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USER GUIDE TO SLIDECONT 2.0

1. generates the starting data to continue the flip bifurcation curve Restarting fromthe end point at label 1 4 we continue the flip curve forward computation 5 PT TY LAB PAR 4 ae MAX U 2 PAR 5 PERIOD 30 15 1 52127E 00 1 08642E 00 2 60775E 00 4 81885E 00 60 16 1 77224E 00 1 20247E 00 2 66918E 00 4 70796E 00 90 1 7 2 03041E 00 1 33688E 00 2 71408E 00 4 63007E 00 120 18 2 28608E 00 1 49759E 00 2 73420E 00 4 59599E 00 73 150 173 180 210 240 270 300 LP ti go 19 20 21 22 23 24 2 5 PNNNNDN NH 50460E 00 56957E 00 56423E 00 46371E 00 30919E 00 13677E 00 95813E 00 and backward computation 6 PT 30 60 90 120 150 180 210 240 270 300 Since MAX U 2 is greater than 1 0 in the forward continuation TY LAB 15 16 17 18 1 9 20 21 22 23 24 PrRRFNwWPSE OI OF PAR 4 131372 232501 34259 76049 40663 30787 449311 807671 35527H 06293I E 00 E 01 E 01 E 01 E 01 E 01 E 01 E 01 E 01 E 01 NONO ON B PR OO 1 1 0 O O J J COO oO 69995E 86351E 90582E 03333E 10546E 14713E 16871E MAX U 21174E 38379E 67585E 10921E 72635E 57456E 71390E 17286E 89069E 76882E 00 00 00 00 00 00 00 HEP CILE LEO ED ER LU N Eg OE E D GEL De OOOO COOOCO fo NN NNN NH PrRrRRFNNNNDND ND 70917E 6
2. PAR 67 2n 1 in the user subroutine SCSTPNT 5 35 An orbit of vector field f connecting a pseudo equilibrium to a saddle with a one dimensional unstable manifold Tf u a H u 0 0 Aif u 0 a AFM u0 a Aic Aj 1 LO a Mean vee w w 1 w y u 1 F i 39 v gt 0 coo feel e 50 Figure 15 Boundary value problems corresponding to A Subsection 5 35 B Subsection 5 36 Ma Piss Oma 2n 3 i j yr V V e Wn V Tj o NICP ICP 1 ICP NICP 2n 5 1 I2 63 A 640 5 671 67 n 1Y ga 67 n Qw1 67 2n 1 w 67 2n v t f 1 boundary equilibrium of fC Aj switch to problem 5 2 for f problem 5 3 for f problem 5 4 for f problem 5 16 for f lt t f 2 boundary equilibrium of f at u 0 Ai switch to problem 5 2 for f 3 n 3 problem 5 3 for f J n 2 problem 5 4 for f problem 5 16 for f t f 3 singular sliding point at u 0 P u 0 a switch to problem 5 2 for f and for f n 3 problem 5 3 for f and for fO n 2 problem 5 19 n 2 problem 5 27 for f n 2 t f 4 boundary equilibrium of f at u 1 H u 1 a i 51 switch to problem 5 1 n 2 problem 5 2 for f n 3 problem 5 3 for fO n 2 problem 5 4 for f problem 5 5 problem 5 9 for f problem 5 16 for f problem 5 29 for f problem 5 33 fo
3. z2 1 44 It is known see Bernardo di et al 2003 that at Q1 Oo 1 5 Q3 0 45 Q4 0 1 Q5 1 7078 a 8z as cycle existing in region 5S touches the discontinuity boundary X see Figure 21 This is a touching 1 5 0 5 Figure 21 A touching grazing cycle of the periodically forced dry friction oscillator 42 grazing bifurcation of the system Starting with the numerical solution corresponding to Figure 21 we continue this bifurcation in two control parameters a4 and as see Section 5 20 To execute the prepared computations simply enter make in the directory 8C DIR examples dfo The list of all commands to be typed is reported in Table 3 69 cp sc fdfo k sc fdfo cp fdfo f k fdfo f cp fdfo dat k fdfo dat scdat fdfo cp q fdfo q dat k sc fdfo sv fdfo k cp sc fdfo 29 sc fdfo cp ftdfo ft 0 fdfo f Cp q dat 2 q fdfo sc fdfo sv fdfo 3 cp sc fdfo 4 sc fdfo cp fdfo f 0 fdfo f cp q fdfo 2 q fdfo sc fdfo sv fdfo 4 cp sc fdfo k sc fdfo cp tdfo f 0 fdfo f cp q fdfo 4 q fdfo sc fdfo sv fdfo k get the constants file get the equations file get the data file convert the data file into the starting solution file save the starting solution file run SLIDECONT save output to p fdfo k q fdfo get the constants file get the equations file get the starting solution file run SLIDECONT save output to p df o 3 q fdfo get the constants file get
4. problem 5 16 for fO Hs u 0 a f 400 0 problem 5 2 for f and for f n 3 problem 5 3 for f and for fO n 2 problem 5 7 for f n 2 problem 5 19 n 2 problem 5 28 for f f n 2 Hu a f u 1 a problem 5 2 for f n 3 problem 5 3 for jo n 2 problem 5 29 for f Hu a O u 1 problem 5 2 for fo n 3 problem 5 3 for riu n 2 problem 5 27 for f n22 problem 5 31 for fO pus H v 1 a problem 5 1 n 2 problem 5 2 for f n 3 problem 5 3 for fO n 2 problem 5 4 for f problem 5 5 problem 5 10 for f 9 r9 problem 5 16 for f problem 5 32 for f fO problem 5 34 for f e pO Rer p argmin Rev Inv 4 0 continuation of a Hopf bifurcation Rev p argmin Rev g Imvg 0 problem 5 4 for f branch switching continuation of a limit point bifurcation note the saddle y is assumed to have a one dimensional unstable manifold the initial values of u 0 and y must be spec ified in PAR 67 PAR 67 2n 1 in the user subroutine SCSTPNT 6 PROGRAMMER S GUIDE A programmer s guide for the software SLIDECONT version 2 0 is presented in this section The reader should refer to the previous sections as well as to Kuznetsov et al 2003 and references therein for the nomenclature and the theory behind the methods 6 1 Disclaimer SLIDECONT is freely available for non commercial use on an as is basis In n
5. Moreover AUTO97 can accurately locate zeros along the solution branch of several test functions see AUTO97 documentation For each SLIDECONT problem we present in a separated subsection the corresponding defining system 6 or 7 8 and some details on its implementation As for the defining system we report its analytical formulation and specify the following informations the state space dimension n for which the defining system is valid the number m of control parameters the list of other active parameters 3 i e different from a1 m and their total number m4 for algebraic problems the dimension ng of the defining system for boundary value problems the number n of differential conditions and the number ny of boundary conditions As for the implementation we specify in accordance with AUTO97 notation the following infor mations the AUTO97 problem type IPS used to perform the computation the order SCIDIFF up to which analytical derivatives of f 1 f 2 and H are required the problem dimension NDIM NDIM ng except for Subsections 5 1 and 5 2 where an extra state variable is used the composition of the state vector U 1 U NDIM the right hand side vector F 1 F NDIM the total number of ac tive parameters NICP and the list of active parameter indexes ICP 1 ICP NICP denoting by Ij the index of the user parameter a i 1 m and reporting other active parameter symbols in parenthesis af
6. SCNDIM X PAR T I 56 SCPVLS SCBCND SCICND SCFOPT defines the starting solution x on input SCNDIM and I contain the vector fields dimension n and the index i of the vector field of interest respectively on output X k and PAR k contain state and parameters starting values for boundary value problems only an analytically known solution X k x4 T can be specified where T denotes the independent time variable which on input takes values in the interval 0 1 in such cases the time length of the solution must be specified in PAR 11 or PAR 60 PAR 62 depending upon the problem see Section 5 and AUTO97 documentation for more de tails Fortran prototype SUBROUTINE SCPVLS SCNDIM X PAR I defines user functions and solution measures see AUTO97 documentation Fortran prototype SCBCND NDIM PAR ICP NBC U0 U1 FB IJAC DBC is meaningful only for AUTO97 problem types Fortran prototype SCICND NDIM PAR ICP NINT U UOLD UDOT F UPOLD FI IJAC DINT is meaningful only for AUTO97 problem types Fortran prototype SCFOPT NDIM U ICP PAR IJAC FS DFDU DFDP is meaningful only for AUTO97 problem types For a fully documented equations file see for example 8C DIR examples hppc hppc f The format of the constants file is the same for all problems An example is listed below NDIM IPS IRS ILP NICP ICP I I 1 NICP 00 NTST NCOL IAD ISP ISW IPLT NBC NINT 100
7. f v a H u 0 a if u 0 0 Aj f u 0 o pif v1 38 Hic Hj NICP ICP 1 ICP NICP 8 i Io 61 7 62 1 63 A 64 A 65 u 661 5 t f 1 boundary equilibrium of f at u 0 Aj switch to problem 5 3 for f problem 5 4 for f problem 5 16 for f t f 2 boundary equilibrium of f at u 0 Ai switch to problem 5 3 for f0 problem 5 4 for f problem 5 16 for f t f 3 singular sliding point at u 0 H u 0 a f u 0 a switch to problem 5 3 for f and for f problem 5 7 for f problem 5 8 for f f problem 5 19 problem 5 26 for f e pO t f 4 tangent point of fC H5 u 1 o f 9 u 1 a switch to problem 5 3 for f problem 5 29 for f t f 5 tangent point of f at u 1 H4 u 1 a f 9 u 1 o switch to problem 5 3 for f problem 5 7 for f0 problem 5 25 for fO problem 5 31 for fO pU 1 i i 1 gt 1 J 49 t f 6 boundary equilibrium of f hj switch to problem 5 3 for f problem 5 4 for f problem 5 16 for f problem 5 30 for f 0 f t f 7 boundary equilibrium of f at v 1 li switch to problem 5 3 for f problem 5 4 for f problem 5 16 for f t f 8 singular sliding point at v 1 H4 v 1 a f v 1 o switch to problem 5 3 for f and for f problem 5 19 problem 5 30 for f 0 f problem 5 32 for f e pO the initial values of u 0 and u 1 must be specified in PAR 67
8. n 2 problem 5 7 for f n 2 problem 5 14 for f problem 5 20 for f t f branch AUTO97 branch t f limit point AUTO97 limit point t period doubling AUTO97 period doubling t f torus AUTO97 torus 5 7 An orbit of vector field f Q connecting a tangent point of f with the boundary 5 Tf ua 0 H u o 0 H u 0 a f u 0 a 0 mi H u o 0 Figure 3 Boundary value problems corresponding to A Subsection 5 7 B Subsection 5 8 15 LUNES RN mam A3 LL t f 1 o A of f H2 u 0 a f ul o switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 2 tangent point of fU j Z i H4 u 0 a f 9 u 0 o switch to problem 5 3 for f problem 5 5 problem 5 9 for f problem 5 19 t f 3 tangent point of f Hs u 1 o f 9 u 1 a switch to problem 5 3 for f problem 5 21 for f t f 4 tangent point of fU j i H4 u 1 a f 9 u 1 a switch to problem 5 3 for f problem 5 8 for f f problem 5 23 for f 0 f t f 5 pseudo equilibrium at u 1 Hz u 1 a g u 1 a switch to problem 5 5 problem 5 25 for f 5 8 A crossing orbit of vector fields f fO j 4 i connecting a tangent point of f with the boundary X U Tif u a 0 T fO v a 0 H u 0 a 0 Hi u 0 a f u 0 a 0 14 H u 1 a 0 u 1 v 0 0 Ao sa 0 Md TD 5 Bma
9. see AUTO97 documentation AII the informations described above are organized in a two columns tabular format where the left column reports symbols or names of the informations while the right column reports their values or 11 expressions Recall that SLIDECONT does not support the automatic switches marked by a star in the right column Finally refer to Figures 3 15 for a graphical representation of boundary value problems in the case of planar systems n 2 5 1 The discontinuity boundary H z a 0 9 EE t f 1 tangent point of fU Hs x a f a t f 2 tangent point of f Hs x a f z a t f 3 pseudo equilibrium Hz x o g x o 5 2 A curve of tangent points of vector field f 0 in three dimensional systems o H 2 0 Hz x a f x o 0 TOJU NDTM H z o Ha 2 o P2 0 i PARCS NICP ICP 1 ICP NICP 1 63 t f 1 tangent point of f 7 Z i H4 x o f x a switch to problem 5 2 for fO 12 5 3 A tangent point of vector field f ao eo oo H a T Ga t f 1 boundary equilibrium of f Hi 2 o f a e switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 2 tangent point of fH 7 Z i Hs x a f V x a switch to problem 5 3 for f problem 5 5 problem 5 19 t double tangency AUTOO7 limit point switch to problem 5 18 for f 5 4 A standard equilibrium
10. 0 NMX RLO RL1 A0 A1 NPR MXBF IID ITMX ITNW NWTN JAC EPSL EPSU EPSS 57 Slae 3 b2e6 Lb9 Qul i DS DSMIN DSMAX IADS 0 NTHL ITHL I THL I I 1 NTHL 0 NTHU ITHU I THU I I 1 NTHU NUZR IUZR I UZR I I 1 NUZR 2 0 33 0 2 SCISTART SCIDIFF i SCNPSI SCIPSI I I 1 SCNPSI 0 SCNFIXED SCIFIXED I I 1 SCNFIXED The first part ending with NUZR is an AUTO97 compliant constants file thus we refer to the AUTO97 documentation for the meaning of single constants In particular IPS is the problem type and DS is the starting step size of the continuation algorithm If DS is positive negative then the computation is performed forward backward with respect to the first active parameter of the ICP list A list of SLIDECONT problem types with alphabetical labels useful for source code inspection is reported in Table 1 Label Problem type Problem description T3D4 200 A curve of tangent points of vector field f in three dimensional sys tems A tangent point of vector field f A double tangency bifurcation of vector field f A standard equilibrium of vector field fY A boundary equilibrium of vector field f V A standard cycle of vector field f An orbit of vector field f connecting a tangent point of f with the boundary X A crossing orbit of vector fields f fU j i connecting a tangent point of f with the boundary X An
11. 00 7 47643E 02 5 54947E 00 8 11215E 02 240 10 1 06399E 00 9 42596E 02 2 68788E 00 6 45378E 02 279 UZ 11 9 00000E 01 1 09860E 01 1 40860E 00 5 23570E 02 319 UZ 32 5 50830E 01 1 36213E 01 1 53073E 01 1 33521E 10 320 B3 5 32750E 01 1 36814E 01 1 23238bE 01 1 05770E 02 327 UZ 14 4 35266E 01 1 38179E 01 2 68822E 09 8 99695E 02 400 EP 15 3 09084E 01 1 36026E 01 8 45254E 02 7 76109E 02 Labels 12 and 14 of the backward computation correspond again to the codimension 2 bifurcation points D and C of Figure 26 respectively zeros of test functions 4 and 1 Starting from the user output point at label 10 we continue the sliding cycle originating from the crossing crossing bifurcation i e a crossing orbit of vector fields f 1 f 2 connecting a tangent point of f with the boundary X for increasing values of z computation 36 PT TY LAB PAR 2 Sa PAR 46 PAR 47 20 16 1 14806E 01 1 76448E 01 2 30574E 01 40 17 1 29605E 01 1 42232E 01 1 45501E 01 60 18 1 48027E 01 9 28119E 02 6 15887E 02 80 19 1 69601E 01 3 54030E 02 2 17204E 03 82 Uz 20 1 70986E 01 3 18574E 02 5 04256E 12 99 LP 21 1 80346E 01 1 74496E 13 7 56008E 03 100 EP 22 1 80301E 01 1 49712E 03 7 28495E 03 Label 20 indicates a pseudo homoclinic bifurcation i e the presence of a crossing orbit of vector fields f 9 f connecting a tangent point of f with a pseudo equilibrium zero of test function
