bifurcation Diagram For growing epsilon the system runs into a strong 1 3 resonance around epsilon 7 05 For detecting resonances one can compute the ratio of the two periods which gives the rotation number or its inverse and do a rational analysis This simply means that we look if this ratio crosses or approaches rational numbers which belong to strong resonances for example 1 1 1 2 wy 1 4 A plot of the inverse rotation number for the stroboscopic map with period Pi It is clearly to see that the system runs into an 1 2 resonance as epsilon tends to zero and into an 1 3 resonance as epsilon tends to values greater than 7 Of
40. ed columns from bifurcation diagram data for instance for plotting Returns the values of the specified columns in a list of lists Parameters data a list of lists containing the bifurcation diagram data X y the two columns to be extracted data read_po_data problem_name run orbit Read the data describing an periodic orbit into a list Parameters problem_name an unique name of your problem as a string e g vdp run the run the orbit was computed in orbit the number of the orbit within the run see the output of pocont E data select po coords data x y z Select the specified coordinates from a periodic orbit for instance for plotting Parameters data the data containing a complete orbit of a periodic orbit dimension gt 3 x y z the columns to be extracted data read torus data problem name run orbit num Read the data describing a 2 torus into a list Parameters problem name an unique name of your problem as a string e g vdp run the run the torus was computed in orbit the number of the torus within the run see the output of torcont copy periodic solution problem name run num new run Copies a the periodic solution num of run run to the start solution of run new run Because pocont does not yet output orbits at specified parameter values this function may not be of much use If you want to restart from a solution at a different parameter value th
41. error estimated error 1 8118e 01 L checking for memory leaks no leaks Create the parameter file expected by torcont with settings problem name vdp run 2 continuation parameter beta b set the value of b beta to the desired value 0 1 mesh 40x40 LFil and Reserve see also ex kawa 500 500 use a Krylov subspace of at most Restart dimensions Restart 50 max number of iterations of gmres IEM X 500 dropping tolerance for ilu 0 005 max value of rel residual of gmres TOL 1 0e 9 continuation interval 0 10 max number of continuation steps 80 max angle between the tangencies of two successive continuation steps degree 10 initial continuation step size 1 max continuation step size 5 min continuation step size 0 1 don t print debugging information continuer outcommented print every npr steps npr 5 and run torcont create tc run vdp 2 continuer param d ode b 0 1 discretisation_pointsl 40 discretisation points2 40 linear_solver LFil 500 linear_solver Reserve 500 linear_solver Restart 50 linear_solver ItMX 500 linear_solver DropTOL 0 005 linear_solver TOL 1 0e 9 continuer param interval 0 10 continuer ItMX 80 continuer Alpha 10 continuer h0 1 continuer h_max 5 continuer h_min 0 1 continuer LogFile clog npr 5
42. ging information The continuer class debugging information for two function calls continuer initialize and continuer step The output has the following meaning continuer initialize h the initial continuation step size the sign decides the direction of the continuation h max the max continuation step size h min the min continuation step size h fac min the min factor for step size adaption f fac max the max factor for step size adaption MaxDiff the max value of llv xll 1 lIxll where v is the predicted and x the corrected solution alpha the maximum angle between to tangencies of two successice solutions gamma a security factor The step size is adapted by h gamma h_fac h with h fac min lt h fac lt h fac max h fac is computed so that two solutions have a rel abs distance of at most MaxDiff and the angle between two successive tangencies is smaller than alpha We do not take the iteration number of the corrector into account Thereby it is possible to use black box correctors for the continuation for instance from the package minpack which do not provide iteration numbers or where they do not make too much sense A clever choice of these parameters can speed up your continuation dramatically For hard problems like this example you should adjust these parameters For checking the success of your adaption use the output of continuer step norm x norm of the solution vecto
43. h argument L gt isol t th gt sin th 0 8 cos th Write the initial solution to disk on a 30x100 mesh The first period is Pi the second is an initial guess The name returned is used in the first call to torfind run 1 Here we raise Digits to full double precision to obtain a most exact mesh in th1 which is fixed through all computations This is usually not necessary but shown in this example This is especially not neccessary for autonomous systems because the periods are determined by the extended system with the required accuracy Parameters problem name pnet run 1 name of the function calculating initial solution values isol T1 Pi T2 6 96 mesh 30x100 gt Digits 19 write_tss pnet 1 isol Pi 6 96 30 100 Digits 10 pnet_qpol dat Run 1 Continuation of Quasiperiodic Orbits eps 1 5 6 5 This run demonstrates how to run torfind and torcont create 3D plots and animations Create the parameter file expected by torfind with settings problem name pnet run 1 initial solution file pnet_qpol dat mesh 30x100 LFil and Reserve see also ex kawa 250 250 use a Krylov subspace of at most Restart dimensions Restart 35 max number of iterations of gmres ItMX 350 dropping tolerance for ilu 0 01 max value of rel residual of gmres TOL 1 0e 8 don t print debugging information of iterative solver and ilu