12. 00 9 08309E 01 1 04565E 02 43 UZ 21 2 75491E 00 1 00000E 00 1 78141E 02 46 EP 22 3 00004E 00 1 10912E 00 2 65706E 02 PT TY LAB PAR 5 PAR 2 PAR 41 1 EP 15 1 01048E 00 3 30000E 01 3 59492E 02 20 16 8 02944E 01 2 59482E 01 4 16078E 02 40 17 1 07293E 01 3 35501E 02 5 97375E 02 60 EP 18 6 15279E 04 1 91480E 04 6 24144E 02 where label 19 in the forward computation identifies a codimension 2 bifurcation namely a boundary Hopf bifurcation see point B in Figure 25 as a zero of test function 1 Starting again from the solution at label 12 we continue the pseudo saddle colliding with the unsta ble focus at the boundary equilibrium bifurcation for increasing values of a5 computation 5 We get the following output PT TY LAB PAR 5 PAR 41 PAR 42 1 BP L5 1 01048E 00 1 25638E 11 1 00000E 00 10 16 1 01568E 00 1 27664E 02 9 87234E 01 20 17 1 04423E 00 8 17597E 02 9 18240E 01 30 18 1 13988E 00 3 05189E 01 6 94811E 01 40 LP 19 1 24360E 00 6 51847E 01 3 48153E 01 50 EP 20 9 04329E 01 9 44763E 01 5 52371E 02 where label 19 indicates a pseudo saddle node bifurcation whose forward and backward continuations computations 6 and 7 are reported below see curve PSN in Figure 25 79 PT TY LAB PAR 5 PEN PAR 2 PAR 41 PAR 42 10 21 1 27118E 00 3 44665E 01 6 23634E 01 3 76366E 01 20 22 1 38762E 00 4 04902E 01 5 13921E 01 4 86079E 01 30 23 1 71214E 00 5 60562E 01 2 69634E
13. 01 7 30366E 01 40 24 2 21424E 00 7 74693E 01 3 60651E 03 9 96393E 01 41 UZ 25 2 22250E 00 7 78000E 01 1 87790E 08 1 00000E 00 50 EP 26 2 53582E 00 8 99199E 01 1 23494E 01 1 12349E 00 PT TY LAB PAR 5 3x PAR 2 PAR 41 PAR 42 10 21 1 21650E 00 3 15428E 01 6 80505E 01 3 19495E 01 20 22 1 11167E 00 2 57473E 01 8 01058E 01 1 98942E 01 28 UZ 23 9 66081E 01 1 72243E 01 1 00000E 00 4 54661E 10 30 24 9 11820E 01 1 38847E 01 1 08607E 00 8 60733E 02 39 UZ 25 7 03037E 01 1 99083E 10 1 50556E 00 5 05564E 01 40 26 6 85973E 01 1 22104E 02 1 54810E 00 5 48096E 0O1 50 EP 27 5 35108bE 01 1 27568E 01 2 00867E 00 1 00867E 00 Notice that point B of Figure 25 is detected again during the forward computation zero of test function 1 at label 25 while label 23 in the backward computation identifies another codimension 2 bifurcation namely a boundary pseudo saddle node bifurcation see point H in Figure 26 as a zero of test func tion 2 Starting from the solution at label 19 of computation 5 we now continue the pseudo node colliding with the pseudo saddle at the pseudo saddle node bifurcation for decreasing values of a5 computation 8 PT TY LAB PAR 5 PAR 41 PAR 42 1 EP 21 1 24360E 00 6 51847E 01 3 48153E 01 10 22 1 24302E 00 6 70003E 01 3 29997E 01 20 23 1 22931E 00 7 34503E 01 2 65497E 01 30 24 1 11341E 00 8 60547E 01 1 39453E 01 40 25 8 39853E 01 9 61723E 01 3 827774E 02 45 U
14. 5 2 for f O n 3 problem 5 3 for f n 22 problem 5 24 for JO fO n 22 t f 7 pseudo equilibrium at v 1 n 2 Hz v 1 a g v 1 o switch to problem 5 5 problem 5 26 for f 0 f 5 16 A boundary equilibrium of vector field FO 22 mnm p Hi _ a 8S 8 RE ll oo Eme eE E E IPS SCIDIFF N 0 0 n 1 M 2 1 hE 15 O HF DY t f 1 Hopf Rer p argmin Rev Inv 4 0 switch to continuation of a Hopf bifurcation t f 2 branch limit point Rer p argmin Rev g Im 0 switch to problem 5 4 for f branch switching continuation of a limit point bifurcation t f 3 tangent point of fH 7 Z i Hs x a f P x a switch to problem 5 2 for f O n 3 problem 5 3 for f n 22 problem 5 19 n 2 x DI U 1 U NDIM 5 17 A pseudo saddle node bifurcation IPS SCIDIFF NDIM 0 0 n 2 U 1 U NDIM 21 En 1 2 27 F k k 1 n Afe a a Arf 2 a F n 1 H 2 a NUES NICP ICP 1 ICP NICP t f 1 boundary equilibrium of f A2 switch to problem 5 2 for f 1 n 3 problem 5 3 for f m 2 problem 5 4 for f problem 5 16 for f t f 2 boundary equilibrium of f switch to problem 5 2 for f 3 n 3 problem 5 3 for jo n 2 problem 5 4 for f problem 5 16 for t f 3 singular sliding point H z o f P x a switch to proble
15. 7 while label 21 indicates the presence of a crossing orbit of vector fields f f connecting a tangent point of f with a tangent point of f 2 zero of test function 6 The forward and backward continuations of these bifurcations computations 37 40 give the following outputs see curves TC P and TCD in Figure 26 87 PT TY LAB PAR 5 alse PAR 2 Tum PAR 46 20 2 7 36964E 01 1 58109E 01 3 76354E 01 40 3 1 01536E 00 2 05810E 01 3 73121E 02 41 LP 4 1 01545E 00 2 06746E 01 3 51155E 02 46 UZ 5 9 92599E 01 2 19545E 01 3 03505E 10 60 6 1 935575 01 5 85610E 02 4 03693E 01 80 7 8 80653E 03 2 84106E 03 5 00675E 01 100 EP 8 4 73232E 04 1 59320E 04 5 05290E 01 PT TY LAB PAR 5 PAR 2 bw PAR 44 20 2 6 64025E 01 1 54660E 01 9 66445E 03 25 UZ 3 6 26627E 01 1 52906E 01 4 79962E 15 40 4 4 92656E 01 1 44396E 01 6 56560E 02 60 5 3 90105E 01 1 30804E 01 9 25824E 02 80 6 1 27499E 01 4 53387E 02 3 19934E 02 100 EP 7 5 46504E 02 1 94337E 02 1 37203E 02 PT TY LAB PAR 5 pe PAR 2 PAR 44 PAR 45 40 23 9 80627E 01 2 03284E 01 1 34864E 02 7 29041E 03 78 LP 24 9 92657E 01 2 18461E 01 1 13581E 03 2 31160E 04 80 25 9 92640E 01 2 19054E 01 5 19980E 04 1 00907E 04 82 Uz 26 9 92599E 01 2 19545E 01 2 L 17031E 10 2 18153E 11 120 27 9 88472E 01 2 219 713E 01 1 5 1 03617E 02 8 41806E 04 160 28 9 82712E 01 2 33226E 01 1 89340E 02
16. Aj t f 1 boundary equilibrium of fC switch to problem 5 2 for f problem 5 3 for f problem 5 4 for f problem 5 16 for f lt t f 2 boundary equilibrium of f at u 0 i switch to problem 5 2 for f n 3 problem 5 3 for f n 22 problem 5 4 for f problem 5 16 for f t f 3 singular sliding point at u 0 H5 u 1 o f 9 u 1 a switch to problem 5 2 for f and for f n 3 problem 5 3 for f and for fO n 2 problem 5 7 for fO n 2 problem 5 19 n 2 t f 4 tangent point of f H4 u 1 o f u 1 a switch to problem 5 2 for f n 3 problem 5 3 for fe n 2 problem 5 29 for f t f 5 tangent point of f at u 1 H4 u 1 o f 9 u 1 o switch to problem 5 2 for f O n 3 problem 5 3 for riu n 2 problem 5 10 for f 0 f problem 5 31 for FO fO t f 6 pseudo equilibrium at u 1 n 2 Hz u 1 o g u 1 a i i i 18 switch to problem 5 5 problem 5 33 for f the initial value of u 0 must be specified in PAR 67 PAR 67 n 1 in the user subroutine SCSTPNT 5 10 A crossing orbit of vector fields f fU j F i connecting a pseudo equilibrium with the boundary 3 16 S000002 D Md 81 eee Orig F k k n 1 2n NICP ICP 1 ICP NICP 0 JA t f 1 boundary equilibrium of f switch to problem 5 4 for f problem 5 16 for f t f 2 boundary equilibrium of f
17. LAB PAR 5 Cn PAR 2 ides PAR 43 PAR 44 20 2 6 97557E 01 1 55978E 01 4 36976E 01 5 63024E 01 40 3 7 67487E 01 1 58273E 01 3 21617E 01 6 78383E 01 60 4 8 40062E 01 1 59322 amp E 21 2 03457E 01 7 96543E 01 80 5 9 07660E 01 1 56286E 01 6 53144E 02 9 34686E 01 88 UZ 6 9 23286E 01 1 51997E 01 3 36276E 08 1 00000E 00 93 LP 7 9 25049E 01 1 49351E 01 2 84730E 02 1 02847E 00 100 EP 8 9 12387E 01 1 40275E 01 9 98941E 02 1 09989E 00 PT TY LAB PAR 5 ee PAR 2 dee PAR 43 PAR 44 20 2 6 09661E 01 1 52064E 01 5 98773E 01 4 01227E 01 40 3 5 56162E 01 1 49051E 01 7 10961E 01 2 89039E 01 60 4 5 08334E 01 1 45688E 01 8 21017E 01 1 78983E 01 80 5 4 65764E 01 1 41796E 01 9 24967E 01 7 50331E 02 97 UZ 6 4 35242E 01 1 38171E 01 1 00000E 00 9 17078E 10 100 EP 7 4 29923E 01 1 37443E 01 1 01281E 00 1 28088E 02 PT TY LAB PAR 5 bus PAR 2 PAR 61 PAR 43 19 LP 10 7 11638E 01 1 12554E 01 1 99485E 01 7 00017E 01 20 11 7 08720E 01 1 07769E 01 1 98371E 01 7 52608E 01 40 12 4 19217E 01 4 97571E 02 7 62366E 01 6 82921E 01 60 T3 4 12679E 01 4 88618E 02 9 72040E 01 6 74863E 01 80 14 4 12442E 01 4 88294E 02 9 84043E 01 6 74569E 01 100 EP 15 4 12440E 01 4 88290E 02 9 84173E 01 6 74566E 01 PT TY LAB PAR 5 PAR 2 PAR 61 PAR 43 20 10 5 23313E 01 1 37062E 01 2 84719E 01 1 08711E 01 25 UZ 11 4 35242E 01 1 38171E 01 3 26994E 01 9 88708E 11 3l UZ 12 2 00000 01 1 29693E 01 4 64692E 01 1 00318E 01 40 13 2 85815E 02 9 18817E 02 7 16601E 01 2 5
18. data file contains NDIM 1 numbers namely the time variable t 0 1 and the AUTO97 state components U 1 U NDIM at f As detailed in Section 5 when only one vector field is involved in the differential equations of the boundary value problem i e when NDIM SCNDIM the time length of the starting solution must be specified in PAR 61 PAR 11 for the continuation of a standard cycle see Subsection 5 6 By contrast when both vector fields are involved i e when NDIM 2 SCNDIM the time lengths of the two connected parts of the starting solution must be specified in PAR 61 and PAR 62 Finally the time length of a sliding segment has to be given in PAR 60 if solutions with such a segment are computed 7 EXAMPLES Each example directory in 8C DIR examples contains constant files sc k k 1 2 forall successive computations several equations files k for computations requiring parameter and state initialization in subroutine SCSTPNT and several data files dat k for manually set up the starting solution of boundary value problems when automatic switch from previous computations is not sup 61 ported 0 is used when no initialization is required Browse dryf f k to see how the equations have been specified in subroutine SCFUNC and SCBOUND and how the starting parameter values have been set in subroutine SCSTPNT To execute each of the prepared computations the user can simply ent
19. f 9 u 0 a switch to problem 5 3 for f problem 5 5 problem 5 9 for f problem 5 10 for f 0 f problem 5 19 problem 5 32 for f 0 f t f 3 tangent point of f H4 u 1 o f u 1 a switch to problem 5 3 for f problem 5 21 for f t f 4 tangent point of f at u 1 H5 u 1 o fP u 1 o switch to problem 5 3 for f problem 5 7 for fO problem 5 21 for f problem 5 23 for f 9 fO t f 5 boundary equilibrium of f at v 1 H v 1 o f 9 v 1 a switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 6 tangent point of f Ha v 1 a f v 1 a switch to problem 5 3 for f problem 5 5 problem 5 19 problem 5 22 for f fO 5 25 An orbit of vector field f Q connecting a tangent point of f witha pseudo equilibrium Tf u a 0 UL 0 0 faa 0 H u 0 a fY u 0 a 0 H u 1 a 0 29 Af 9 u 1 a A fH u 1 a 0 j i M A 1 0 35 A Figure 10 Boundary value problems corresponding to A Subsection 5 25 B Subsection 5 26 U 1 U NDIM U1 Un IUNII NICP ICP 1 ICP NIGP t f 1 boundary equilibrium of f HE u 0 a f CORE n 2 switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 2 tangent point of f at u 0 H u 0 a f 9 u 0 o switch to problem 5 2 for f O n 3 problem 5 3 for f n 2 problem 5 5 n 2 problem 5 9 for f n 22 proble
20. for f n 2 problem 5 7 for f n22 problem 5 8 for f 9 fO n 2 problem 5 19 n 2 problem 5 22 for f 0 f n 2 H u 1 a fO u 1 a problem 5 2 for fo n 3 problem 5 3 for fO n 2 problem 5 29 for f H0 a O u 1 n i switch to problem 5 2 for f O n 3 problem 5 3 for f n 2 2 problem 5 7 for f O n problem 5 23 for fH FO n 2 problem 5 31 for fO fH t f 6 boundary equilibrium of fC H v 1 a f v 1 a n 2 switch to problem 5 4 for f problem 5 5 problem 5 16 for f problem 5 34 for f 0 f t f 7 tangent point of f at v 1 H4 v 1 o fV v 1 a switch to problem 5 2 for f O n 3 problem 5 3 for fo n 2 problem 5 5 n 2 problem 5 19 n 2 problem 5 32 for f 0 f the initial value of u 0 must be specified in PAR 67 PAR 67 n 1 in the user subroutine SCSTPNT 5 31 An orbit of vector field f connecting a pseudo equilibrium with a tangent point of f j z i Tf u a H u 0 o Auf 9 u 0 a AFM u 0 a 1 iua s 35 Hu a GI u 1 o f u 1 a oooooo u 1 v 0 A B Figure 13 Boundary value problems corresponding to A Subsection 5 31 B Subsection 5 32 44 U 1 U NDIM Ui Un TE NICP ICP 1 ICP NICP t f 1 boundary equilibrium of f switch to n problem 5 3 for f n problem 5 4 for
21. for f n 3 problem 5 3 for f and for f n 2 problem 5 7 for f n 2 problem 5 19 n 2 problem 5 21 for f n 22 t f 4 boundary equilibrium of fC HZ u 1 a f n 2 switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 5 tangent point of f at u 1 H u 1 o fV switch to problem 5 2 for f O n 3 problem 5 3 for riu n 2 problem 5 5 n 2 problem 5 10 for fO fO problem 5 19 n 2 problem 5 31 for f 0 fU the initial value of u 0 must be specified in PAR 67 PAR 67 n 1 in the user subroutine SCSTPNT 42 5 30 A crossing orbit of vector fields f fU j F i connecting a pseudo equilibrium with a tangent point of f i 0 0 H u 0 a 0 0 Aic Aj 1 0 34 A u 1 a 0 u 1 v 0 0 A v 1 a 0 0 Ma B1 m4 t f 1 boundary equilibrium of f switch to t f 2 boundary equilibrium of f at u 0 switch to t f 3 singular sliding point at u 0 switch to t f 4 tangent point of f switch to t f 5 tangent point of f at u 1 43 4 Tj i Aj 6 U1 T2 61 7 62 15 63 Ai 64 05 problem 5 2 for f n problem 5 3 for f problem 5 4 for f problem 5 16 for i problem 5 2 for f J n 3 problem 5 3 for f 3 n 2 problem 5 4 for f problem 5 16 for f I u 1 9 u 1 a problem 5 2 for f and for f n 3 problem 5 3 for f and
22. in two control parameters continue a standard orbit possibly crossing the boundary X connecting two tangent points of the same vector field in planar systems n 2 e g a crossing crossing bifurcation of sliding periodic solutions in two control parameters continue a standard orbit possibly crossing the boundary X connecting two tangent points of different vector fields in planar systems n 2 e g a buckling switching or sliding crossing bifurcation of sliding periodic solutions in two control parameters continue a standard orbit possibly crossing the boundary X connecting a tangent point with a pseudo equilibrium e g the standard part of a sliding homoclinic orbit to a pseudo saddle in planar systems n 2 in two control parameters continue a standard orbit possibly crossing the boundary X connecting a tangent point with a standard saddle in planar systems n 2 e g the standard part of a sliding homoclinic orbit to a saddle in two control parameters continue a standard orbit possibly crossing the boundary X connecting a pseudo equilibrium with a tangent point in two control parameters continue a standard orbit possibly crossing the boundary X connecting two pseudo equilibria in planar systems n 2 in two control parameters continue a standard orbit possibly crossing the boundary X connecting a pseudo equilibrium with a standard saddle with one dimensional unstable manifol