44. he Hopf points which depend on delta gt display anl insequence true display an2 insequence true KW RAV vs SS muss SOY d Z Did A R Ord A eni LZ Z IY 7 TAY Wiha SF 77 AA gt N Run 2 Continuation of Quasiperiodic Orbits b 0 1 d 0 10 This run demonstrates the same as run 1 without debugging information and in addition how to obtain a start solution from a previous run at a specified parameter value Only this additional point is commented here at length Create a start solution for run 2 from a solution of run 1 Parameters problem name vdp old run 1 number of solution 1 new run 2 Because torcont does not print solutions at user specified parameter values yet this function may not be of much use at this time See the next execution group on how to obtain the required start solution gt 4 copy torus solution vdp 1 1 2 test d data vdp 2 mkdir data vdp 2 OK L cp data vdp 1 qpol dat data vdp 2 qpo0 dat OK This alternative way of obtaining start solutions should become obsolete in the future We simply misuse torfind to create the solution we require We create a parameter file with the settings problem name vdp run 2 initial solution format data lt name gt lt run gt qpo lt num gt dat data vdp 1 qpol dat mesh 40x40 LFil and
45. he solution x the unknown basic frequencies and and an estimation of the error The latter is calculated as the L norm of the difference of solutions of methods of order 2 and order 4 at the same mesh The solution of order 4 is used therefore the error is largely overestimated Executing this worksheet creates the file contpack m which is required if you want to use its functions This worksheet is best executed with the menu item Edit gt Execute gt Worksheet Please note the next paragraph before executing restart with process Change the path to the location of the maple file contpack mws at your system gt currentdir export fschild maple L gt Documentation All functions of this package are described briefely in the following sections Please refere to the Maple example files also This documentation gives hints for using the command line programs without Maple The expected format of the input and the parameter files are described in the sections of the preprocessing functions Please read the script file demo contained in each example s subdirectory on how to call the programs Postprocessing can be done with e g gnuplot or Maple Continuation of invariant tori is usually done in two steps At first one has to compute a suitable initial solution because one has usually crude approximations only Then one can do a parameter continuation beginning at this start solution The
46. i 0 1 000e 03 2 9027e 00 vdp 1 qpo0 dat continuer step norm x 1 1659e 02 norm v x 7 6098e 01 beta 8 7267e 00 h_facl 3 8630e 01 h 1 9000e 01 continuer step norm x 1 1664e 02 norm v x 3 0422e 00 beta 1 8852e 01 h_facl 9 6676e 00 h 1 9138e 01 continuer step norm x 1 1488e 02 norm v x 4 5831e 00 beta 3 0713e 01 h_facl 6 3209e 00 h 1 1922e 01 3 1 183e 01 vdp 1 qpol dat 2 8717e 00 h 2 1007e 00 6 1 716e 01 vdp 1 qpo2 dat 2 7202e 00 h 5 5237e 00 9 1 784e 01 vdp 1 qpo3 dat 2 7097e 00 continuer step norm x 1 0861e 02 norm v x 5 0460e 01 beta 9 6509e 00 h_fac1 5 4304e 01 h 1 0495e 01 continuer step norm x 1 0935e 02 norm v x 1 3745e 00 beta 1 4659e 01 h_facl 2 0072e 01 h 1 357le 01 continuer step norm x 1 1050e 02 norm v x 2 2317e 00 beta 1 8470e 01 h_facl 1 2490e 01 h 1 3949e 01 12 1 878e 01 vdp 1 qpo4 dat 2 7621e 00 1 1089e 02 5 1927e 01 continuer step norm x 1 1140e 02 norm v x 5 0576e 00 beta 5 3340e 01 h_fac1 5 5559e 00 h 5 6629e 00 continuer step norm x 1 0951e 02 norm v x 2 5570e 00 beta 3 4501e 01 h_fac1 1 0805e 01 h 3 1502e 00 continuer step norm x 1 0882e 02 norm v x 8 0444e 01 beta 2 8645e 01 h_facl 3 4129e 01 2 2653e 00 continuer step norm x 1 0855e 02 norm v x 3 8154e 01 be
47. ified coordinates from a 2 torus for instance for plotting Parameters data the data containing a complete torus of a periodic orbit phase space dimension gt 3 x y z the columns to be extracted data cut1 data begin end Cuts out a section of a 2 torus with respect to thl from the mesh point begin to the mesh point end Parameters data the data of a 2 torus begin end the start and end segment data cut2 data begin end Cuts out a section of a 2 torus with respect to th2 from the mesh point begin to the mesh point end Parameters data the data of a 2 torus begin end the start and end segment data section1 data idx Extracts a cross section of a 2 torus for a fixed value of th1 at the mesh point idx This is the incariance curve of the stroboscopic map of the quasiperiodic solution with period T 1 22 Pi omega 1 locally defined near the torus Parameters data the data of a 2 torus idx the index of the mesh point data section2 data idx Extracts a cross section of a 2 torus for a fixed value of th2 at the mesh point idx This is the incariance curve of the stroboscopic map of the quasiperiodic solution with period T 2 22 Pi omega 2 locally defined near the torus Parameters data the data of a 2 torus idx the index of the mesh point save write poss write tss xtd system create ode create fpf run create fpc run create pof run create
48. its documentation is distributed under the terms of the GNU General Public License as published by the Free Software Foundation either version 2 of the License or at your opinion any later version You should have received a copy of the GNU General Public License version 2 together with this documentation file tt LICENSE 2 Download This package will move to SourceForge http sourceforge net projects nlstools In the meantime you may download it from http www mathematik tu ilmenau de fschild nlsanalyzer 3 Introduction This continuation package consists of finder and continuer pairs of programs It contains algorithms for computation finder and continuation continuer of fixed points fpfind fpcont periodic solutions of autonomous and periodically forced systems pofind pocont and quasiperiodic invariant 2 tori of autonomous and periodically forced systems torfind torcont torfind4 torcont4 The continuer use pseudo arclength continuation pofind pocont torfind and torcont use a finite difference method of order 4 For torfind and torcont error estimation is done by computing the difference of solutions obtained by methods of order 2 and 4 Therefore th rror is largely overestimated because it is in fact an estimation for the solution of order 2 torfind4 and torcont4 use a finite element method of order 1 There is no error estimation implemented yet This p
49. l and Reserve see also ex kawa 250 300 use a Krylov subspace of at most Restart dimensions Restart 50 max number of iterations of gmres ItMX 350 dropping tolerance for ilu 0 01 max value of rel residual of gmres TOL 1 0e 7 don t print debugging information of iterative solver and ilu LogFile NULL continuation interval 0 0 22 max angle between the tangencies of two successive continuation steps degree 20 initial continuation step size 10 max continuation step size 25 min continuation step size 1 max number of continuation steps 25 print debugging information continuer LogFile clog print every npr steps npr 3 and run torcont gt create tco run vdp 1 continuer param b discretisation_pointsl 40 discret isation points2 40 inear solver LFil 250 linear solver Reserve 300 linear solver Restart 50 linear solver ItMX 350 linear solver DropTOL 0 01 linear solver TOL 1 0e 7 linear solver LogFile NULL clog continuer param interval 0 0 22 solve 64 94 126 158 NOBFO continuer Alpha continuer h0 10 continuer h_max 25 continuer h_min 1 continuer ItMX 25 continuer LogFile clog npr 3 torcont vdp 1 continuer initialize h 1 0000e 01 h_max 2 5000e 01 h fac min 5 0000e 01 e 01 gamma 9 5000e 01 STEP PAR file Ix
50. lay n n n n n n n n n n n n n n n n n n n OO 1 OY O1 4S CO NO ES CO an insequence true 7 AZZ A LJ Two coupled Van der Pol Oscillators This famous system must appear as an example of a package for computation and continuation of invariant tori Here it comes o 9 T e x 1 3 x B0 x i 9 59 reo Do Jembe n 5 The system is of interest for different fixed values of epsilon and for beta delta 0 Here we do only two simple continuations of tori with respect to either beta run 1 or delta run 2 Attention Before starting the computations please set the paths in the read and the currentdir commands correctly Ignore the error message issued by mkdir restart with plots with process read export fschild maple contpack m currentdir export fschild examples vdp mkdir data currentdir Warning the name changecoords has been redefined Error in mkdir directory exists and is not empty export fschild examples vdp PAN RV Lu OE N V V V V N Definition of the System and Creation of the Shared Object The constants and parameters are defined as a list of name value pairs The system is defined as a function taking a list and a number as arguments and returning a list of expressions ini gt Constants Params s e 0 3 d 0 2 b 0 001 VDP x t gt x 2 x 1 b
51. les and maple the package contpack and its documentation You should strip the executables because the symbol information occupies very much of their size If gmake fails and you have the software versions listed above or newer please send me a log file which you obtain by the csh commands gmake cleaner gmake i amp tee make log gzip make log Please send me the file make log gz by mail subject make torcont Include into this mail the version info which you obtain by the commands gcc v g v g77 v flex V bison V gmake v I will try to eliminate the errors and probably ask for further help or information Note It is known that colpilation with newer versions of gcc fails A new version is under development Do not send bug reports regarding this issue They will not be dealt with Documentation of the Maple Package contpack mws General Remarks We consider two kinds of equations periodically forced case d 3 x f x t where fis 2 r periodic with respect to t and autonomous case d FASI y 79 with x element of R and t element of R We seek periodic solutions x t u Q t or quasiperiodic solutions x t 2 u Q t f In the periodically forced case the frequency 1 is equal to the forcing frequency In addition in the automomous case it is possible to compute and continue fixed points The software computes and continues the torusfunction u instead of t
52. lver ItMX 350 linear solver DropTOL 0 0025 linear solver TOL 1 0e 12 linear solver LogFile clog torfind lang 1 Iterat D mpfung Normen Rechenzeit I SI gamma IuIxiI I gamma d F x DF x solve 0 0 0 0000e 00 4 8645e 01 4 6590e 00 0 0000e 00 0 0 0 PGMRES ilutp time 7 6 nnz A 72004 nnz LU 1292166 max nnz 3081301 PGMRES pgmres 32 time 4 2 1 1 1 0000e 00 4 7493e 01 1 0849e 01 1 9569e 00 0 3 Sa 31 4 PGMRES ilutp time 7 2 nnz A 72004 nnz LU 1282896 max nnz 3081301 PGMRES pgmres 24 time 3 2 2 1 1 0000e 00 4 7473e 01 2 5990e 04 6 5099e 02 OFS 6 1 42 2 PGMRES ilutp time 7 2 nnz A 72004 nnz LU 1283517 max nnz 3081301 PGMRES pgmres 27 time 3 6 3 1 1 0000e 00 4 7473e 01 7 5179e 05 4 5610e 02 0 8 9 53 4 PGMRES ilutp time 7 2 nnz A 72004 nnz LU 1283536 max nnz 3081301 PGMRES pgmres 26 time 3 4 4 1 1 0000e 00 4 7473e 01 1 5044e 08 6 4515e 04 1 0 12 2 64 5 PGMRES ilutp time 7 2 nnz A 72004 nnz LU 1283510 max nnz 3081301 PGMRES pgmres 26 time 3 5 5 1 1 0000e 00 4 7473e 01 2 3314e 12 5 1449e 08 T 3 15 2 75 37 period T1 1 7951594779940944768e 00 period T2 4 3115354542660266901e 00 solution written to file data lang 1 qpo0 dat estimating error PGMRES ilutp time 2 05 nnz A 72004 nnz LU 682361 max nnz 3081301 PGMRES pgm
53. m number of continuation steps in both directions continuer MaxDiff 0 25 set the maximum rel abs difference between predicted and corrected solution for step size control continuer Alpha 7 0 set the maximum angle between the tangent vectors of two solution points continuer h0 0 1 set the initial continuation stepsize continuer h max 0 5 set the maximal continuation stepsize continuer h min 0 01 set the minimal continuation stepsize continuer LogFile for printing debugging information of the continuer set LogFile to clog this helps very much to tune the continuer npr 5 print solution every npr steps N Procedures to call External Programs El fpfind problem_name run Runs the program fpfind and prints its output into the worksheet Parameters problem_name an unique name of your problem as a string e g vdp run number of the run e g 1 2 E fpcont problem name run Runs the program fpcont and prints its output into the worksheet The run is timed Parameters problem_name an unique name of your problem as a string e g vdp run number of the run e g 1 2 pofind problem_name run Runs the program pofind and prints its output into the worksheet Parameters problem name an unique name of your problem as a string e g vdp run number of the run e g 1 2 pocont problem name run Runs the program pocont and prints its output i