23. of vector field f LY NNNM MNT Poa ONT mueum O t f 1 boundary equilibrium of f H 2 0 switch to problem 5 1 n 2 problem 5 2 for f n 3 problem 5 3 for f n 22 problem 5 5 problem 5 16 for f t f branch AUTO97 branch t f limit point AUTO97 limit point 13 switch to continuation of a limit point bifurcation t f Hopf AUTO97 Hopf switch to continuation of a Hopf bifurcation 5 5 A pseudo equilibrium A x a Af x a A f O a a 10T22 1 0 C e 12 AG ffo Ho t f 1 boundary equilibrium of fU switch to problem 5 2 for f 0 n 3 problem 5 3 for f n 2 problem 5 4 for f problem 5 16 for f t f 2 boundary equilibrium of f switch to problem 5 2 for f 3 n 3 problem 5 3 for jo n 2 problem 5 4 for f 2 problem 5 16 for f 2 t f 3 singular sliding point H z o f P x a switch to problem 5 2 for f and for f n 3 problem 5 3 for f and for fO n 2 problem 5 19 n 2 t pseudo saddle node bifurcation AUTO97 limit point switch to problem 5 17 gt 14 5 6 A standard cycle of vector field f U 1 U NDIM 21 Zn ER Km locom fi x a NICP IGP 1 ICP NICP t f 1 touching bifurcation min NTST NCOL AC 1 H x 0 switch to problem 5 1 n 2 problem 5 2 for f n 3 problem 5 3 for fO
24. orbit of vector field f connecting a pseudo equilibrium with the boundary X A crossing orbit of vector fields f fU j i connecting a pseudo equilibrium with the boundary X A touching grazing cycle of vector field fC An orbit of vector field f connecting two tangent points of f A crossing orbit of vector fields f fU j i connecting two tangent points of f An orbit of vector field f connecting a tangent point of f with a tangent point of f G i A crossing orbit of vector fields f fY j Z i connecting a tangent point of f with a tangent point of f rj LEE a E el Problem type uz soi i ci ao pe peque n ME Bi MER eas ren as i ol p por Table 1 continue 58 TGP2 i20 An orbit of vector field connecting a tangent point of f with a pe pseudo equilibrium TCP2 721 A crossing orbit of vector fields f fU j Z i connecting a tangent puc point of f with a pseudo equilibrium An orbit of vector field f connecting a tangent point of f to a saddle T i23 A crossing orbit of vector fields f fU j Z i connecting a tangent MEE point of f to a saddle of PETi An orbit of vector field f connecting a pseudo equilibrium with a per tangent point of f D 6 An orbit of vector field f connecting a pseudo equilibrium with a S 4 5 PT 1 tangent point of fO G i PC i27 A crossing orbit of vector fields f fU j 4 i conne
25. problem 5 22 for 9 fO n 2 H u 0 a f u 0 o 2 n 3 2 n 2 problem 5 2 for f problem 5 3 for f Hs u 1 a f u problem 5 2 for f problem 5 3 for f Hu a f u problem 5 2 for f problem 5 3 for f problem 5 7 for f problem 5 8 for f I a 2 n 3 2 m 2 2 fU n 2 problem 5 15 for f m problem 5 22 for f fO n 2 5 14 A sliding cycle with a standard orbit of vector field f a 0 Tog s a 0 H u 0 a 0 a 0 s 0 u 1 0 s 1 u 0 0 20 Figure 7 Boundary value problems corresponding to A Subsection 5 14 B Subsection 5 15 24 TUR cu n EO es CNN NICP E bua em m ICE NICP t f 1 boundary squitibrumi of f CAL a 0 a fe u 0 n 2 switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 2 tangent point of fU j i H u 0 o f u 0 o switch to problem 5 2 for f n 3 problem 5 3 for fD n 2 2 problem 5 5 n 2 problem 5 9 for f n22 problem 5 19 n 2 t f 3 tangent point of fC H u 1 o f u 1 a switch to problem 5 2 for f m 3 problem 5 3 for f n 22 problem 5 21 for fe n 2 t f 4 tangent point of fU j i Hs u 1 o fP u 1 o switch to problem 5 2 for f O n 3 problem 5 3 for f n 22 problem 5 8 for JO f n 2 problem 5 23 for f f n 2 problem 5 15 for f 0 fO t f 5 ps
26. the equations file get the starting solution file run SLIDECONT save output to p df o 4 q fdfo get the constants file get the equations file get the starting solution file run SLIDECONT save output to p dfo k q fdfo Eod Table 3 Command list sv is an AUTO97 command We get the following output computation 1 PT 30 35 60 90 120 150 180 210 236 240 270 273 300 The computed family o TY PD PD BP EP LAB PAR 4 2 1 46553E 01 3 1 59166E 01 4 2 27099E 01 5 3 18761E 01 6 4 28346E 01 7 5 62395E 01 8 7 22631E 01 9 9 05530E 01 10 1 07114E 00 11 1 09846E 00 12 1 30693E 00 1 3 1 32427E 00 14 1 50283E 00 Sc f os PRPPRPRPPRPRPPHRP PE EB MAX U 2 PAR 5 PAR 61 00000E 00 1 79104E 00 1 40324E 01 00000E 00 1 80658E 00 1 39118E 01 00000E 00 1 87385E 00 1 34123E 01 00000E 00 1 94700E 00 1 29084E 01 00000E 00 2 02670E 00 1 24008E 01 00000E 00 2 12073E 00 1 18510E 01 00000E 00 2 22943E 00 1 12732E 01 00000E 00 2 34532E 00 1 07161E 01 00000E 00 2 43853E 00 1 03065E 01 00000E 00 2 45252E 00 1 02477E 01 00000E 00 2 54231E 00 9 88578E 00 00249E 00 2 54812E 00 9 86325E 00 18394E 00 2 58419E 00 9 72559E 00 periodic touching grazing cycles is shown in Figure 22 where the end solution with label 14 is omitted The corresponding bifurcation curve T C H in the o4 a5 plane is present
27. the solution at label 38 of the output file q hppc 12 produced by cycle continuation Though not explicitly pointed out in the following a similar remark holds for each 81 non automatically supported switch between problems The forward and backward continuations com putations 13 and 14 give the following outputs see curve TCH in Figure 26 PT TY LAB PAR 5 Re PAR 2 PAR 61 PAR 41 10 2 6 78992E 01 1 63219E 01 3 42591E 01 1 17523E 01 20 3 7 87620E 01 1 67518E 01 3 20644E 01 8 36044E 02 30 4 8 96762E 01 1 71199E 01 3 05823E 01 3 69883E 02 38 UZ 5 9 66082E 01 1 72243E 01 3 02124E 01 1 00995E 08 40 6 9 82029E 01 1 72158E 01 3 02418E 01 9 58993E 03 50 EP 7 1 03897E 00 1 69287E 01 3 13145E 01 5 00037E 02 PT TY LAB PAR 5 P4 PAR 2 PAR 61 PAR 41 40 2 5 44655E 02 1 11966E 01 6 80716E 01 3 70434E 02 80 3 1 28314E 05 3 66698E 02 1 75100E 02 1 17004E 05 120 4 7 38073E 11 1 84162E 02 3 37676E 02 7 05412E 11 160 5 3 09171E 17 1 14904E 02 5 36404E 02 3 00635E 17 200 EP 6 3 46004E 24 8 15577E 03 7 53186E 02 3 39223E 24 where point H of Figure 25 is detected again during the forward computation zero of test function 1 at label 5 At point H the touching grazing cycle degenerates to a point boundary Hopf bifurca tion Thus the solutions obtained in computation 13 after this point are spurious because the touching grazing cycle of vector field f lies in region S2 Startin
28. value the sliding cycle is located entirely in the domain 5 portrait 1 in Figure 18 while just above this parameter value the sliding cycles have also a standard segment in S portrait 2 in Figure 18 and therefore can be continued by means of the boundary value problem 21 Starting from the data file dryf dat 3 corresponding to label 3 computation 3 produces see Fig 19 PT TY LAB PAR 2 PAR 60 PAR 61 PAR 62 PAR 45 10 2 5 71491E 02 1 56748E 00 5 21147E 00 6 48471E 02 7 83742E 01 20 3 7 67732E 02 1 04430E 00 5 08803E 00 9 72593E 01 5 22152E 01 27 UZ 4 1 02312E 01 6 98102E 10 4 97317E 00 1 90398E 00 3 49050E 10 30 5 1 28304E 01 1 83052E 00 4 89559E 00 2 46968E 00 9 15258E 01 40 6 1 80925E 01 1 05903E 01 4 83254E 00 3 04894E 00 5 29517E 00 50 EP 7 2 05960E 01 1 97537E 01 4 84202E 00 3 22755E 00 9 87687E 00 Figure 19 A family of sliding cycles of system 41 The thick solutions correspond to a switching bifurcation at ag 0 0557 and a crossing bifurcation at a2 0 1023 Label 4 indicates a crossing bifurcation zero of test function 5 At a5 0 1023 a tangent point of f is detected at the end point v 1 that coincides with the starting tangent point T see portrait CC 66 in Figure 18 According to Kuznetsov et al 2003 this implies that for a slightly bigger than a5 a stable crossing cycle replaces the sliding cycle Numer
29. 01 6 24439E 02 11 9 UZ 5 1 22753E 00 2 51248E 01 2 39698E 13 120 6 1 20857E 00 2 25765E 01 1 60261E 03 130 UZ 7 1 02940E 00 1 83584E 02 4 03145E 13 132 UZ 8 1 01048E 00 2 30242E 09 2 47076E 11 160 9 8 11376E 01 1 52349E 01 1 73077E 02 200 EP 10 6 69562E 01 2 15265E 01 2 58086E 02 thus detecting a pseudo homoclinic bifurcation zero of test function 5 at labels 5 and 7 whose forward and backward continuations computations 28 and 29 are reported below see curve TGP in Figure 25 PT TY LAB PAR 5 PAR 2 vs PAR 43 PAR 44 10 2 1 22947E 00 3 33549E 01 5 17583E 01 4 82417E 01 14 LP 3 1 23038E 00 3 37747E 01 4 85347E 01 5 14653E 01 20 4 1 21954E 00 3 48173E 01 3 75685E 01 6 24315E 01 30 5 1 13889E 00 3 48593E 01 1 76992E 01 8 23008E 01 40 6 9 78219E 01 3 18569E 01 2 65354E 03 9 97346E 01 41 UZ 7 9 74779E 01 S TII57ESOlL 22 3 38507E 09 1 00000E 00 50 EP 8 8 52887E 01 2 86163E 01 7 73273E 02 1 07733E 00 PT TY LAB PAR 5 er PAR 2 s PAR 43 PAR 44 10 2 1 22462E 00 3 26290E 01 5 66373E 01 4 33627E 01 20 3 1 19145E 00 3 01936E 01 6 96532E 01 3 03468E 01 30 4 1 07533E 00 2 49321E 01 9 04037E 01 9 59632E 02 36 UZ 5 9 92599E 01 2 19545E 01 1 00000E 00 2 20176E 09 40 6 9 35376E 01 2 00814E 01 1 05534E 00 5 53443E 02 50 EP 7 7 98581E 01 1 60300E 01 1 16489E 00 1 64888E 01 85 and identify the codimension 2 bifurcation points A and J in Figures 2
30. 0E 04 200 EP 22 8 41273E 05 2 21150E 05 2 78255E 01 5 06800E 05 where label 18 in the backward computation corresponds again to the codimension 2 bifurcation point I in Figure 26 zero of test function 3 We now continue the sliding cycle originating from the buckling switching bifurcation label 14 of computation 30 for decreasing values of a2 computation 33 PT TY LAB PAR 2 3x PAR 61 PAR 62 PAR 45 20 2 2 18770E 01 6 27164E 01 1 51874E 01 3 08010E 01 40 3 1 88132E 01 7 44769E 01 1 06016E 00 2 92309E 01 60 4 1 03584E 01 1 37591E 02 2 02648E 00 1 81107E 01 72 UZ 5 5 45143E 02 2 57363E 02 1 84682E 00 8 49024E 11 80 6 2 33203E 02 5 94075E 02 1 61535E 00 6 61825E 02 100 EP 7 3 951T0E 03 3 47969E 03 1 47370E 00 6 66000E 02 86 thus detecting a crossing crossing bifurcation zero of test function 5 at label 5 whose forward and backward continuations computations 34 and 35 are reported below see curve T C T in Figure 26 PT TY LAB PAR 5 Ped PAR 2 re E PAR 41 PAR 44 20 8 1 52984E 00 5 16540E 02 1 25818E 01 1 01595E 01 40 9 1 73297E 00 3 08930E 02 2 85271E 01 1 19799E 01 60 10 1 92386E 00 8 44640E 03 1 34447E 02 1 38984E 01 80 T3 1 96698E 00 2 88164E 03 4 15863E 02 1 43693E 01 100 EP 12 1 97580E 00 1 72740E 03 7 01312E 02 1 44674E 01 PT TY LAB PAR 5 PAR 2 T PAR 41 PAR 44 80 8 1 41617E 00 6 23651E 02 8 60407E 00 9 20870E 02 160 9 1 27960E
31. 3 08632E 01 2 98943E 01 2 70292E 01 1 09306E 01 30 31 2 57424E 01 4 80366E 01 6 22558E 02 2 85792E 01 40 32 1 92281E 01 7 72133E 01 1 01750E 02 3 36026E 01 44 HB 33 1 72243E 01 8 82183E 01 3 45071E 02 3 12231E 01 50 EP 34 1 32406E 01 1 14522E 00 1 46268E 01 2 10076E 01 detecting a Hopf bifurcation label 33 For the same arguments used before the Hopf bifurcation curve of vector field f is a straight line see curve Ho in Figure 26 therefore we avoid its numerical computation We now continue the stable limit cycle originating from the Hopf bifurcation which can be shown to be supercritical by means of suitable software e g CONTENT Kuznetsov amp Levitin 1995 1997 see Kuznetsov 1998 for an analytical proof for decreasing values of a2 computation 12 obtaining the following output PT TY LAB PAR 2 PERIOD PAR 41 10 35 1 72111E 01 3 02580E 01 2 90096E 01 20 36 1 70870E 01 3 07029E 01 2 30798E 01 30 37 1 67490E 01 3 20771E 01 1 33031E 01 40 UZ 38 1 62226E 01 3 48569E 01 6 10623E 14 50 EP 39 1 57846E 01 3 79726E 01 1 13806E 01 where label 38 indicates a touching grazing bifurcation zero of test function 1 Notice that SLIDECONT does not support the automatic switch from cycle to touching grazing continuation Thus the user must provide a data file hppc dat 13 which specifies numerically the starting solution This file can be easily constructed from
32. 32591E 00 240 10 2 45441E 00 1 00000E 00 2 70230E 00 4 65025E 00 270 TI 2 23336E 00 1 00000E 00 2 50128E 00 5 02397E 00 300 EP 12 2 01106E 00 1 00000E 00 2 30227E 00 5 45826E 00 The computed family of lt periodic touching grazing cycles is shown in Figure 24 where solutions with labels 9 12 are omitted The corresponding bifurcation curve TC H in the a4 a5 plane can be seen in Figure 23 The flip grazing point A is more accurately detected in the backward continuation at 72 1 5 T Figure 24 A family of lt periodic touching grazing cycles terminating in a period halving bifurca tion at a4 a5 2 814 3 0345 The starting solution is o marked while the thick solution corresponding to another flip grazing is traced twice label 2 where the periodic touching grazing cycle exhibits the flip PD bifurcation Notice that a branching point BP is detected at label 8 where the touching grazing cycle undergoes a period halving bifurcation at A2 a4 a5 2 8138 3 0344 Finally we compute the flip bifurcation curve where the standard 27 cycle exhibits a period doubling bifurcation This curve can be obtained by switching at point A detected in computation 2 as the PD point at label 2 A dummy continuation computation 4 PT TY LAB PAR 4 eA MAX U 2 TE PAR 5 PERIOD 5 EP 14 1 32089E 00 1 00000E 00 2 54701E 00 4 93377E 00
33. 4333E 61787E 50886E 40856E 31599E 22938E Dg uw Pe DG BN PAR 5 47978E 39323E 30263E 21067E 11974E 03179E 94823E 87000E 79TTTE 73204E REDE DID DQIODpomE D gu bifurcation curve is meaningfull It is depicted in Figure 23 as PD 7 3 Harvesting a prey predator community 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 O101010 1 5 BBs 2oOoo000010101 01 O1 638461 753981 800231 008801 217381 425921 636721 DDR OEPMTIIBAS DI PERIOD 06753l 250811 4577401 684421 92826 18489 450161 719981 989981 25523l Ed bd Ed Ed Dd Db DH EH DH DH 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 only the backward part of the flip This example illustrates a variety of one and two parameter calculations in a planar Filippov system System 5 with fO s a fO s o zi 1l 21 v z1 z2 V z1 xr2 322 zi l 21 v z1 z2 V x1 xr2 03133 0422 v zi1 H x a 2 o5 ri ag 2i 45 46 47 48 models an harvested prey predator community where x and x are prey and predator population den sities and harvesting of the predator population at constant effort a4 is allowed only if predator are 74 sufficiently abundant i e if 2 gt o5 see Kuznetsov et al 2003 Dercole et al 2003 for more de t