54. mputed torus See the example kawa for the format of the data structure gt data read_torus_data lang 1 0 data select_torus_coords data 3 4 5 surfdata data ANN ANN X Create the parameter file expected by torcont with settings problem name lang run 1 continuation parameter rho mesh 40x40 LFil and Reserve see also ex kawa 200 300 use a Krylov subspace of at most Restart dimensions Restart 50 max number of iterations of gmres ItMX 350 dropping tolerance for ilu 0 001 max value of rel residual of gmres TOL 1 0e 12 don t print debugging information of iterative solver and ilu LogFile clog continuation interval 0 15 0 7 max number of continuation steps 200 print every npr steps npr 5 and run torcont create tc run lang 1 continuer param rho discretisation pointsl1 40 discretisation points2 40 linear solver LFil 200 linear solver Reserve 300 linear solver Restart 50 linear solver ItMX 350 linear solver DropTOL 0 001 linear solver TOL 1 0e 12 linear solver LogFile NULL clog continuer param interval 0 15 0 7 continuer ItMX 200 npr 5 torcont lang 1 STEP PAR IuIxI I TOL Period T1 P file 0 5 500e 01 1 1830e 00 3 3538e 03 1 7952e 00 4 lang 1 qpo0 dat 5 5 627e 01 1 1764e
55. n taking a list and a number as arguments and returning a list of expressions gt Constants s k2 0 05 B 0 22 BO 0 03 Params s PRET 09 Kawa x t gt x 2 k1 x 2 1 8 x 1 2 3 x 3 2 x 1 B cos t 1 8 k2 3 x 1 2 x 3 2 x 3 B0 L li Create a shared object by calling codegen and then compiling and linking The compiler options used are for Solaris You may need different options gt create ode kawa Kawa Constants Params po compiler linker codege gcc fl L gcc fPIC gece P gcc fP1 n lt kawa ode gt kawa c OK PIC c o kawa o kawa c OK shared o kawa so kawa o E gt 3 Definition of the Start Solutions Define initial approximations to periodic psol q quasiperiodic dtsol solutions This functions i e IC shared OK uasiperiodic tsol and the doubled the coefficients were obtained by Fourier analysis of orbits which were the result of numerical simulation Unfortunately there are no branch switching algorithms available yet so we need this initial t 8 05828E 01 l cos i t 5 20006E 01 l cos 1 guesses gt psol t gt I 1 70583E 03 5 17355E 01 sin i 1 47861E 04 7 99765E 01 sin i 1 32976E 00 1 91934E 04 sin i t 3 91183E 05 cos t Ja tsol t th gt 1 70583E 03 8 05828E 01 cos 1 1 08938E 04 sin os t sin th 13 364
56. name kawa run 2 initial solution file kawa qpo2 dat mesh 20x40 min number of nonzero entries per row of L and U 50 max number of additional nonzero elements is rows reserve reserve 100 dropping tolerance for the ilu factorisation 0 02 don t print debugging information LogFile NULL and then compute the initial torus Note You should always provide enough space for nonzeros generated by ilu by setting LFil and Reserve to resonable high values because the linear systems are very hard to solve The iteration methods are very sensitive to dropping too much This is valid for all of the computations gt create tf run kawa 2 isol kawa qpo2 dat discretisation_pointsl 20 discret isation_points2 40 inear_solver LFil 50 linear_solver Reserve 100 linear_solver DropTOL 0 02 linear_solver LogFile NULL clog torfind kawa 2 Iterat D mpfung Norme I SI gamma I Ix I1 1 gamma dl 0 0 0 0000e 00 4 8714e 01 1 6097e 00 0 0000e 00 1 1 1 0000e 00 4 8993e 01 1 5010e 01 4 4725e 00 2 1 1 0000e 00 4 8789e 01 5 5154e 03 8 3250e 01 3 1 1 0000e 00 4 8781e 01 2 8667e 04 2 6383e 02 4 1 1 0000e 00 4 8781e 01 4 8317e 06 3 8652e 03 5 1 1 0000e 00 4 8781e 01 5 0101e 07 7 9926e 05 period T2 6 1497738681980379738e 01 solution written to file data kawa 2 qpo0 dat checking for memory leaks no leaks Create the parameter file expected b
57. next execution group for an other way to obtain the required start solution gt copy_torus_solution lang 1 18 2 test d data lang 2 mkdir data lang 2 OK L cp data lang 1 qpo18 dat data lang 2 qpo0 dat OK This alternative way of obtaining start solutions should become obsolete in the future We simply misuse torfind to create the solution we require We create a parameter file with the settings problem name lang new run 2 start solution format data lt name gt lt run gt qpo lt num gt dat data lang 1 qpo18 dat set the value of rho to the required value 0 25 mesh 40x40 LFil Reserve 200 250 restart gmres after Restart iterations 50 max number of iterations gmres 350 dropping tolerance 0 0025 max rel residual gmres 1 0e 12 print debugging information LogFile clog and run torfind to compute the solution torfind automatically writes the solution to the start solution of run 2 gt create tf run lang 2 isol data lang 1 qpol8 dat ode rho 0 25 discretisation_pointsl 40 discretisation points2 40 linear solver LFil 200 linear solver Reserve 250 linear solver Restart 50 linear solver ItMX 350 linear solver DropTOL 0 0025 linear solver TOL 1 0e 12 linear solver LogFile clog torfind lang 2 Iterat D mpf
58. nique name of your problem as a string e g vdp run number of the run e g 1 2 solution a function expecting two real numbers th1 and th2 as arguments and returning a vector or list containing the value of u th1 th2 the two periods of this function must be normalised to 2 Pi period1 period2 the initial guesses of the periods of the starting solution for periodically forced systems period1 must be the exact value N1 N2 the number of discretisation points on each axis This function returns a string containing the unique name of the created file start solution f ile name This name must be used in the call to create tf run create ode problem name ode constants params options Creates a shared library containing the right hand side of the ode Parameters problem name an unique name of your problem as a string e g vdp ode the rhs of the ode the function always expects the arguments x and t and returns a list of expressions resp values constants a list of name value pairs for each constant of the ode params a list of name value pairs for each free parameter of the ode an initial value must be provided options a sequence of name value pairs Possible Options and Default Values codegen codegen name des C file generators use always codegen compiler gcc name and options of the C compiler to be used linker gcc shared name and op
59. nto the worksheet The run is timed Parameters problem name an unique name of your problem as a string e g vdp run number of the run e g 1 2 torfind problem name run Runs the program torfind and prints its output into the worksheet Parameters problem name an unique name of your problem as a string e g vdp run number of the run e g 1 2 torcont problem name run Runs the program torcont and prints its output into the worksheet The run is timed Parameters problem name an unique name of your problem as a string e g vdp run number of the run e g 1 2 info odeinfo problem name Runs the program odeinfo and returns two values in info info 1 true or false wether or not the ode system is autonomous info 2 dimension of the ode system Parameters problem name an unique name of your problem as a string e g vdp E print odeinfo problem name Runs the program odeinfo and prints its output into the worksheet Parameters problem name an unique name of your problem as a string e g vdp N Postprocessing Procedures data read bd problem name run run2 runN Reads the bifurcation diagrams of several runs into a list of lists Parameters problem name an unique name of your problem as a string e g vdp run run2 runN a list of the runs of interest ES data select bd cols data x y Selects specifi
60. od T1 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 Period Tl 7952e 00 7952e 00 data lang 2 bd dat Period T2 7941e 00 7941e 00 7942e 00 7946e 00 7956e 00 7971e 00 7990e 00 8012e 00 8035e 00 6 6 6 6 6 6 6 6 6 6 8058e 00 6 8080e 00 6 8100e 00 6 8115e 00 6 8123e 00 6 8122e 00 6 8107e 00 6 8071e 00 6 8012e 00 6 7951e 00 Period T2 6 7941e 00 6 7941e 00 data lang 2 torcont log no leaks 10 3 4 5 data data data data data data data data data data data data data data data data data data data data data data data an NULL for i from 1 by 1 to nops nums do prin da da gt nums 0 18 tf reading graph sd n nums il ta read torus data lang 2 nums i ta select torus coords data 3 4 5 an an od readi readi readi readi readi readi readi readi readi readi readi readi readi readi readi readi readi readi readi ng ng ng ng ng ng ng ng ng ng ng ng ng ng ng ng ng ng ng surfdata data nums 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 grap grap grap grap grap grap grap grap grap grap grap grap grap grap grap grap grap grap grap gt disp
61. ply install the package and enter ldd torcont If there are problems ldd will most probably issue an error message 4 3 Compiling the source distribution Compiling the package is expensive You will need 250MB RAM and approximately 250MB free disc space The source code distribution consists of the files nlsanalyzer 1l 1 tar gz and examples tar gz It was compiled using the following compilers and tools program version SOLARIS version LINUX gcc 2 95 3 2 953 g 2495 3 2 95 3 g77 Da 95 3 2 95 3 flex 2 5 4 2 5 4 bison 1 35 1 28 gmake 3299 61 3 79 1 You will need all this programs You may try older or newer versions of flex bison and gmake You may try newer versions of gcc g g77 Move both files to a location of your choice Change to this directory Enter the commands gunzip examples tar gz gunzip nlsanalyzer 1l 1l tar gz tar xf examples tar tar xf nlsanalyzer l l tar This will create the directories examples and nlsanalyzer 1 1 The directory examples contains the fully documented examples of use Change to nlsanalyzer 1 1 Enter the command gmake or make if gmake is the default on your system This will take a while The package will compile with gmake only In addition the parser files are not lex and yacc compatible so you will need flex and bison as listed above If the package is compiled successfully then the subdirectory bin contains the executab
62. qpol4 dat 40 3 399e 01 1 2847e 00 6 2052e 03 lang 1 qpo15 dat 45 3 101e 01 1 2989e 00 6 6068e 03 lang 1 qpol6 dat 50 2 814e 01 1 3128e 00 7 1439e 03 lang 1 qpol7 dat 55 2 545e 01 1 3265e 00 7 8840e 03 lang 1 qpo18 dat 60 2 297e 01 1 3398e 00 8 9780e 03 lang 1 qpol9 dat 65 2 074e 01 1 3528e 00 1 4906e 02 lang 1 qpo20 dat 70 1 877e 01 1 3654e 00 6 9333e 02 lang 1 qpo21 dat 75 1 706e 01 1 3776e 00 3 6438e 01 lang 1 qpo22 dat 80 1 560e 01 1 3895e 00 1 7298e 00 lang 1 qpo23 dat 83 1 485e 01 1 3964e 00 3 5604e 00 lang 1 qpo24 dat bifurcation diagramm written to file output written to file checking for memory leaks no leaks real 1 12 31 9 user 1 12 08 9 L sys Tya data read_torus_data lang 1 SA ZS SSS SSS STI ee WIZE M data lang data lang 19 Zid 5 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 7952e 00 1 bd dat Oo O0 O O O1 CO CO n 3263e 00 5978e 00 9147e 00 2837e 00 7123e 00 2090e 00 7831e 00 4457e 00 2095e 00 0090e 01 0683e 01 1 torcont log data data data data data data data data data data data Here we create an animation of the evolution of the torus under changes of the parameter rho At first we read in all solutions given in a list nums The for loop creates a set of plot structures gt nums 7 0 8 2