34. 4400E 02 60 14 8 18660E 04 5 36179E 02 1 17558E 02 8 08507E 04 80 15 1 43765E 05 3 60365E 02 1 71494E 02 1 43008E 05 100 EP 16 1 41020E 07 2 62269E 02 2 33775E 02 1 40492E 07 Labels 6 in computations 16 and 17 correspond to the codimension 2 bifurcation points G and C in Figure 26 respectively zeros of test functions 3 and 4 Point C is also detected during computation 19 zero of test function 3 at label 11 Starting from the user output point at label 12 of computation 19 we continue the sliding cycle originating from the buckling switching bifurcation i e a crossing orbit of vector fields f e gu connecting a tangent point of f 2 with the boundary X for decreasing values of a computation 20 PT TY LAB PAR 2 PAR 61 PAR 62 PAR 45 20 2 1 25296E 01 4 50573E 01 2 27843E 00 3 63682E 02 40 3 9 43993E 02 3 80478E 01 1 64804E 01 5 38328E 04 41 UZ 4 9 38293E 02 3 79463E 01 1 67780E 01 2 34665E 12 50 EP 5 5 93235E 02 3 28817E 01 4 36469E 01 1 62700E 02 83 detecting a crossing crossing bifurcation zero of test function 5 at label 4 whose forward and backward continuations computations 21 and 22 give the following outputs see curve TCT in Figure 26 PT TY LAB PAR 5 Piso PAR 2 Ue PAR 44 20 6 2 40485E 01 9 81182E 02 1 05289E 02 40 7 3 36752E 01 1 08317E 01 1 41588E 02 60 8 4 55907E 01 1 21933E 01 1 48638E 02 78 UZ 9 5
35. 5 9 for jo n 2 problem 5 10 for f 0 f n 2 problem 5 19 n 2 37 t f 3 tangent point of f H5 u 1 o f 9 u 1 a switch to problem 5 2 for f m 3 problem 5 3 for f n 22 problem 5 21 for f n 2 t f 4 tangent point of f at u 1 H5 u 1 o fP u 1 o switch to problem 5 2 for f n 3 problem 5 3 for riu n 2 problem 5 7 for riu n 2 problem 5 23 for f fO n 2 problem 5 25 for fO t f 5 boundary equilibrium of fC j switch to problem 5 2 for f problem 5 3 for fC problem 5 4 for f problem 5 16 for f problem 5 22 for f 0 f n 2 din n 3 2 t f 6 boundary equilibrium of f at v 1 switch to problem 5 2 for f O n 3 problem 5 3 for fO n 2 problem 5 4 for f problem 5 16 for f t f 7 singular sliding point at v 1 H4 v 1 a f v 1 a switch to problem 5 2 for f and for f n 3 problem 5 3 for f and for f n 2 problem 5 19 n 2 problem 5 22 for f 0 f n 2 problem 5 24 for f 0 f n 2 the initial value of v 1 must be specified in PAR 67 n PAR 67 2n 1 inthe user subroutine SCSTPNT 5 27 An orbit of vector field f connecting a tangent point of f to a saddle I I ooooooco ti 8 g gt e IL c D g gt Sa em e x Sete 31 y gt 0 38 Cai v 0 B Figure 11 Boundary value problems corresponding to A Subs
36. 5 and 26 zeros of test functions 3 and 4 at labels 7 and 5 in the forward and backward computations respectively Continuing again the sliding cycle of computation 27 starting from the user output point at label 4 for decreasing values of gt computation 30 PT TY LAB PAR 2 PAR 61 PAR 44 10 ng 3 27180E 01 3 79075E 01 6 52592E 02 20 12 3 00261E 01 4 23003E 01 4 28902E 02 30 L3 2 51521E 01 5 29961E 01 1 36618E 02 35 UZ 14 2 23157E 01 6 12665E 01 9 2019 5E 13 40 155 1 84635E 01 7 60351E 01 1 78660E 02 50 EP 16 1 40280E 01 1 01650E 02 4 03611E 02 we detect a buckling switching bifurcation zero of test function 4 at label 14 whose forward and backward continuations computations 31 and 32 give the following outputs see curve T T D in Fig ure 26 PT TY LAB PAR 5 PAR 2 PAR 61 PAR 43 20 17 1 57430E 00 2 09054E 01 7 00127E 01 4 55407E 01 40 18 1 83312E 00 1 29146E 01 1 35543E 02 1 45478E 00 60 19 1 96568E 00 3 64351E 02 4 99926E 02 9 28855E 00 80 20 1 98489E 00 6 36240E 03 2 86109E 03 6 27551E 01 100 EP 21 1 98719E 00 2 57161E 03 7 07785E 03 1 58493E 02 PT TY LAB PAR 5 PAR 2 PAR 61 PAR 43 40 17 1 35673E 00 2 39149E 01 4 95065E 01 2 27234E 01 68 UZ 18 9 92599E 01 2 19545E 01 3 62614E 01 2 91412E 11 80 19 4 65709E 01 1 16683E 01 3 01776E 01 1 40896E 01 120 20 2 42453E 02 6 36225E 03 2 79175E 01 1 42195E 02 160 21 1 42387E 03 3 74265E 04 2 78305E 01 8 5651
37. 50830E 01 1 36213E 01 1 81976E 11 80 10 5 57225E 01 1 37788E 01 3 09460E 03 OF OEP a EI 5 66527E 01 1 42303E 01 1 32836E 02 OO EP 12 5 42426E 01 1 46714E 01 2 38698E 02 PT TY LAB PAR 5 e PAR 2 PAR 44 20 6 1 46530E 01 8 78584E 02 6 48039E 03 40 7 1 05962E 02 5 87733E 02 4 70094E 04 60 8 1 56275E 04 3 79024E 02 6 93305E 06 80 9 1 23652E 06 2 68820E 02 5 48539E 08 100 EP 10 6 39789E 09 2 04284E 02 2 83807E 10 where label 9 of the forward computation corresponds to the codimension 2 bifurcation points D of Figure 26 zero of test function 4 We now restart from the supercritical Hopf bifurcation of vector field f label 4 of computa tion 1 and continue the stable limit cycle originating from it for decreasing values of a2 computa tion 23 PT TY LAB PAR 2 Pe PERIOD PAR 41 10 9 7 77537E 01 3 38224E 01 7 30472E 01 20 10 7 58064E 01 3 45201E 01 4 5 10637E 01 30 II 6 63927E 01 3 86395E 01 3 22890E 01 40 12 4 91392 01 5 07445E 01 3 76301E 01 49 UZ 13 3 30000E 01 7 18753E 01 5 59307E 01 50 EP 14 3 14964E 01 7 47881E 01 5 79500E O1 and starting from the user output at label 13 for decreasing values of a5 computation 24 PT TY LAB PAR 5 Sue PERIOD PAR 41 10 15 2 48900E 00 7 18753E 01 4 82933E 02 11 UZ 16 2 44071E 00 7 18753E 01 4 63707E 08 14 EP 17 5 59294E 01 7 18753E 01 3 00000E 00 thus detecting at label 16 a touch
38. 7 16653E 04 200 EP 29 9 77730E O1 2 36848E 01 2 57838E 02 4 89891E 04 PT TY LAB PAR 5 PAR 2 T PAR 44 PAR 45 20 23 8 59208E 01 1 74863E 01 2 32613E 02 5 70129E 02 40 24 7 58033E 01 1 66882E 01 1 63950E 02 9 42467E 02 60 25 6 74568E 01 1 63044E 01 4 81413E 05 1 18639E 01 61 UZ 26 6 74401E 01 1 63037E 01 1 01363E 09 1 18681E 01 80 27 6 10048E 01 1 60579E 01 2 55614E 02 1 32471E 01 100 EP 28 5 64458E 01 1 58824E 01 5 34391E 02 1 39407E 01 and identify the codimension 2 bifurcation points E J and F of Figure 26 zeros of test functions 4 5 and 4 at labels 3 26 and 26 of computations 38 39 and 40 respectively 8 ACKNOWLEDGMENTS We are grateful to S Rinaldi for useful suggestions and to A Gragnani for her help in the analysis of the ecological example We are also thankful to E Doedel and O De Feo for their assistance on AUTO97 technicalities The research was partially supported by MIUR under project FIRB RBNE01CW3M 88 LITERATURE CITED Bernardo di M Budd C amp Champneys A 2001 Unified framework for the analysis of grazing and border collision in piecewise smooth systems Physical Review Letters 86 2554 2556 Bernardo di M Champneys A amp Budd C 1998a Grazing skipping and sliding analysis of the nonsmooth dynamics of the DC DC buck converter Nonlinearity 11 858 890 Bernardo di M Feigin M I Hogan S amp Homer M 1999 Loca
39. D SCICND and SCFOPT the constants file sc lt name gt and possibly the data file lt name gt dat which specifies numerically the starting solution for boundary value problems lt name gt is a user selected name The user supplied subroutines may be regarded as higher level input routines that are called by the standard AUTO97 routines contained in the SLIDECONT library The purpose of the user supplied sub routines is the following SCFUNC SCBOUND SCSTPNT Fortran prototype SUBROUTINE SCFUNC SCNDIM X PAR SCIDIFF FI DFIDX DFIDP DFIDXDX DFIDXDP I defines the vector fields fO a a and fO a a on input SCNDIM X k PAR k SCIDIFF and I contain the vector fields dimension n the actual state and parameter values x and a the order up to which derivatives must be analytically specified and the index of the vector field of interest respec tively as in AUTO97 only parameters PAR 1 PAR 9 can be used by the user on output FI k contains the k th component of f while DFIDX k p DFIDP k p DF IDXDX k p q and DFIDXDP k p q contain the derivatives of f if provided with respect to p Qp Cou q and p amp q respectively Fortran prototype SUBROUTINE SCBOUND SCNDIM X PAR SCIDIFF H DHDX DHDP DHDXDX DHDXDP defines the discontinuity boundary function H z o Input and output argu ments are analogous to those of SCFUNC Fortran prototype SUBROUTINE SCSTPNT
40. IPS SCIDIFF NDIM U 1 U NDIM U1 U2 V1 V2 16 F k k 1 2 F k k 3 4 NICP ICP 1 ICP NICP t f 1 boundary equilibrium of fC switch to t f 2 tangent point of fO switch to t f 3 tangent point of f switch to t f 4 tangent point of fU switch to t f 5 tangent point of fC switch to t f 6 tangent point of fU switch to t f 7 pseudo equilibrium at v 1 switch to 5 9 Anorbit of vector field f u Tf H u o Af A Af 9 u 0 17 u H u T u a Tif v a 3 1 61 T 62 Hz u 0 o f u 0 4 problem 5 4 for ff i problem 5 5 problem 5 16 for f Hi u 0 o f u 0 a problem 5 3 for f problem 5 5 problem 5 9 for f problem 5 10 for f 0 f problem 5 19 GL u 1 a f u 1 o problem 5 3 for f problem 5 21 for f Hz u 1 a f u 1 a problem 5 3 for f problem 5 7 for f problem 5 23 for f fO Hz v 1 o f v 1 o problem 5 3 for f i problem 5 22 for f 0 f Ha v 1 a f v 1 a problem 5 3 for f G problem 5 24 for f fO Hz v 1 a g v 1 o problem 5 5 problem 5 26 for f fO connecting a pseudo equilibrium with the boundary 5 Ou a a a 0 0 Aj 1 jd ocooooco 15 Figure 4 Boundary value problems corresponding to A Subsection 5 9 B Subsection 5 10 Ma Biss Bma 3 07 A5
41. USER GUIDE TO SLIDECONT 2 0 Fabio Dercole and Yuri A Kuznetsov October 20 2005 Department of Electronics and Information Politecnico di Milano Italy dercole elet polimi it Department of Mathematics Utrecht University The Netherlands kuznetsov8math uu nl ABSTRACT SLIDECONT an AUTO97 driver for sliding bifurcation analysis of discontinuous piecewise smooth au tonomous systems known as Filippov systems is described in detail Sliding bifurcations are those in which some sliding on the discontinuity boundary is critically involved The software allows for detec tion and continuation of codimension 1 sliding bifurcations as well as detection of some codimension 2 singularities with special attention to planar systems n 2 Some bifurcations are also supported for n dimensional systems This document gives a brief introduction to Filippov systems describes the structure of SLIDECONT and all computations supported by SLIDECONT 2 0 provides a user guide and describes several tutorial examples which are distributed together with the source code of SLIDECONT 2 0 Key words amp Phrases Sliding bifurcations discontinuous piecewise smooth systems Filippov systems continuation techniques AUTO97 1 INTRODUCTION SLIDECONT version 2 0 is a suite of routines accompanying AUTO97 Doedel amp Kern vez 1986 Doedel et al 1997 which allow one to perform bifurcation analysis of generic discontinuous piecewise smooth au
42. a file into the starting solution file 13 save the starting solution file get the starting solution file get the starting solution file convert the data file into the starting solution file 16 save the starting solution file C1 6aq hppe get the starting solution file 15 q hppc get the starting solution file 15 q hppe get the starting solution file dat 20 hppc dat get the data file convert the data file into the starting solution file 20 save the starting solution file convert the data file into the starting solution file save the starting solution file pc get the starting solution file pc cp q dat 25 q hp get the starting solution file convert the data file into the starting solution file save the starting solution file Cp q dat 28 q hppc get the starting solution file cp q hppc 27 q hppc get the starting solution file Table 4 continue TI scdat hppc convert the data file into the starting solution file E cp q hppe q dat 31 save the starting solution file Cp q dat 31 q hppc get the starting solution file 33 cp hppc dat 33 hppc dat get the data file H scdat hppc convert the data file into the starting solution file cp q hppc q dat 33 save the starting solution file C pc 33 q hppc get the starting solution file p gq hppc 2 i cp q hppc 33 q hppc get the starting solution file 36 cp q hppc 35 q hppc get the starting solution file cp hppc dat 37 hppc dat get the data file scdat hppc conve
43. ails The analysis is performed with respect to o and as for the following other parameter values a 0 3556 a3 0 0444 a 0 2067 The results are shown in Figures 25 and 26 Q2 Figure 25 Bifurcation diagram of model 5 45 48 in the a5 2 plane Bifurcation curves BEQ boundary equilibrium of vector field f 2 PSN pseudo saddle node bifurcation TCH touching grazing bifurcation of vector field U TCT crossing orbit of vector fields f f connecting two tangent points of fU TTD orbit of vector field f connecting a tangent point of fU with a tangent point of f TGP orbit of vector field f connecting a tangent point of f with a pseudo equilibrium H Hopf bifurcation of vector field f Points A and B are codimension 2 bifurcation points detected by SLIDECONT Notice that in order to avoid numerical problems when x1 or x2 are very small new state variables 21 22 with z In z i 1 2 are introduced To execute all prepared computations simply enter make in the directory 8C DIR examples hppc In the following we execute all computations separately by means of SLIDECONT commands see Sec tion 6 The list of all commands to be typed is given in Table 4 75 0 L L 0 0 5 1 Q5 Figure 26 Magnified view of the shaded region in Figure 25 Bifurcation curves T C H touching grazing bifurcation of vector field f TCT crossing orbit
44. alue problems corresponding to A Subsection 5 11 B Subsection 5 12 t f 1 boundary equilibrium of fC H u 0 a switch to problem 5 1 problem 5 3 for f problem 5 4 for f problem 5 5 problem 5 7 for f problem 5 16 for f t f 2 branch limit point Rev p arg min Revg switch to problem 5 4 for f branch switching continuation of a limit point bifurcation t f 3 tangent point of f H u 1 o f 9 u 1 a switch to problem 5 3 for f problem 5 27 for f t f 4 tangent point of fU 7 i at u 1 H u 1 a f 9 u 1 a switch to problem 5 3 for f t f 5 pseudo equilibrium at u 1 Hz u 1 o g u 1 a switch to problem 5 5 problem 5 35 for f the initial value of y must be specified in PAR 67 PAR 67 n 1 in the user subroutine SCSTPNT 21 5 12 A crossing orbit of vector fields f fO j 4 i connecting a saddle of f with the boundary X T fO uo T fO vo fy o 2 e w vw w w 1l w y u 0 H u 1 a u 1 w0 H v 1 a v lt 0 18 OXqOgO oo coo mq 51 m4 6 T5 Y1 Y2 W1 W2 V Nd Np 4 10 IPS SCIDIFF NDIM U 1 U NDIM F k k 1 2 F k k 3 4 NICP ICP 1 ICP NICP t f 1 boundary equilibrium of fC switch to t f 2 branch limit point switch to t f 3 tangent point of fC switch to t f 4 tangent point of fU at u 1 switch to t f 5 t