63. r The binary distribution consists of the files nlsa 1 1 linux bin tar gz for LINUX on an Intel PC and nlsa 1 1 sun solaris bin tar gz for SUN SOLARIS on a SUN Workstation respectively and the fil xamples tar gz both operating systems Move both files to a location of your choice Change to this directory Enter the commands gunzip examples tar gz gunzip nlsa 1 1 bin tar gz tar xf examples tar tar xf nlsa 1 1 bin tar This will create the directories bin maple and examples These contain bin Scripts and binaries maple maple package contpack examples four examples of use kawa lang pnet vdp The binaries are linked against the following dynamic libraries You can use the binaries if these libs are available on your system ldd torcont produced the following output LINUX libdl so 2 lib libdl so 2 0x40024000 libstdc libc6 2 2 s0 3 gt usr lib libstdc libc6 2 2 s0 3 0x40028000 libm so 6 gt lib libm so 6 0x40075000 libc so 6 lib libc so 6 0x40097000 lib ld linux so 2 lib ld linux so 2 0x40000000 SOLARIS libdl so 1 gt usr lib libdl so 1 libstdc s0 2 10 0 gt usr local lib libstdct so 2 10 0 libm so 1 gt usr lib libm so 1 libc so 1 gt usr lib libc so 1 usr platform SUNW Ultra 80 lib libc psr so 1 You can use ldd to check wether the libs are available or not Sim
64. r this is the non normalized euclidian norm norm v x norm of the difference between predicted and corrected solution rel abs diff norm v x 1 norm x this value is used to estimate h_fac1 in comparison with MaxDiff beta the actual angle between two successive tangencies this value is used to estimate h_fac2 in comparison with alpha h facl the factor for changing the step size estimated so that rel abs err may become equal to MaxDiff h fac2 the factor for changing the step size estimated so that beta may become equal to alpha h the new step size which will be used in the next step Hints MaxDiff probably needs not to be changed 0 25 is mostly never met The angle is a quite good measure of the nonlinearity of your problem and usually dominates the step size control Change the value of Alpha with care If you enlarge Alpha always set h max to a value wich is not too large Usually the debigging information helps you very much to find out why the continuer chooses a small step size In this case you may need to adapt the default values gt Here we create an animation of the evolution of the torus under changes of the parameter b beta At first we read in all solutions given in a list nums The for loop creates two sequences of plot structures g d For plotting we choose 0 x x for anl and 0 y y for an2 gt nums 0 9 anl NULL an2 NULL for i from 1 by 1 to nops
65. r method breaks down which is shown by the peak of the error near epsilon 7 gt data2 select bd cols datal 1 3 plot data2 2 7 05 0 1 1 color blue plot data2 2 7 05 0 0 1 color blue 1 0 8 0 6 0 44 i 4 I J l 0 24 J 02 7 7 4 iy a 7 0 1 0 08 0 06 0 04 0 02 4 The Example of W F Langford W F Langford considered a dynamical system of Hydrodynamics under the following transformations 1 Reduction to the 3 dimensional center manifold and 2 Transformation into the Poincare Birkhoff Normal Form Omitting higher order terms he derived the system d ay C Toy d 3 7 9 tG Dy 9 A di 3 Stg Y y 1 pz Ezx which is considered here for the parameter values epsilon 0 0 1 rho 0 15 0 7 and omega 3 5 For epsilon 0 and rho gt 0 615446 the system posesses a periodic orbit which for rho 0 615446 undergoes a Hopf bifurcation and an attractive invariant 2 torus emerges for rho gt 0 615446 For smaller values of rho a Shilnikov type attractor is born in a global bifurcation involving the torus see run 1 Values of epsilon gt 0 lead to strong resonances see run 2 Attention Before starting the computations please set the paths in the read and the currentdir commands correctly Ignore the error message issued by mkdir restart with plots with process read export fschild maple contpack m
66. refore for all the considered kinds of solutions there is a finder and a continuer solution type finder continuer fixed point fpfind fpcont periodic sol pofint pocont quasiperiodic sol torfind torcont The finder saves the solution found to a location where it is expected by the corresponding continuer N Preprocessing Procedures B ssf_name write_poss problem_name run solution period N write periodic orbit start solution Creates a file containing the discretisation of the start solution in the format expected by pofind Parameters problem_name an unique name of your problem as a string e g vdp run number of the run e g 1 2 solution a function expecting a real number t as argument and returning a vector or list containing the value of x t the period of this function must be normalised to 2 Pi period the initial guess of the period of the starting solution for periodically forced systems this period must be the exact value N the number of mesh points This function returns a string containing the unique name of the created file start solution f ile name This name must be used in the call to create pof run E ssf name write tss problem name run solution periodl period2 N1 N2 write torus start solution Creates a file containing the discretisation of the start solution in the format expected by torfind Parameters problem name an u
67. res 17 time 1 25 estimated error 3 3538e 03 L checking for memory leaks no leaks Interpretation of the debugging information The iterative solver class issues a line of debugging information for each call to the preconditioner ilutp and the solver pgmres These lines provide the following information PGMRES ilutp time consumed processor time in seconds nnz A structural nonzero elements of matrix A nnz LU lt structural nonzero elements used by the factors L and U of A gt max nnz maximum number of structural nonzero elements available with the current settings gt PGMRES pgmres lt number of iterations gt time lt consumed processor time in seconds gt Hints Always provide enough space for fill in by setting LFil and Reserve to reasonable high values Play with DropTOL so that the sum of consumed processor time of ilutp and gmres becomes approximately a minimum A good choice seem to be values for which torfind both times are almost equal torcont the time of ilutp is almost equal to the sum of the times of pgmres of each newton iteration in torcont for each corrector step the incomplete LU factorisation is calculated only once i e the different linear systems of each step are solved with the same preconditioner Set the tolerance of the linear solver to the maximal possible value for which the newton process still converges nicely gt Create s plot of the co