45. angent point of f at v 1 switch to t f 6 tangent point of f at v 1 switch to 4 1 4 U1 U2 U1 V2 Tif u a Tj fy v a 7 01 62 17 67 y1 68 y2 69 w1 70 w2 7Y v H u 0 a problem 5 1 problem 5 3 for f problem 5 4 for f problem 5 5 problem 5 7 for f problem 5 8 for f problem 5 16 for f Rev p argming Revjl problem 5 4 for f branch switching continuation of a limit point bifurcation H0 a f u 1 a problem 5 3 for f problem 5 27 for f Hu a O u 1 problem 5 3 for f problem 5 7 for fO Hz v 1 o f v 1 problem 5 3 for f i Hz v 1 o f P v 1 o problem 5 3 for f j problem 5 28 for f f 22 t f 7 pseudo equilibrium at v 1 Hz v 1 o g v 1 o switch to problem 5 5 problem 5 36 for f f the initial value of y must be specified in PAR 67 PAR 67 n 1 in the user subroutine SCSTPNT 5 13 A crossing cycle 19 ll Il ocoooooco Mad 81 share Pmg DA uso T a 5 d 6T CTD t f 1 tangent point of fU at u 0 H5 u 0 a f u 0 o 23 switch to t f 2 tangent point of f switch to t f 3 tangent point of f switch to t f 4 tangent point of f switch to problem 5 2 for f problem 5 3 for f problem 5 7 for f problem 5 8 for f D n 3 1 n 2 0 f n 2 problem 5 15 for f 0 f
46. apabilities and limitations of SLIDECONT Section 3 and we describe its structure Section 4 as well as the problems it can solve Section 5 Then we present a brief programmer s guide Section 6 including the information on availability of the software and its installation Finally we consider several tutorial examples from mechanics and ecological modelling Section 7 2 PRELIMINARIES We consider a generic Filippov system Filippov 1964 fO a xe Sy f a x So where x R Sj reR H z lt 0 S2 x R H z gt 0 H is a smooth scalar function with non vanishing gradient H x on the discontinuity boundary X x E R H x 0 and f R R i 1 2 are smooth functions For n 2 we denote Ht Ha x Hz Solutions of 1 can be constructed by concatenating standard solutions in S 2 and sliding solutions on X obtained with the Filippov convex method Filippov 1964 1988 Kuznetsov et al 2003 described below Let a z H z f 2 Us a f 2 2 where denotes the standard scalar product in R The crossing set Xe C X is defined by Xa cae vole SO By definition at points in X the orbit of 1 crosses X The sliding set 31 is the complement to X in X Le Lea lees fole 0 5 FY a A Figure 1 Filippov construction Points z X where Hs a f a f a 0 are called singular sliding points At such points eit
47. at u 0 Ai switch to problem 5 2 for f 3 n 3 problem 5 3 for f J n 2 problem 5 4 for f problem 5 16 for f t f 3 singular sliding point at u 0 H5 u 1 o f 9 u 1 a switch to problem 5 2 for f and for f n 3 problem 5 3 for f and for f n 2 problem 5 7 for f n 2 problem 5 8 for f f n 2 problem 5 19 n 2 19 t f 4 tangent point of f H u 1 o f u 1 a switch to problem 5 2 for f n problem 5 3 for f n 22 problem 5 29 for f t f 5 tangent point of fO H u 1 o fV switch to problem 5 2 for f n 3 problem 5 3 for riu n 2 problem 5 7 for riu n 2 problem 5 31 for f f t f 6 tangent point of fC Ha v 1 a f v 1 a switch to problem 5 2 for f m 3 problem 5 3 for fO n 2 problem 5 30 for f 0 f t f 7 tangent point of f H4 v 1 o f v 1 a switch to problem 5 2 for f O n 3 problem 5 3 for riu n 2 problem 5 32 for f 0 f t f 8 pseudo equilibrium at v 1 n 2 Hz v 1 o g v 1 a switch to problem 5 5 problem 5 34 for f 0 f the initial value of u 0 must be specified in PAR 67 PAR 67 n 1 in the user subroutine SCSTPNT 511 An orbit of vector field f connecting a saddle of f with the boundary X Tf u a fO y a i aw Pts wre w w 1 w y u 0 H u 1 a oooo oo 20 Figure 5 Boundary v
48. cting a pseudo equilibrium with a tangent point of f an 0 1 An orbit of vector field f connecting two pseudo equilibria A crossing orbit of vector fields f fU j 4 i connecting two pseudo equilibria P 129 PES An orbit of vector field f connecting a pseudo equilibrium to a saddle Pepi ize 13 with a one dimensional unstable manifold 13 pem A crossing orbit of vector fields f fU j i connecting a pseudo PCS pen equilibrium to a saddle with a one dimensional unstable manifold 132 A sliding cycle with a standard orbit of vector field f A sliding cycle with standard orbits of vector fields f and fU j F i An orbit of vector field f connecting a saddle of f with the bound ary 3 SCBi i35 A crossing orbit of vector fields f fU j i connecting a saddle of f with the boundary X The discontinuity boundary PEO 301 A pseudo equilibrium ANNE RI Table 1 SLIDECONT problem labels types and descriptions 12 i2 PCT4 32 A crossing orbit of vector fields f fU j 4 i connecting a pseudo equilibrium with a tangent point of f 12 12 Si a Di Pi Pi D SLIDECONT problem types are composed of three digits with the following meaning the first digit is either 1 or 2 if the problem refers to vector field f with i 1 or i 2 in the defining system see Section 5 while it is equal to 3 if the problem does not refer to a particular vector field The remaining two digits
49. d in two control parameters Accurate detection of additional local degeneracies along these computations is supported together with switching possibilities between different types of problems see Section 5 for details 4 STRUCTURE OF SLIDECONT In this section the structure of SLIDECONT is presented together with some comments on its imple mentation which is further described in the next two sections SLIDECONT solves through numerical continuation techniques several problems a list of which has been summarized in the previous section The general idea is that SLIDECONT sets up the proper defining equations of the user selected problem in AUTO97 format so that the computation can be performed by means of standard AUTO97 routines This is why SLIDECONT is an AUTO97 driver The driving process and the overall structure of SLIDECONT are illustrated in Figure 2 lt name gt f sc lt name gt lt name gt dat t 21 x2 user supplied AUTOOT constants w Us 3 user subroutines SLIDECONT constants sclib o scprep SLIDECONT SLIDECONT library SLIDECONT preprocessor y preprocessor execution scprob f r lt name gt preprocessing SLIDECONT constants AUTO97 constants autlib o fortran compiler execution AUTO97 library AUTO97 lt name gt exe executable file comp
50. dary value problems corresponding to A Subsection 5 33 B Subsection 5 34 47 n m Nd Nb 2 8 Md Bio Bma 5 T Nis j Bis H3 Te u a 7 41 Io 61 T 63 A 64 A 65 ui 66 u problem 5 3 for f problem 5 4 for f problem 5 16 for f U 1 U NDIM F k k 1 2 NICP ICP 1 ICP NICP t f 1 boundary equilibrium of f switch to t f 2 boundary equilibrium of fU at u 0 switch to t f 3 singular sliding point at u 0 switch to t f 4 boundary equilibrium of fC switch to t f 5 boundary equilibrium of f at u 1 switch to t f 6 singular sliding point at u 1 switch to problem 5 3 for f problem 5 4 for f problem 5 16 for f s u 0 a 9 u 0 a problem 5 3 for f and for f problem 5 7 for f problem 5 19 problem 5 25 for f Hj problem 5 3 for f problem 5 4 for f problem 5 16 for f Hi problem 5 3 for f problem 5 4 for f problem 5 10 for fO fO problem 5 16 for fO problem 5 31 for fO fO I u 1 a f0 u 1 a problem 5 3 for f and for fO problem 5 10 for f fO problem 5 19 problem 5 29 for f problem 5 31 for f f the initial values of u 0 and u 1 must be specified in PAR 67 PAR 6742n 1 in the user subroutine SCSTPNT 48 5 34 A crossing orbit of vector fields f fO j i connecting two pseudo equilibria T f u a Qm T
51. dryf sv dryf k save output to p dryf k q dryf k d dryf k get the constants file get the equations file get the starting solution file run SLIDECONT save output to p dryf k q dryf k d dryf k Table 2 Command list sv is an AUTO97 command 0 5 X2 0 5r 5 En 0 5 0 0 5 1 X1 Figure 16 A sliding cycle of system 41 First we continue the sliding cycle of Fig 16 with respect to o see Fig 17 Data files with the initial solutions to the boundary value problem 20 can be easily obtained by numerical integration of the vector field f and of the Filippov vector field g 3 Computation 1 for decreasing a2 gives 63 PT TY LAB PAR 2 ER PAR 60 PAR 61 PAR 43 10 2 2A LZ5SBIE07 Qu 9 62352E 01 5 49895E 00 4 81176E 01 20 3 4 16217E 03 3 31482E 01 5 96249E 00 1 65741E 01 24 LP 4 1 46358E 15 7 99840E 09 6 28319E 00 3 99920E 09 30 5 4 56498E 02 1 57261E 00 J 25355ET U0 7 86307E 01 40 6 1 36890E 01 8 03478E 00 7 75782E 00 4 01739E 00 50 EP 7 1 80686E 01 1 74344E 01 7 97053E 00 8 71721E 00 1 T T T S 0 5 0 X2 0 5r 1 9 4 1 0 5 0 0 5 1 X1 Figure 17 A family of sliding cycles of system 41 The right thick solution corresponds to a grazing bifurcation at a2 0 the left thick solution corresponds to a switching bifurcation at a2 0 0557 The solution family ends at a touching grazing bifu
52. e collides with the stable inner crossing cycle born at the C C bifrcation at a5 and disappears via the fold bifurcation so that no periodic motion is present above the critical parameter value al portrait 4 in Figure 18 This completes construction of the one parameter bifurcation diagram of 41 7 2 Forced dry friction oscillations This example illustrates the two parameter continuation of a touching grazing cycle in a 4 dimensional Filippov system Following Yoshitake amp Sueoka 2002 and Bernardo di et al 2003 consider a dry friction oscillator 67 Figure 20 A family of crossing cycles of 41 The thick solutions correspond to the crossing bifurcation at as 0 1023 and the fold bifurcation of crossing cycles at al 0 1044 described by the equation z ay sgn 1 1 ag 1 31 ag 1 ii ag cos ast 42 Here positive constants 0 2 a3 are the coefficients of the kinematic friction characteristics a4 is the amplitude of the forcing and o is its angular frequency Since cos a5t is the x3 component of a stable periodic solution to the planar system 2 n2 t3 3 504 313 11 TEN 2 2 LA 0523 4 z4 x3 24 68 equation 42 is equivalent to a 4 dimensional Filippov system 5 with T2 an 3 f03 s i 213 01 a2 1 23 a3 1 2 A423 43 43 A504 z3 22 z O43 24 wars 73 and H x a
53. ection 5 27 B Subsection 5 28 ICE ICP d LCP NICP 1 t f 1 boundary equilibrium of f switch to t f 2 tangent point of fO switch to t f 3 boundary equilibrium of fC switch to t f 4 branch limit point switch to 7 in 67 y1 68 y2 69 w1 70 w2 7Y v Hz u 0 a f u 0 a problem 5 4 for f problem 5 5 problem 5 16 for f Hs u 0 a f 9 u 0 a problem 5 3 for f problem 5 5 problem 5 19 problem 5 35 for f H u 1 a problem 5 1 problem 5 3 for f problem 5 4 for f problem 5 5 problem 5 7 for f problem 5 16 for f problem 5 21 for f problem 5 25 for f Rev p arg min Revg problem 5 4 for f branch switching continuation of a limit point bifurcation the initial value of y must be specified in PAR 67 n PAR 67 2n 1 in the user subroutine SCSTPNT 5 28 A crossing orbit of vector fields f f j i connecting a tangent point of f to a saddle of f 32 v 0 Il oOoooooooooo U 1 U NDIM F k k 1 2 F k k 3 4 NICP ICP 1 ICP NICP 8 1 Io 61 T 67 y1 68 y2 69 w1 70 w2 71 v t f 1 boundary equilibrium of f Hz u 0 a f u 0 a switch to problem 5 4 for f problem 5 5 problem 5 16 for f problem 5 35 for fO t f 2 tangent point of f at u 0 H5 u 0 o f u 0 o switch to problem 5 3 for f problem 5 5 problem 5 9 for f problem 5 19 probl
54. ed in Figure 23 70 The solution at label 13 is close to a touching grazing solution with period i that is traced twice The continued touching grazing cycle undergoes a period halving bifurcation that is detected as a branch point BP In other words the periodically forced dry friction oscillator exhibits a codimension Ar 2 bifurcation when a nonhyperbolic an cycle touches the discontinuity boundary at Ay a4 a5 1 32 2 55 Notice also two flip PD bifurcations of the Sn periodic touching grazing cycle 5 at labels 3 and 10 These are another codimension 2 flip garzing points in the system We skip their analysis Figure 22 A family of 82 periodic touching grazing cycles terminating in a period halving bifurcation at a4 05 1 3208 2 5470 The thick orbit is traced twice To continue the touching grazing bifurcation of the 27 cycle the data file fdfo either constructed manually from the output at label 13 or obtained by simulations continuation of the touching grazing PT TY LAB PAR 4 5 BD 2 1 32089 30 3 1 156311 60 4 9 61275E 90 5 7 82608E 120 6 6 22837E Ed Ed Dd Dd Fl f i t t 4 a PRPrPRPP TU 5 MAX U 2 00000E 00000E 00000E 00000E 00000E ry a ry 00 54 O00 24 oe 0 side 71 NONO OPNS IO PD PAR 5 54701 E 44812E 318581 189481 cycle gives computa
55. em 5 36 for fO fo t f 3 tangent point of fC H u 1 o f 9 u 1 a switch to problem 5 3 for f problem 5 21 for f 40 t f 4 tangent point of f at u 1 H u 1 o f u 1 o switch to problem 5 3 for f problem 5 23 for f 0 f problem 5 27 for fO t f 5 boundary equilibrium of fC H v 1 a switch to problem 5 1 problem 5 3 for fO problem 5 4 for fO problem 5 5 problem 5 8 for fe ro problem 5 16 for f problem 5 24 for f 0 f problem 5 26 for f e pO t f 6 branch limit point Rer p arg min Revg switch to problem 5 4 for f branch switching continuation of a limit point bifurcation the initial value of y must be specified in PAR 67 n PAR 67 2n 1 in the user subroutine SCSTPNT 5 29 An orbit of vector field f Q connecting a pseudo equilibrium with a tangent point of f 33 Figure 12 Boundary value problems corresponding to A Subsection 5 29 B Subsection 5 30 41 U 1 U NDIM Ui Un TE NICP ICP 1 ICP NICP t f 1 boundary equilibrium of f j switch to problem 5 2 for f n 3 problem 5 3 for f n 2 problem 5 4 for f problem 5 16 for f t f 2 boundary equilibrium of f at u 0 Ai switch to problem 5 2 for f O n 3 problem 5 3 for f n 22 problem 5 4 for f problem 5 16 for f t f 3 singular sliding point at u 0 H4 u 1 a f D a switch to problem 5 2 for f and