68. rograms give much nicer results than torfind and torcont which is due to the fact that the FEM does not rely as much on smoothness If you want to use the FEM within Maple change the procedures torfind and torcont in contpack mws accordingly and execute contpack mws for updating the library contpack m All algorithms are still experimental and not adaptive yet 4 Installation This is a stable pre release of the continuation package TORCONT written by Frank Schilder at the TU Ilmenau Germany Contents of directory nlsanalyzer 1 1 INSTALL this file LICENSE copy of the FSF GPL V2 Readme very brief overview of abilities nlsanalyzer l l tar gz the complete source tree examples tar gz examples of use Maple worksheets nlsa 1 1 linux bin tar gz binary distribution for LINUX nlsa 1 1 sun solaris bin tar gz binary distribution for SOLARIS doc tar gz documentation only 4 1 Documentation You need gunzip and tar If you don t want to install the package or just want to read trough the documentation then you may install the documentation only Move the file doc tar gz to a location of your choice Change to this directory Enter the commands gunzip doc tar gz tar xf doc tar This will create the directory doc which contains the documentation of the Maple package contpack and of the examples in postscript format 4 2 Installing the binary distribution You need gunzip and ta
69. rror message issued by mkdir restart with plots with process read export fschild maple contpack m currentdir export fschild examples pnet mkdir data currentdir Warning the name changecoords has been redefined Error in mkdir directory exists and is not empty export fschild examples pnet V V V V 3 Definition of the System and Creation of the Shared Object The constants and parameters are defined as a list of name value pairs The system is defined as a function taking a list and a number as arguments and returning a list of expressions Constants alpha epsilon B beta epsilon 2 B Params epsilon 3 0 B 0 1 PNet x t gt x 2 alpha x 2 3 beta x 2 1 B sin 2 t x 1 l 3 PNet x t gt x ax Bx 1 Bsin 2 t x Create a shared object by calling codegen and then compiling and linking The compiler options used are for Solaris You may need different options gt gt create_ode pnet PNet Constants Params compiler gcc fPIC linker gcc fPIC shared codegen pnet ode pnet c OK gcc fPIC c o pnet o pnet c OK L gcc fPIC shared o pnet so pnet o OK 3 Definition of the Start Solutions gt Define initial approximations to the quasiperiodic solution This function is a guess This torus function s must always be 2 Pi periodic in eac
70. s 4 0 th 1 4964e 01 sin t c os 4 0 th 9 3327e 02 cos t cos 4 0 th 1 7377e 03 sin 5 0 t h 4 2339e 02 sin t sin 5 0 th 1 1509e 02 cos t sin 5 0 th 9 3014e 04 cos 5 0 th 1 6307e 02 sin t cos 5 0 th 4 4285 e 02 cos t cos 5 0 th 1 7772e 04 8 1657e 01 sin t 4 7682e 01 cos t 7 2020e 04 si n 1 0 th 1 1031e 02 sin t sin 1 0 thn 9 9793e 02 cos t sin 1 0 th 7 2372e 04 cos 1 0 th 3 3203e 03 sin t cos 1 0 th 4 2824e 02 cos t cos 1 0 th 9 0885e 04 sin 2 0 th 4 5432e 01 sin t sin 2 0 th 3 0015e 01 cos t sin 2 0 th 1 2109e 03 cos 2 0 th 7 2740e 02 sin t cos 2 0 th 5 3331le 01 cos t cos 2 0 th 8 1945e 04 sin 3 0 th 6 1829e 02 sin t sin 3 O th 5 4459e 02 cos t sin 3 0 th 1 8827e 03 cos 3 0 th 6 6952e 02 sin t cos 3 0 th 5 5852e 02 cos t cos 3 0 th 2 2 792e 04 sin 4 0 th 1 3274e 01 sin t sin 4 0 th 6 9366e 02 cos t sin 4 0 th 2 1520e 03 cos 4 0 th 7 0248e 02 sin t c os 4 0 th 1 1002e 01 cos t cos 4 0 th 2 5432e 04 sin 5 0 t h 8 3238e 03 sin t sin 5 0 th 2 9773e 02 cos t sin 5 0 th 2 1189e 03 cos 5 0 th 3 2202e 02 sin t cos 5 0 th 9 3574 e 03 cos t cos 5 0 th 1 3092e 00 1 091le 04 sin t 5 1342e 05 cos t 1 7796e 02 si n 1 0 th 1 0779e 04 sin t sin 1 0 th 1 5922e 05 cos t sin 1 0 th 4 194le 02 cos 1 0 th 1 6124e 04 sin t cos 1 0 th 9 3525e 05 cos t cos 1 0 th 5 9965e 02 sin 2 0 th 1 2945e 04 sin t sin 2 0 th 3 246le 05 cos t sin 2 0 th 9 9713e 02 cos 2 0 th 3
71. sin th x LL d Write the initial solution to disk on a 40x40 mesh The first period is exactly known from analysis of the system the second is an initial guess The name returned is used in the first call to torfind run 1 Parameters problem name lang run 1 name of the function calculating initial solution values isol T1 2 Pi 3 5 T2 431 mesh 40x40 write tss lang 1 isol 2 Pi 3 5 4 31 40 40 lang qpol dat Run 1 Continuation of Quasiperiodic Orbits rho 0 15 0 7 This run demonstrates how to run torfind and torcont obtain and interpret debugging information of the linear solver create 3D plots and animations Create the parameter file expected by torfind with settings problem name lang run 1 initial solution file lang_qpol dat mesh 40x40 LFil and Reserve see also ex kawa 200 250 use a Krylov subspace of at most Restart dimensions Restart 50 max number of iterations of gmres IEMX 350 dropping tolerance for ilu 0 0025 max value of rel residual of gmres TOL 1 0e 12 print debugging information of iterative solver and ilu LogFile clog and run torfind B en create tf run lang 1 isol lang qpol dat discretisation_pointsl 40 discretisation points2 40 linear solver LFil 200 linear solver Reserve 250 linear solver Restart 50 linear so