56. er make in the corresponding example directory It is also possible to execute all computations separately by means of SLIDECONT commands see Tables 2 4 Some differences in the resulting screen outputs are to be expected on different machines 71 A simple dry friction oscillator Consider a linear dumped oscillator with dry friction proposed by Tondl 1970 t z1 aya3sgn a5 t a204 a5 t 41 This defines a piecewise linear Filippov system 5 with x9 1 H x a 2 as S x R 2 a5 lt 0 So x R 2 a5 gt 0 and fO fO k 21 0103 aga4 a5 T2 X1 a103 Q amp 204 a5 2 Set Qi 0 1 a2 0 03 o Q4 gt 4 a5 0 5 At these parameter values the system has a stable sliding cycle starting at the visible tangent point T of f and composed of a standard segment in S and a horizontal sliding segment see Fig 16 To execute the prepared computations simply enter make in the directory 8C DIR examples dryf The list of all commands to be typed is reported in Table 2 62 get the constants file get the equations file get the data file convert the data file into the starting solution file save the starting solution file run SLIDECONT cp sc dryf k sc dryf cp dryf f k dryf f cp dryf dat k dryf dat scdat dryf cp q dryf q dat k sc dryf sv dryf k cp sc dryf k sc dryf cp dryf f 0 dryf f cp q dat k 1 q dryf sc
57. eudo equilibrium at u 1 n 2 z u 1 a g u 1 o switch to problem 5 5 problem 5 25 for f 25 5 15 A sliding cycle with standard orbits of vector fields f and f j i u Tfu a 0 Tj f v o 0 Tog s a 0 H u 0 a 0 Hx u 0 a f u 0 0 0 21 H u 1 a 0 v 0 w 1 0 s 0 v 1 0 s 1 u 0 0 Mads Bi y bma 25 Ti Tj To nan BB MENEE FU E inkl dm t f 1 boundary equilibrium of f at u 0 Hz u 0 o f u 0 n 2 switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 2 tangent point of f H5 u 0 o f u 0 o switch to problem 5 2 for f n 3 problem 5 3 for f 9 n 22 problem 5 5 n 2 problem 5 9 for f n22 problem 5 10 for fO fO n 22 problem 5 19 n 2 t f 3 tangent point of f H4 u 1 o f u 1 a switch to problem 5 2 for f n 3 problem 5 3 for jo n 2 problem 5 21 for fO n22 t f 4 tangent point of f H4 u 1 a f 9 u 1 a switch to problem 5 2 for f n 3 problem 5 3 for riu n 2 problem 5 7 for f n 2 problem 5 23 for f fO n 2 t f 5 tangent point of f at v 1 H4 v 1 a f v 1 a 26 switch to problem 5 2 for f n 3 problem 5 3 for f n 22 problem 5 22 for f 9 fO n 2 t f 6 tangent point of f at 1 H4 v 1 o f v 1 a switch to problem
58. f problem 5 16 for f t f 2 boundary equilibrium of f at u 0 Ai switch to problem 5 2 for f n 3 problem 5 3 for f 3 n 2 problem 5 4 for f problem 5 16 for f t f 3 singular sliding point at u 0 H5 u 1 a f 9 u 1 a switch to problem 5 2 for f and for f n 3 problem 5 3 for f and for f n 2 problem 5 7 for fO n 2 problem 5 8 for f 9 fO n 2 problem 5 19 n 2 problem 5 23 for fO fO n 22 t f 4 boundary equilibrium of f at u 1 Hz u 1 o f u 1 a n 2 switch to problem 5 4 for f problem 5 5 problem 5 16 for f problem 5 33 for f t f 5 tangent point of fC Hs u 1 o f 9 u 1 a switch to problem 5 2 for f n 3 problem 5 3 for fO n 2 problem 5 5 n 2 problem 5 19 n 2 problem 5 29 for f the initial value of u 0 must be specified in PAR 67 PAR 67 n 1 in the user subroutine SCSTPNT 45 5 32 A crossing orbit of vector fields f fU j F i connecting a pseudo equilibrium with a tangent point of f G T f u a 0 Tj f P v o 0 H u 0 0 0 Af u 0 0 AFM u 0 a 0 MN Az 1 0 36 B u 1 d 0 u 1 v 0 0 H v1 o 0 Hz v 1 f v 1 e 0 Ma B1 Bmg 4 Ti Tj i Aj Dn In 5 4 1 2n Uls Uns Ulysse Un Tfi usa TP a 6 d Is ONT GT 8800 640 t f 1 boundary equilibrium of f Aj switch to problem 5 2 for
59. f problem 5 3 for f problem 5 4 for f problem 5 16 for t f 2 boundary equilibrium of f at u 0 Ai switch to problem 5 2 for f J n 3 problem 5 3 for f 3 n 2 problem 5 4 for f problem 5 16 for f t f 3 singular sliding point at u 0 H4 u 1 o f u 1 o switch to problem 5 2 for f and for f n 3 problem 5 3 for f and for f n 2 problem 5 7 for f n22 problem 5 8 for f 9 fO n 2 problem 5 19 n 2 problem 5 24 for f 0 f n 2 t f 4 tangent point of f H4 u 1 a f u 1 a switch to problem 5 2 for f n 3 problem 5 3 for fO n 2 problem 5 29 for f t f 5 tangent point of f at u 1 H4 u 1 o f 9 u 1 a n n i 46 switch to problem 5 2 for f O n 3 problem 5 3 for f n 22 problem 5 7 for f n 2 problem 5 21 for fo n 2 problem 5 31 for f 0 f t f 6 boundary equilibrium of fU at v 1 Hz v 1 o fP v 1 a n 2 switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 7 tangent point of f IT v 1 o f v 1 a switch to problem 5 2 for f m 3 problem 5 3 for fO n 2 problem 5 5 n 2 problem 5 19 n 2 problem 5 30 for f f the initial value of u 0 must be specified in PAR 67 PAR 67 n 1 in the user subroutine SCSTPNT 5 33 Anorbit of vector field f connecting two pseudo equilibria 37 Figure 14 Boun
60. form a progressive number Thus SLIDECONT problem types do not overlap with AUTO97 59 ones so that all AUTO97 facilities are accessible through SLIDECONT In fact if IPS is an AUTO97 problem type then the rest of the constants file starting with SCI START is ignored the vector field f is assumed to be the right hand side of the AUTO97 problem and the user supplied subroutines SCBCND SCICND and SCFOP T assume the role of the analogous AUTO97 subroutines By contrast for SLIDECONT problems the constants NBC NINT JAC are ignored while the signif icance of the remaining SLIDECONT constants is described below SCISTART type of initial solution when switching from a boundary value to an algebraic problem see Section 5 1 start from u 0 2 start from u 1 3 start from v 1 SCIDIFF order up to which derivatives are provided SCNPSI SCIPSI number and numerical labels of monitored test functions see Section 5 for the list of the test functions for each problem if the test function k is monitored then its value is assigned to PAR 40 k which is an overspecified parameter that appears in the output thus zeros of the test function k are detected as ze ros of PAR 40 k and classified by AUTO97 as user specified solution points alphabetical code UZ numerical code 4 SCNFIXED SCIFIXED dummy constants SLIDECONT can be run only in command mode through the commands described below However for applications in
61. g from the touching grazing cycle at label 38 we continue the sliding cycle originating from the touching grazing bifurcation i e an orbit of vector field f 2 connecting a tangent point of fO with the boundary 27 for decreasing values of gt computation 15 PT TY LAB PAR 2 ne PAR 61 PAR 44 PAR 45 10 2 1 62194E 01 3 44617E 01 1 31463E 01 1 71899E 02 20 3 1 60021E 01 3 15805E 01 1 00953E 01 1 38230E 02 26 UZ 4 1 54167E 01 2 85774E 01 6 56335E 02 3 45955E 12 30 5 1 42164E 01 2 50970E 01 2 46167E 02 2 65791E 02 34 UZ 6 1 29442E 01 2 27310E 01 8 43333E 11 5 21760E 02 40 7 9 76398E 02 1 99475E 01 2 10734E 02 1 15167E 01 48 UZ 8 8 87051E 02 2 11274E 01 1 50894E 09 1 15456E 01 50 EP 9 8 75947E 02 2 17676E 01 2 76574E 02 9 15012E 02 Label 4 indicates a pseudo homoclinic bifurcation i e the presence of an orbit of vector field f 2 con necting a tangent point of with a pseudo equilibrium zero of test function 5 while labels 6 and 8 indicate buckling switching bifurcations i e the presence of an orbit of vector field f connecting a tangent point of f 2 with a tangent point of f U zeros of test function 4 Starting from the solutions at labels 4 and 6 we continue the corresponding bifurcations forward and backward computations 16 19 thus obtaining the outputs reported below see curves TGP and TT D in Figure 26 82 PT TY
62. header and source files are contained in include and src respectively cmds contains SLIDECONT commands see Subsection 6 4 and the environment file slc env while examples contains a few tutorial examples three of which are described in Section 7 The directories bin and 1ib are empty initially The environment variables AUTO_DIR and SC_DIR must be set to the absolute paths of the AUTO97 and SLIDECONT directories and the directories SAUTODIR cmds SAUTO_DIR bin SC_DIR cmds and SC_DIR bin must be added to the system search path list For this the environment file s1c env should be appropriately edited and sourced before running SLIDECONT For example the following lines tc shell setenv AUTO DIR usr local auto 97 setenv SC_DIR local scratch SlideCont 2 0 set path SAUTO_DIR cmds SAUTO_DIR bin Spath set path SC DIR cmds SC DIR bin Spath set up the necessary paths to both AUTO97 and SLIDECONT under the Unix shell csh assuming that AUTO97 is installed in usr 10cal auto 97and SLIDECONT 2 0in local scratch SlideCont 2 0 Then SLIDECONT should be compiled by typing make in the SC_DIR src directory 77 is assumed to be the Fortran compiler command name This produces SLIDECONT preprocessor and libraries and places them into bin and 1ib repectively 55 6 4 Running SLIDECONT The user must provide three files The equations file lt name gt f containing the Fortran subroutines SCFUNC SCBOUND SCSTPNT SCPVLS SCBCN
63. her both vectors f x and f 2 a are tangent to X or one of them vanishes while the other is tangent to X or they both vanish The Filippov method see Fig 1 associates the following convex combination g x of the two vectors f a to each nonsingular sliding point x X D dfn X3 OG xs Ms gfe Af Oz 3 996 9 vg c e OG 6 Thus i g x x XM 4 is a smooth system of differential equations in codimension 1 domains of X which are composed of nonsingular sliding points Solutions of this system are called sliding solutions Equilibria of 4 where the vectors f x are transversal to X and anti collinear are called pseudo equilibria of 1 An equilibrium X of 4 where one of the vectors f X vanishes is called a boundary equilibrium In this setting all isolated singular sliding points are equilibria of 4 The boundary of a sliding domain is composed of tangent points T where both vectors f G T are nonzero but one of them is tangent to X i e boundary equilibria and singular sliding points Tangent points are called visible invisible if the orbits of i f x starting from them at time t 0 belong to S S j i for all sufficiently small t 4 0 Orbits of 1 can overlap when sliding Three types of periodic orbits can occur in 1 standard crossing i e passing through both domains S but with no points in 2 and sliding i e with at least one point in 2 Two Filippov system
64. ical integration at az 0 0716 however reveals an unstable crossing cycle Starting from the corresponding data file dryf dat 4 we can continue the crossing cycle see Section 5 13 forward computation 4 PT TY LAB PAR 2 PAR 61 PAR 62 PAR 41 10 2 7 10239E 02 3 81133E 00 2 56183E 00 1 09438E 00 20 3 9 09305E 02 4 13420E 00 2 34613E 00 5 71175E 01 26 LP 4 1 04455E 01 4 73036E 00 2 02413E 00 1 20594E 01 27 UZ 5 1 02312E 01 4 97317E 00 1 90398E 00 1 34521E 08 30 6 1 87623E 02 6 13818E 00 4 96953E 01 6 72345E 01 31 EP 7 4 36612E 03 6 31774E 00 1 22892E 01 8 30773E 01 and backward computation 5 PT TY LAB PAR 2 PAR 61 PAR 62 PAR 41 10 8 6 92846E 02 3 78902E 00 2 57819E 00 1 14855E 00 20 9 5 38503E 02 3 61272E 00 2 71469E 00 1 74921E 00 30 10 3 30625E 02 3 41422E 00 2 88372E 00 3 32320E 00 40 11 1 63250E 02 3 27216E 00 3 01443E 00 7 33706E 00 50 EP 12 8 60323E 03 3 20972E 00 3 07440E 00 1 43668E 01 The result is shown graphically in Figure 20 Label 4 in the forward continuation corresponds to a fold bifurcation of crossing cycles when two such cycles collide and disappear at ol 0 1044 Label 5 corresponds to the already detected crossing bifurcation zero of test function 1 Therefore the unstable crossing cycle exists for all 0 lt ag lt al see portraits 1 SW 2 C C and 3 in Figure 18 Its amplitude grows to infinity as a2 0 This crossing cycl
65. ield f does not depend on as the Hopf bifurcation curve in the a5 2 plane is a straight line see curve H in Figure 25 therefore we skip its numerical computation By inspection of the output file q hppc 1 one can check that after the Hopf bifurcation the equilibrium is an unstable focus Starting from the solution at the user output point label 6 we continue the equilibrium for decreas ing values of a5 computation 2 78 PT TY LAB PAR 5 SR U 1 U 2 PAR 41 1 EP 9 3 00000E 00 3 05586E 00 1 04302E 02 1 98952E 00 10 10 2 67200E 00 3 05586E 00 1 04302E 02 1 66152E 00 20 11 1 67200E 00 3 05586E 00 1 04302E 02 6 61515E 01 271 UZ 12 1 01048E 00 3 05586E 00 1 04302E 02 3 84850E 09 30 13 7 10485E 01 3 05586E 00 1 04302E 02 3 00000E 01 38 EP 14 8 95154E 02 3 05586E 00 1 04302E 02 1 10000E 00 Label 12 indicates a zero of test function 1 i e a boundary equilibrium bifurcation Starting from this solution we continue the bifurcation in the plane o5 a2 forward computation 3 and backward computation 4 thus obtaining the following outputs see curve B EQ in Figure 25 PT TY LAB PAR 5 PAR 2 PAR 41 1 EP 15 1 01048E 00 3 30000E 01 3 59492E 02 10 16 1 04966E 00 3 43488E 01 3 48669E 02 20 T 1 24872E 00 4 12928E 01 2 92948E 02 30 18 1 80786E 00 6 16903E 01 1 29270E 02 36 UZ 19 2 22250E 00 7 78000E 01 4 94230E 11 40 20 2 54049E
66. ing grazing bifurcation zero of test function 1 whose forward and backward continuations computations 25 and 26 are reported below see curve TC H in Figure 25 84 PT TY LAB PAR 5 labs PAR 2 PAR 61 PAR 41 20 2 2 46383E 00 3 47550E 01 6 87607E 01 3 09391E 00 40 3 2 53907E 00 4 08450E 01 5 97995E 01 2 52081E 00 60 4 2 62710E 00 4 95379E 01 5 03757E 01 1 89425E 00 80 LP 5 2 68762E 00 6 21707E 01 4 09735E 01 1 16644E 00 97 UZ 6 2 22250E 00 7 78000E 01 3 38064E 01 5 06351E 08 100 EP 7 1 88932E 00 7 60450E 01 3 44321E 01 2 23574E 01 PT TY LAB PAR 5 lt a PAR 2 PAR 61 PAR 41 20 2 2 40026E 00 3 00123E 01 7 79182E 01 3 67640E 00 40 3 2 18753E 00 1 49210E 01 1 41240E 02 7 83949E 00 60 4 2 03316E 00 3 68003E 02 5 12072E 02 3 29342E 01 80 5 1 99537E 00 6 28300E 03 2 91247E 03 1 95212E 02 100 EP 6 1 99139E 00 2 55826E 03 7 12930E 03 4 80409E 02 where label 6 of the forward computation corresponds to the codimension 2 bifurcation points B of Figure 25 zero of test function 1 After this point the solution has no sense As already done for vector field f we continue the sliding cycle originating from the touching grazing bifurcation label 16 of computation 24 for decreasing values of a5 computation 27 PT TY LAB PAR 5 Rs PAR 41 PAR 45 40 2 2 28039E 00 2 73091E 00 7 65388E 01 80 3 1 81794E 00 1 38427E 00 1 91760E 01 103 UZ 4 1 50000E 00 6 92441E
67. inuation of a boundary value problem one should check if the critical solution satisfies the defining system of some problems not listed among the switching possibilities A common practice in AUTO97 is parameter overspecification namely the number of active pa rameters NICP is allowed to be greater than the number required by the specified problem In such cases the extra activated parameters located at the end of the ICP list are not true continuation parameters but their values appear in the output Overspecified parameters are denoted in the parameter index list ICP by indicating in parenthesis the value at which they are set see Subsections 5 16 and 5 19 Parameter overspecification is also used by SLIDECONT for the implementation of test functions and by the user for defining user test functions see Section 6 so that the actual number of active parameters NICP can be greater than the value specified here Notice that the defining systems for standard equilibrium and cycle continuation see Subsections 5 4 and 5 6 are not reported since they correspond to AUTO97 built in problems Similarly the defin ing systems for the continuation of pseudo saddle node and double tangency bifurcations see Subsec tions 5 17 and 5 18 are not reported since they are implemented indirectly by using the defining systems for pseudo equilibrium and tangent point continuation Subsections 5 5 and 5 3 and enabling AUTO97 limit point continuation ISW 2
68. l analysis of C bifurcations in n dimensional piecewise smooth dynamical systems Chaos Solitons and Fractals 10 1881 1908 Bernardo di M Garofalo F Glielmo L amp Vasca F 1998b Switchings bifurcations and chaos in DC DC converters IEEE Trans Circuits Systems I Fund Theory Appl 45 133 141 Bernardo di M Kowalczyk P amp Nordmark A 2002 Bifurcations of dynamical systems with sliding Derivation of normal form mappings Physica D 11 175 205 Bernardo di M Kowalczyk P amp Nordmark A 2003 Sliding bifurcations A novel mechanism for the sudden onset of chaos in dry friction oscillators Int J Bifurcation and Chaos 13 2935 2948 Dercole F Gragnani A Kuznetsov Yu A amp Rinaldi S 2003 Numerical sliding bifurcation analysis An application to a relay control system JEEE Trans Circuits Systems I Fund Theory Appl 50 1058 1063 Doedel E Champneys A Fairgrieve T Kuznetsov Yu A Sandstede B amp Wang X 1997 AUTO97 Continuation and Bifurcation Software for Ordinary Differential Equations with Hom Cont User s Guide Concordia University Montreal Canada Doedel E amp Kern vez J 1986 AUTO Software for continuation problems in ordinary dif ferential equations with applications Applied Mathematics California Institute of Technology Pasadena CA Feigin M I 1994 Forced Oscillations in Systems with Discontinuous Nonlinearities Nauka Mo
69. lem 5 3 for f problem 5 5 n problem 5 8 for f 3 n 3 J n 2 2 J n 2 problem 5 15 for f fO problem 5 9 for f 9 n 2 problem 5 10 for f f n 2 problem 5 19 n 2 problem 5 23 for f fO n 2 problem 5 25 for f nm 2 problem 5 29 for f O n 2 problem 5 31 for fO fO n 2 t branch AUTO97 branch t f limit point AUTO97 limit point t period doubling AUTO97 period doubling branch switching AUTO97 torus continuation of a torus bifurcation 5 21 An orbit of vector field f Q connecting two tangent points of f Tf u Hz u 0 o f Ha u 1 o f a H u 0 o u 0 a a a A u 1 u 1 ooooco 30 25 A B Figure 8 Boundary value problems corresponding to A Subsection 5 21 B Subsection 5 22 nmm NICP ICP 1 ICP NICP 3 L In 61 T t f 1 boundary equilibrium of f Hi u 0 a f u 0 a switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 2 tangent point of f H u 0 a f 9 u 0 o switch to problem 5 3 for f problem 5 5 problem 5 9 for f problem 5 19 problem 5 29 for f t f 3 boundary equilibrium of fC Hi u 1 o f u 1 a switch to problem 5 4 for f problem 5 5 problem 5 16 for f problem 5 25 for f t f 4 tangent point of f Hs u 1 o fP u 1 o switch to problem 5 3 for f problem 5 5 pr
70. m 5 19 n 2 t f 3 boundary equilibrium of f switch to problem 5 2 for f n problem 5 3 for f n problem 5 4 for f problem 5 16 for f t f 4 boundary equilibrium of f at u 1 switch to problem 5 2 for f O n 3 problem 5 3 for f O n 2 problem 5 4 for fO problem 5 8 for f 9 fO n 2 problem 5 16 for f problem 5 23 for f fO n 2 t f 5 singular sliding point at u 1 H5 u 1 o f 9 u 1 a 36 switch to problem 5 2 for f and for f n 3 problem 5 3 for f and for fO n 2 problem 5 8 for f f n 2 problem 5 19 n 2 problem 5 21 for f 6 n 2 problem 5 23 for f fO n 2 the initial value of u 1 must be specified in PAR 67 n PAR 67 2n 1 inthe user subroutine SCSTPNT 5 26 A crossing orbit of vector fields f f j i connecting a tangent point of f with a pseudo equilibrium T f u a 0 d Tf wa 0 H u 0 a 0 Hz u 0 a f u 0 a 0 H u 1 a 0 30 u 1 v 0 0 H w1 a 0 Xf 9 v 1 0 Ajf v 1 0 0 Ait Aj 1 0 Ma B1 1Pma 6 s Ia ICT ONT 6808 OGD t f 1 boundary equilibrium of f at u 0 Hz u 0 o f u 0 a n 2 switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 2 tangent point of f at u 0 H4 u 0 a f 9 u 0 o switch to problem 5 2 for f O n 3 problem 5 3 for fD n 22 problem 5 5 n 2 problem
71. m 5 2 for f and for f n 3 problem 5 3 for f and for fO n 2 problem 5 19 n 2 5 18 A double tangency bifurcation of vector field f H s a Fea t f 1 boundary equilibrium of f Hi x a f a a switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 2 tangent point of fU 7 Z i Hs x a f P x a switch to problem 5 3 for f problem 5 5 problem 5 19 28 5 19 Coinciding tangent points A x a 0 UT x o f x o 0 23 H a o f x o 0 n Md Nd U F m fay His His a fra Hs a f 2 2 2 hy BEART O 3 t f 1 boundary equilibrium of f GITE a o f a a switch to problem 5 4 for f U problem 5 16 for f t f 2 boundary equilibrium of f Hi x a f x a switch to problem 5 4 for f 2 problem 5 16 for t double tangency AUTO97 limit point switch to problem 5 18 for f D or for f 5 20 A touching grazing cycle of vector field f Q Tf u a 0 H u 0 a 0 Hz u 0 a fF u 0 0 0 24 u 0 u 1 0 ma c ag 5 D 6T t f 1 boundary equilibrium of f at u 0 Hz u 0 o f u 0 a n 2 29 switch to t f 2 tangent point of fU switch to t f torus switch to problem 5 4 for f problem 5 5 problem 5 16 for f i H u 0 a f u 0 o problem 5 2 for f prob
72. nt point of f j i T f O u o H u 0 o H u 0 a f u 0 27 I oooococ H u 1 a Hz u 1 a f u 1 a Figure 9 Boundary value problems corresponding to A Subsection 5 23 B Subsection 5 24 NICP ICP 1 ICP NICP 3 L In 61 T t f 1 boundary equilibrium of f switch to roblem 5 4 for f problem 5 5 problem 5 16 for f t f 2 tangent point of f at u 0 H5 u 0 a f 9 u 0 o 33 switch to problem 5 3 for f problem 5 5 problem 5 9 for f problem 5 10 for fO fO problem 5 19 problem 5 31 for fO fO t f 3 boundary equilibrium of f atu 1 Hz u 1 a f u 1 o switch to problem 5 4 for f problem 5 5 problem 5 16 for f problem 5 25 for f t f 4 tangent point of f H4 u 1 o f u 1 a switch to problem 5 3 for f problem 5 5 problem 5 19 problem 5 21 for f problem 5 25 for f 5 24 A crossing orbit of vector fields f fP j F i connecting a tangent point of f with a tangent point of f a TfO u a 0 b Dfwa 0 oe A H4 u 0 a f u 0 o 0 0 0 9 0 0 i u 1 v 0 0 H v 1 a 0 Hz v 1 f v1 0 0 ma Gi n u a 4 Ii D 61T 62005 t f 1 boundary equilibrium of f at u 0 H u 0 a f u 0 o switch to problem 5 4 for f problem 5 5 problem 5 16 for f 34 t f 2 tangent point of f0 H u 0 a
73. o circumstances can the authors be held liable for any deficiency fault or other mishappening with regard to the use or perfor mance of SLIDECONT 6 2 System requirements SLIDECONT requires that AUTO97 is installed under UNIX Please follow AUTO97 documentation to install and test the package A pre defined maximum value SCNDIMX of the user problem dimen sion SCNDIM is set in the header file s1idecont h This maximum affects the run time memory requirements and should not be set to unnecessarily large values The restriction can be changed by editing s1idecont h followed by recompilation Notice that the effective dimension of the defining systems and the number of active parameters depend upon the user problem dimension and they may exceed AUTO97 restrictions on problem size requiring AUTO97 recompilation In particullar AUTO97 should be compiled with NPARX gt 100 in both auto h and con h see AUTO97 documentation If the SLIDECONT maximum user problem dimension or the AUTO97 restrictions on problem size are 54 exceeded in a SLIDECONT run then the computation halts with an error message 6 3 Installation SLIDECONT is freely available for download at http www math uu nl people kuznet cm slidecont tar gz The software can be extracted by running tar xzvf slidecont tar gz in the directory in which SLIDECONT should be installed where the directories bin cmds examples include lib and src are created The SLIDECONT
74. oblem 5 8 for FO fO problem 5 19 problem 5 23 for f e po problem 5 25 for f 31 5 22 A crossing orbit of vector fields f f j i connecting two tangent points of f T f u a 0 Tj f P v o 0 d Pd 0 z u 0 a fY u 0 a 0 Hs u 0 ror E 26 u l v 0 0 H v 1 o 0 H v 1 a f v 1 0 ma fA Bg 2 4 r Ia 6T Tp t f 1 boundary equilibrium of f at u 0 H u 0 f 9 u 0 a switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 2 tangent point of f at u 0 Hz u 0 o f u 0 o switch to problem 5 3 for f problem 5 5 problem 5 9 for f problem 5 10 for f 0 f problem 5 19 problem 5 30 for f 0 f t f 3 tangent point of fC H4 u 1 o f u 1 a switch to problem 5 3 for f problem 5 21 for f t f 4 tangent point of f at u 1 H5 u 1 o fP u 1 a switch to problem 5 3 for f problem 5 7 for f problem 5 23 for fO fO problem 5 23 for f f t f 5 boundary equilibrium of f Hi v 1 o f v 1 a switch to problem 5 4 for f problem 5 5 problem 5 16 for f t f 6 tangent point of f at v 1 H4 v 1 o f v 1 a 32 switch to problem 5 3 for f problem 5 5 problem 5 19 problem 5 24 for f 0 f problem 5 26 for f 0 f 5 23 An orbit of vector field fO connecting a tangent point of f with a tange
75. of vector fields f f connecting two tangent points of f 2 TT Ds orbit of vector field f connecting a tangent point of f 2 with a tangent point of f TCD crossing orbit of vector fields U f 2 connecting a tangent point of f with a tangent point of f 2 TGP orbit of vector field f connecting a tangent point of f with a pseudo equilibrium TC P crossing orbit of vector fields f f 2 connecting a tangent point of f with a pseudo equilibrium H Hopf bifurcation of vector field f 2 Points C I are codimension 2 bifurcation points detected by SLIDECONT 76 EP O y O Q y W c f k hppc C o o opc k O O O 5 o QO Q Q Q gt A Ee O O alja FAN 3 y O Q y hppc pc q dat pc q dat aja aa Ko Q O Q Q RT O Q h hp Q O Q hppc pc q dat pc dat 25 hn hppc q dat o q dat 25 q hp GIOGO KO y O Q y GLO O Q Q y O Q y a lala 5 O Q Q Q gt n s y O O ala J wo aJa O o o Q Q W EA Oo o QA Hn m Q 1 q hp hppc 2 q hp hppc 2 q hp hppc 2 q hp hppc 5 q hp hppc 5 q hp hppc 8 q hp hppc 8 q hp hppc 8 q hp hppc h A h hppc k sc hppc get the constants file af get the equations file run SLIDECONT save output to p hppc k q hppc k d hppc k pc pc pc pc pc pc pc pc pc pc ppc ppc dat get the data file convert the dat
76. planar systems n 2 for fixed parameter values compute a curve of tangent points in three dimensional systems n 3 for fixed parameter values continue a tangent point in planar systems n 2 in one control parameter continue a standard equilibrium in one control parameter continue a pseudo equilibrium in one control parameter continue a standard periodic solution in one control parameter continue a standard orbit possibly crossing the boundary X connecting a tangent point with the boundary X in planar systems n 2 e g the standard part of a sliding cycle in one control parameter continue a standard orbit possibly crossing the boundary X connecting a pseudo equilibrium with the boundary X in one control parameter continue a standard orbit possibly crossing the boundary X connecting a standard saddle with the boundary X in planar systems n 2 in one control parameter continue a crossing periodic solution in one control parameter continue a sliding periodic solution possibly crossing the boundary X in one control parameter continue a boundary equilibrium in two control parameters continue a pseudo saddle node bifurcation in two control parameters continue a double tangency bifurcation in planar systems n 2 in two control parameters continue coinciding tangent points in planar systems n 2 in two control parameters continue a touching grazing bifurcation of periodic solutions
77. r f n 2 nnn oH ean switch to continuation of a Hopf bifurcation t f 6 branch limit point Rev p argmin Revg Imr 0 switch to problem 5 4 for f branch switching continuation of a limit point bifurcation note the saddle y is assumed to have a one dimensional unstable manifold 0 the initial values of u 0 and y must be spec ified in PAR 67 PAR 67 2n 1 in the user subroutine SCSTPNT 5 36 A crossing orbit of vector fields f f j i connecting a pseudo equilibrium to a saddle with a one dimensional unstable manifold 40 v 0 Il cOooooooooocoo Md Oise bma 2n 4 Ti i j basso Yn Wij saga V NUES UC U NDIM F k k 1 5n 52 F k k n 1 2n NICP ICP 1 ICP NICP t f 1 boundary equilibrium of f switch to t f 2 boundary equilibrium of f at u 0 switch to t f 3 singular sliding point at u 0 switch to t f 4 tangent point of f switch to t f 5 tangent point of f at u 1 switch to t f 6 boundary equilibrium of f switch to t f 7 Hopf t f 8 branch limit point switch to 53 2n 6 I1 I2 61 7 63A 64 67 y1 o5 67 n Da 67 nw 67 2n 1 wn 67 2n v problem 5 2 for f n problem 5 3 for f n problem 5 4 for f problem 5 16 for f problem 5 35 for fO i problem 5 2 for f n 3 problem 5 3 for riu n 2 problem 5 4 for f