72. ta 1 8878e 01 h_fac1 7 1783e 01 h 2 1131e 00 continuer step norm x 1 0842e 02 norm v x 2 9877e 01 beta 1 3773e 01 h_facl 9 1562e 01 h 2 9072e 00 continuer step norm x 1 0840e 02 norm v x 2 5602e 01 beta 9 0787e 00 h_fac1 1 0683e 02 3 4224e 00 1 2184e 01 h_min 1 0000e 00 h_fac_max 2 0000e 00 MaxDiff 2 5000e 01 alpha 2 0000 TOL Period T1 Period T2 data 6 3151e 00 5 7600e 00 data rel_abs_diff 6 4717e 03 h_fac2 2 2824e 00 rel abs diff 2 5860e 02 h_fac2 1 0603e 00 rel abs diff 3 9551e 02 h fac226 5572e 01 6 0004e 00 5 4889e 00 data rel_abs_diff 4 4997e 02 h_fac2 3 8688e 01 rel_abs_diff 2 3138e 02 h_fac2 5 8556e 01 rel abs diff 7 3251e 03 h fac227 0195e 01 5 9185e 00 5 3240e 00 data rel_abs_diff 3 4827e 03 h_fac2 1 0589e 00 rel_abs_diff 2 7304e 03 h_fac2 1 4482e 00 rel_abs_diff 2 3402e 03 h_fac2 2 1941e 00 5 9041e 00 5 2836e 00 data rel abs diff 4 6037e 03 h_fac2 2 0643e 00 rel_abs_diff 1 2455e 02 h_fac2 1 3611e 00 rel_abs_diff 2 0016e 02 h_fac2 1 0820e 00 5 8412e 00 5 2039e 00 data continuer step norm x 1 1159e 02 norm v x 2 3874e 00 rel abs diff 2 1205e 02 beta 1 9306e 01 h_facl 1 1790e 01 h_fac2 1 0356e 00 h 1 3723e 01 continuer step norm x 1 1243e 02 norm v x 2 3620e 00 rel abs diff 2 0823e 02 beta 1 9235e 01 h_facl 1 2006e 01 h_fac2 1 0394e 00 h 1 3551le 01 continuer step norm x 1
73. tions of the linker to produce a shared object create fpf run problem name run isol x0 options Creates the parameter file expected by fpfind Parameters problem name an unique name of your problem as a string e g vdp run number of the run e g 1 2 isol the initial solution given as a list of values if isol is not given x0 is assumed to be the zero vector options a sequence of name value pairs Possible Options and Default Values ode lt parname gt lt value gt sets the value of the free parameter lt parname gt to lt value gt overwrites the default value nonlinear_solver ItMX 10 max number of Newton steps nonlinear_solver SubItMX 8 max number of damping steps per Newton step nonlinear_solver TOL 1 0e 4 stopping criterion for the Newton iteration create fpc run problem name run options Creates the parameter file expected by fpcont Parameters problem name an unique name of your problem as a string e g vdp run number of the run e g 1 2 options a sequence of name value pairs Possible Options and Default Values ode lt parname gt lt value gt sets the value of the free parameter lt parname gt to lt value gt overwrites the default value nonlinear_solver ItMX 10 max number of Newton steps nonlinear_solver SubItMX 8 max number of damping steps per Newton step nonlinear_solver TOL 1 0e 4 stopping criterion for the Ne
74. ung Normen Rechenzeit TSI gamma Ixi I gamma d F x DF x solve 0 0 0 0000e 00 5 3183e 01 1 5007e 01 0 0000e 00 0 0 0 PGMRES ilutp time 6 7 nnz A 72004 nnz LU 1024023 max nnz 3081301 PGMRES pgmres 20 time 2 2 1 1 1 0000e 00 5 3278e 01 238 1023e 03 4 3433e 01 03 od 28 5 PGMRES ilutp time 6 4 nnz A 72004 nnz LU 998435 max nnz 3081301 PGMRES pgmres 18 time 1 9 2 1 1 0000e 00 5 3277e 01 1 1367e 06 5 1322e 03 0 5 6 1 Shel PGMRES ilutp time 6 4 nnz A 72004 nnz LU 1003641 max nnz 3081301 PGMRES pgmres 19 time 2 0 3 1 1 0000e 00 5 3277e 01 1 1644e 12 4 5306e 07 0 8 9 2 46 1 period T1 1 7951594779940942548e 00 period T2 6 7940976510276360756e 00 solution written to file data lang 2 qpo0 dat estimating error PGMRES ilutp time 3 27 nnz A 72004 nnz LU 737751 max nnz 3081301 PGMRES pgmres 18 time 1 43 estimated error 8 0399e 03 checking for memory leaks no leaks create tc run lang 2 continuer param epsilon ode rho 0 25 discretisation_pointsl 40 discretisation points2 40 linear_solver LFil 200 linear_solver Reserve 250 linear_solver Restart 50 linear_solver ItMX 350 linear_solver DropTOL 0 0025 linear solver TOL 1 0e 12 continuer param interval 0 0 1 continuer ItMX npr 1 torcont lang STEP file PAR 0 0 000e 00 2 qpo0 dat 1 9
75. wton iteration continuer param lt parname gt set the primary continuation parameter continuer param_interval 1 1 set the parameter interval continuer ItMX 50 set the maximum number of continuation steps in both directions continuer MaxDiff 0 25 set the maximum rel abs difference between predicted and corrected solution for step size control continuer Alpha 7 0 set the maximum angle between the tangent vectors of two solution points continuer h0 0 1 set the initial continuation stepsize continuer h_max 0 5 set the maximal continuation stepsize continuer h_min 0 01 set the minimal continuation stepsize continuer LogFile for printing debugging information of the continuer set LogFile to clog this helps very much to tune the continuer npr 5 print solution every npr steps create_pof_run problem_name run isol file_name options Creates the parameter file expected by pofind Parameters problem_name an unique name of your problem as a string e g vdp run number of the run e g 1 2 isol the name of the file containing the initial solution see write poss options a sequence of name value pairs Possible Options and Default Values ode lt parname gt lt value gt sets the value of the free parameter lt parname gt to lt value gt overwrites the default value discretisation_points 20 set the number of mesh points linear_solver LFil 10 set the
76. y torcont with settings problem name kawa run 2 continuation parameter k1 mesh 20x40 dropping tolerance 0 01 max norm of relative residual for gmres TOL 1 0e 8 don t print dedugging information LogFile NULL continuation interval 0 04 0 12 max continuation step size 1 0 and then continue the invariant torus gt create tc run kawa 2 continuer param kl discretisation pointsl 20 discret isation_points2 40 inear_solver DropTOL 0 01 linear_solver TOL 1 0e 8 linear_solver LogFile NULL clog continuer param_interval 0 04 0 12 continuer h_max 1 0 torcont kawa 2 STEP PAR LIxi I TOL Period T2 0 9 000e 02 1 7247e400 1 0136e 01 6 1498e 01 at 5 9 902e 02 1 7221e 00 1 1373e 01 6 3415e 01 at 10 1 128e 01 1 7183e 00 1 4946e 01 6 6012e 01 at 15 1 197e 01 1 7167e 00 1 9703e 01 6 7101e 01 at 16 1 204e 01 1 7166e 00 1 9810e 01 6 7206e 01 at STEP PAR LIxi I TOL Period T2 0 9 000e 02 1 7247e400 1 0136e 01 6 1498e 01 at 5 7 953e 02 1 7277e400 9 1782e 02 5 9108e 01 at 10 5 752e 02 1 7339e 00 8 1960e 02 5 3408e 01 at 15 3 828e 02 1 7396e 00 7 9678e 02 4 7345e 01 Rechenzeit F x DF x solve olelelete O OI 4s PN F O CO CO SO IE CO NO i gt OY CO O o 1 OY O1 QO BOO NO H OO data file data kawa 2 qpo0 data kawa 2 qpol data kawa 2 qpo2 data kawa 2 qpo3 data kawa 2 qpo4 data file d

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