78. rcation at a2 0 see label 4 where test function 3 vanishes This bifurcation is degenerate since at c 0 the systems in both S4 and S2 become linear oscillators with closed orbits around a103 0 0 4 0 and a a3 0 0 4 0 respectively portrait O in Figure 18 Computation 2 for increasing az gives PT TY LAB PAR 2 e ci PAR 60 PAR 61 PAR 44 10 2 3 28003E 02 1 08725E 00 5 43205E 00 2 56373E 01 19 UZ 3 5 57355E 02 1 60000E 00 5 22189E 00 4 33287E 08 20 4 6 22664E 02 1 75086E 00 5 17563E 00 7 54277E 02 30 5 1 49842E 01 4 47741E 00 4 85572E 00 1 43870E 00 40 6 2 44375E 01 1 14592E 01 4 90712E 00 4 92959E 00 50 EP 7 2 94622E 01 2 07868E 01 5 10207E 00 9 59339E 00 64 Figure 18 Bifurcation diagram of 41 O centers a2 0 1 a sliding cycle with the standard arc in S and a crossing cycle 0 lt ag lt o 0 0557 SW buckling switching bifurcation at a5 2 a sliding cycle with the standard arcs in both S and 5 and a crossing cycle o lt ag lt o CC crossing bifurcation at a5 0 1023 3 two crossing cycles a5 lt o lt al LP limit point of the crossing cycles at al 0 1044 4 no attractors ag gt al 65 Label 3 indicates a buckling switching bifurcation zero of test function 4 At oj 0 0557 an invisible tangent point 75 of f 2 is detected at the end point u 1 see Figure 18 SW Just below this parameter
79. rt the data file into the starting solution file cp q hppc q dat 37 save the starting solution file Cp q dat 37 q hppc get the starting solution file Cp q dat 36 q hppc get the starting solution file Cp q dat 36 q hppc get the starting solution file Table 4 Command list the first last two commands of com putation 1 must precede follow the commands listed for k 2 40 the equations file hppc f k if not present must be replaced with hppc f 0 sv is an AUTO97 command For ag 1 and as 3 top right corner of Figure 25 vector field f 1 has a stable focus at z1 1 9472 29 1 0134 Starting from this solution we continue the equilibrium for decreasing values of a2 computation 1 We get the following output PT TY LAB PAR 2 U 1 U 2 PAR 41 1 EP T 1 00000E 00 1 94720E 00 1 01338E 00 2 45095E 01 10 2 9 64219E 01 1 98363E 00 9 82883E 01 3 27852E 01 20 3 8 28333E 01 2 13554E 00 8 53210E 01 6 52830E 01 23 HB 4 7 78000E 01 2 19823E 00 7 98633E 01 7 77500E 01 30 5 5 75949E 01 2 49893E 00 5 29837E 01 1 30134E 00 39 UZ 6 3 30000E 01 3 05586E 00 1 04302E 02 1 98952E 00 40 7 3 07518bE 01 3 12642E 00 5 67696E 02 2 05519E 00 50 EP 8 1 51267E 01 3 83590E 00 7 43205E 01 2 52441E 00 where label 4 LAB 4 indicates a Hopf bifurcation while label 6 is a user output point at ag 0 33 see constants file sc hppc 1 Since vector f
80. s are called topologically equivalent if there is a homeomorphism R R that maps the discontinuity boundary of one system onto the discontinuity boundary of the other and that maps orbits of one system onto the corresponding orbits of the other preserving the time direction and mapping standard and sliding segments of any orbit onto the corresponding segments of its image Now consider a Filippov system depending on parameters fO z o 2 Si a fO z o 2 Sa where x R a R and f i 1 2 are smooth functions of x a while S la x R H z a lt 0 So a x R H z a gt 0 for some smooth function H x a with H z a 4 0 for all x such that H x o 0 System 5 exhibits a bifurcation at a ag if by an arbitrarily small parameter perturbation we get a topologically nonequivalent system All bifurcations of 5 are classified as local or global A local bifurcation can be detected by looking at an arbitrarily small neighborhood of a point in the state space All other bifurcations are called global 3 OVERVIEW SLIDECONT can be used to perform a partial bifurcation analysis of n dimensional Filippov systems 5 and a much more complete bifurcation analysis in the planar case n 2 No more than two control parameters are allowed m lt 2 Specifically using the terminology introduced in Kuznetsov et al 2003 SLIDECONT can compute the boundary X in
81. scow in Russian 89 Filippov A F 1964 Differential equations with discontinuous right hand side In American Math ematical Society Translations Series 2 AMS Ann Arbor pp 199 231 Filippov A F 1988 Differential Equations with Discontinuous Right Hand Sides Kluwer Aca demic Dordrecht Kuznetsov Yu A 1998 Elements of Applied Bifurcation Theory 2nd ed Springer Verlag New York Kuznetsov Yu A amp Levitin V V 1995 1997 CONTENT A multiplatform environment for ana lyzing dynamical systems tp ftp cwi nl pub CONTENT Kuznetsov Yu A Rinaldi S amp Gragnani A 2003 One parameter bifurcations in planar Filip pov systems Int J Bifurcation and Chaos 13 2157 2188 Tondl A 1970 Self Excited Vibrations Monographs and Memoranda No 9 National Research Institute for Machine Design B chovice Yoshitake Y amp Sueoka A 2002 Forced self excited vibration with dry friction In Applied Non linear Dynamics and Chaos of Mechanical Systems with Discontinuities eds Wiercigroch M amp de Kraker B World Scientific Singapore pp 237 259 90
82. ter the corresponding indexes the list of test functions for detecting additional local de generacies with switching possibilities a switch to a problem is denoted by a reference to the problem subsection indicating the vector field s at which the problem is applied and adding a star if the 10 switch is not automatic namely the user must set up the starting solution manually Notice that bound ary conditions FB 1 FB NBC NBC np are not reported since their definition is always clear By x we denote computed points on a standard cycle at all principle and intermediate mesh points v are the eigenvalues of a standard or boundary equilibrium It is worth to remark that for boundary value problems there can be more switching possibilities than those listed which however are not possible with certainty when the corresponding test function vanishes For example continuing for n 2 an orbit of vector field f connecting a tangent point of f 9 with the boundary X see Subsection 5 7 a zero of test function 3 detects a tangent point of f at the right boundary value If this tangent point is the same tangent point present at the left boundary value a condition that does not imply a codimension 2 bifurcation then left and right boundary values coincide and one can switch to the continuation of a standard cycle Subsection 5 6 or of a touching bifurcation Subsection 5 20 Thus when a test function vanishes during the cont
83. tion 2 067721 LI r a DI qp i f i f f t t t Nou oes PAR 61 933771 133071 41986E 73942E 07742E I DAE CEDERE Ty f fi f i f t dat 2 can be The backward 00 00 00 00 3 2 T T T T Q5 Figure 23 Bifurcation curves of 42 TC H touching grazing bifurcation of the 4 cycle P D flip period doubling bifurcation of the i Aj 2 codimension 2 flip grazing points cycle TC H touching grazing bifurcation of the sz cycle 150 7 4 83843E 01 1 00000E 00 1 95931E 00 6 41366E 00 180 8 3 64053E 01 1 00000E 00 1 86614E 00 6 73388E 00 210 9 2 57025E 01 1 00000E 00 1 78451E 00 7 04190E 00 240 10 1 50801E 01 1 00000E 00 1 70534E 00 7 36883E 00 255 PD 11 8 76020E 02 1 00000E 00 1 65844E 00 7 57721E 00 270 12 3 85974E 02 1 00000E 00 1 62087E 00 7 75285E 00 30 0 BP 13 2 17858E 02 1 00000E 00 1 60619E 00 7 82372E 00 while the forward continuation computation 3 results in PT TY LAB PAR 4 MAX U 2 PAR 5 PAR 61 30 2 1 49773E 00 1 00000E 00 2 64225E 00 4 75593E 00 60 3 1 72744E 00 1 00000E 00 2 74919E 00 4 57094E 00 90 4 1 96735E 00 1 00000E 00 2 84171E 00 4 42212E 00 120 5 2 21504E 00 1 00000E 00 2 91835E 00 4 30599E 00 150 6 2 46787E 00 1 00000E 00 2 97864E 00 4 21883E 00 180 7 2 72335E 00 1 00000E 00 3 02272E 00 4 15730E 00 191 BP 8 2 81386E 00 1 00000E 00 3 03447E 00 4 14121E 00 210 9 2 67440E 00 1 00000E 00 2 90491E 00 4
84. tonomous systems Filippov 1964 1988 here called Filippov systems with special attention to planar systems Bifurcation analysis of Filippov systems is important in many applications in various fields of science and engineering Unfortunately the complete catalogue of sliding bifurcations in n dimensional systems is not yet available There is a growing number of interesting results on bifurcations of periodic solutions in specific 3 dimensional and in general n dimensional Filippov systems see for example Feigin 1994 Bernardo di et al 1998a b 1999 2001 and in particular Bernardo di et al 2002 For the case of planar systems n 2 codimension 1 sliding bifurcations have been recently com pletely analyzed Kuznetsov et al 2003 and suitable defining systems have been proposed for the numer ical computation of bifurcation curves with standard continuation techniques We provide their imple mentation in AUTO97 indicating explicitly when they are also applicable to general n dimensional sys tems Moreover defining systems for continuing periodic orbits with a sliding segment in n dimensional systems are implemented The document is organized as follows In the next section we recall the definition of Filippov systems and some of their properties for details and references see Kuznetsov et al 2003 Then we focus on SLIDECONT assuming that the reader is acquainted with AUTO97 In particular we give an overview of the c
85. uces the corresponding AUTO97 constants file r lt name gt and a problem specific Fortran file scprob f containing the definition of all SLIDECONT constants as global variables and an initialization subrou tine which sets these variables at the values specified by the user in the constants file The initialization subroutine is called by the subroutine of the SLIDECONT library first called by AUTO97 so that during the computation SLIDECONT constants are well defined The library contains the standard AUTO97 user subroutines for each problem and additional support routines The compilation of the user and problem specific Fortran files and the linking with the SLIDECONT and AUTO97 libraries produce the executable file lt name gt exe whose execution finally gives the standard AUTO97 output files fort 5 PROBLEM DESCRIPTION Among other things AUTO97 computes curves of solutions to algebraic problems F U u 0 U F R u c Rl which we rewrite as F x a 6 0 rcR FcR ocR g8cR 6 as well as paths of solutions to boundary value problems with non separated boundary conditions U r F U 7 o B 0 U F R o R 8 R r 0 1 7 b U 0 U 1 a 8 0 bc R 8 In both cases control parameters a i 1 2 7 are allowed to vary and the following conditions on dimensions are imposed n Mm mq Nna 1 for equation 6 and m mg ny nq 1 for equations 7 8
86. utation execution computation fort output output files Figure 2 SLIDECONT implementation structure As in AUTO97 the user must provide three files An equations file lt name gt f where lt name gt is a user selected name a constants file sc lt name gt and possibly a data file lt name gt dat see Fig 2 The equations file contains a set of Fortran subroutines specifying model 5 namely the two vector fields f U and f 2 and the scalar function H the starting solution either analytically or numer ically and possible state and parameter user functions to be monitored during continuation Analytical derivatives of fU f 2 and H are required by some problems see Section 5 The constants file specifies all parameters qualifying the AUTO97 continuation algorithms plus some SLIDECONT specific constants including of course the problem type namely a constant indicating the problem to be solved As in AUTO97 problem types are coded by means of integer numbers This coding is done in such a way that SLIDECONT problem types do not overlap with those of AUTO97 so that the SLIDECONT user can access all AUTO97 facilities Finally the data file is required to numerically specify the starting solution of boundary value problems As shown in Figure 2 SLIDECONT is composed of two parts the SLIDECONT preprocessor scprep and the SLIDECONT library sclib o Preprocessing takes the user constants file and prod
87. volving several computations a more flexible approach is to use a program e g make for directing recompilation The commands scprep and scexe are specially useful for this purpose sc Type sc lt name gt to run SLIDECONT Starting data if needed must be in q lt name gt and SLIDECONT constants in sc name This is the simplest way to run SLIDECONT 60 scprep Type scprep lt name gt to run the SLIDECONT preprocessor SLIDECONT constants must be in sc lt name gt The corresponding AUTO97 constants file r name and the problem specific Fortran file scprob f see Section 4 are created scexe Type scexe lt name gt to produce the executable file lt name gt exe see Section 4 SLIDECONT constants must be in sc name The correspond ing AUTO97 constants file r name is also created scdat Type scdat lt name gt to convert the user supplied data file lt name gt dat into the AUTO97 formatted file q lt name gt SLIDECONT constants must be in sc name The command scdat is necessary for each problem whose defining system consists of a boundary value problem to start the computation from a numerically known solution not obtained by previous computations or as a solution of a different problem from which automatic switch is not supported In such cases the user must provide a data text file containing numerical data representing the starting solution of the boundary value problem Each row in the
88. z 26 6 53850E 01 1 00000E 00 2 99377E 13 50 EP 27 4 53636E 01 1 03049E 00 3 04875E 02 detecting a boundary equilibrium of vector field f 2 zero of test function 2 at label 26 The forward and backward continuations of the boundary equilibrium bifurcation computations 9 and 10 give the following outputs see curve B EQ in Figure 25 PT TY LAB PAR 5 ne PAR 2 PAR 41 1 EP 28 6 53850E 01 3 30000E 01 0 00000E 00 20 29 7 76769E 01 3 05643E 01 0 00000E 00 34 LP 30 9 95619E 01 2 08084E 01 7 34674E 02 39 gt LUZ 31 9 66081E 01 1 72243E 01 1 93017E 08 40 32 9 54478E 01 1 65785E 01 1 32372E 02 60 33 2 39419E 01 2 67371E 02 0 00000E 00 80 EP 34 8 85139E 05 9 24969E 06 0 00000E 00 80 PT TY LAB PAR 5 DEC PAR 2 PAR 41 1 EP 28 6 53850E 01 3 30000E 01 0 00000E 00 20 29 5 41730E 01 3 48582E 01 0 00000E 00 40 30 1 12237E 01 4 04090E 01 0 00000E 00 60 EP 31 1 59648E 06 4 16169E 01 0 00000E 00 where label 31 in the forward computation identifies a second boundary Hopf codimension 2 bifurcation see again point H in Figure 26 as a zero of test function 1 Restarting from the solution at label 26 of computation 8 we continue the stable focus of vector field f for decreasing values of z computation 11 PT TY LAB PAR 2 U 1 U 2 PAR 41 1 EP 28 3 30000E 01 2 31998E 01 4 24877E 01 5 73852E 12 10 29 3 25483E 01 2 45781E 01 4 3 87566E 01 2 48564E 02 20 30

USER GUIDE TO SLIDECONT 2.0

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