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1. Software for computing unstable manifolds in delay differential equations Kirk Green Bernd Krauskopf Dirk Roose Report TW 877 December 2003 BA eN SR Katholieke Universiteit Leuven a gt amp Department of Computer Science M N Pie 3 Celestijnenlaan 200A B 3001 Heverlee Belgium KATE A ans Software for computing unstable manifolds in delay differential equations Kirk Green Bernd Krauskopf Dirk Roose Report TW 877 December 2003 Department of Computer Science K U Leuven Abstract This report demonstrates how to compute 1D unstable manifolds in delay differential equations DDEs with discrete fixed delays Specifically using the Matlab continuation package DDE BIFTOOL to compute the necessary starting data we first show how to com pute unstable manifolds of saddle steady states using time integration Secondly we use a recently developed algorithm to compute the unstable manifold of a saddle periodic orbit in a DDE As illustration we consider two DDE models describing semiconductor lasers subject to phase conjugate feedback and conventional optical feedback Department of Engineering Mathematics University of Bristol UK 1 Introduction This report is a user manual for computing unstable manifolds in systems of delay differential equations DDEs with constant delays One can compute 1D unstable manifolds of saddle steady states and saddle periodic orbits with one unstable Floqu2. GMRUN Alternatively execute in Matlab using unix GMRUN As a check we output to screen the values of the parameters the file in which the manifold data will be written and the section value parameter 0 2 5 parameter 1 3 57e 08 parameter 2 1 64e 08 parameter 3 1190 parameter 4 1 4e 12 parameter 5 3 parameter 6 2e 09 parameter 7 0 0651 parameter 8 1 6e 19 1 Unstable manifold of a fixed point 2 Unstable manifold of a saddle periodic orbit Select option from menu 2 pefdemo data manikt2p5int19 dat Section value 7 62 Check correct branches are being computed Continue y or n y To check that one is computing the correct branch of the manifold ten iterations of the local Poincar map are performed The data is stored in the file pcfdemo data manikt2p5int19 dat This can be plotted the result giving an indication as to whether or not you want the manifold computation to continue Typing y the output file will be overwritten with the data computed using the algorithm Typing n the program will exit Assuming we wish to continue the option of displaying screen output will be made Display screen output y or n y 8 7394633e 01 2 6296566e 00 7 6200000e 00 FP 1 1 8146162e 02 8 5604148e 01 2 6326061e 00 7 6200000e 00 ED 2 1 8146162e 02 8 5260698e 01 2 6333790e 00 7 6200000e 00 IOPP 3 1 8646162e 02 8 5208801e 01 2 6334573e 00 7 6200000e 00 Algm 4 1 9646162e 02 8 5109899e
3. Jo NE Pao Gro NG Nw BWP 3 for the evolution of the complex electric field E t and the inversion N t 6 Equations 2 and 3 have S symmetry under the transformation E N cE N where the symmetry group St c C e 1 When computing unstable manifolds of the LK equations we exploit their rotational symmetry and write the complex electric field in a rotating frame of reference E t A t exp ibt 5 resulting in the following system a Go 1 ia N t Ny A t ibA t KA t r exp a C b7 4 A Jo 70N 8 Pato Gro NG Nw AP 5 In this way periodic solutions known as external cavity modes of the COF laser appear as steady states We will compute an unstable manifold of one of these saddle steady states namely cofdemo cp31p5stst mat First we must define the system in C as follows 3 dimension 0 22 tau 2500 subintervals with right hand side 12 include ClassDefinitions h include GlobalVariables h include Prototypes h include lt cmath gt include lt vector gt void Datapt RHS Datapt first Datapt last Lang Kobayashi Equations parO kappa pari Cp par2 bb par3 alpha par4 GMO par5 NO par6 JO par7 gamma0 par8 GammaMO par9 GNO par10 tau state 0 0 5 parameter 4 last state 2 parameter 5 last state 0 parameter 2 parameter 3 0 5 parameter 4 last state 2 parameter 5 last
4. along the saddle periodic orbit and along the unstable eigenfunctions on either side of the orbit 4 The manifold is then computed as a sequence of points of length 7 with headpoints in a suitable section X transverse to the flow The distance between computed points is governed by four accuracy parameters Qmin Ymax Aa min and A max where a denotes the approximate angle between three consecutive points and A is the distance between two successive points on the manifold To reduce the number of bisection steps in a calculation a small parameter is specified To identify and adjust for tangencies to the flow with 4 during a computation the integration time between consecutive points on the manifold is tracked We allow a given difference in the integration time between two such points A computation stops when convergence to a fixed point is detected an end tangency is detected or when a prescribed arclength of the manifold is reached see Ref 7 for full details 7 66 0 8 7 62 3 L z 0 a 7 58 0 8 1 0 1 2 R u Figure 2 Left Saddle periodic orbit A shown in projection onto the E N plane The plane of intersection X E N N 7 62 is indicated by a dashed line the end of the periodic profile is indicated by the large dot Right Stability information 3 1 Computing the starting data We will assume that we have computed a branch of periodic orbits from which we have extracted a saddle periodic orbi
5. Datapt first Datapt last Phase conjugate feedback laser parameter O kt 1 epsilon 2 NO 3 GN 4 taup 5 alpha 6 taue 7 I 8 q double Escale 1 0e 2 double Nscale 1 0e 8 double tauscale 2 0 3 0 1 0e 9 double kappa parameter 0 tau tauscale double power last state 0 Escale last state 0 Escale last state 1 Escale last state 1 Escale double nonlingain 1 0 parameter 1 power double Nsol parameter 2 1 0 parameter 3 parameter 4 double lingain parameter 3 last state 2 Nscale parameter 2 double lingainsol parameter 3 last state 2 Nscale Nsol double stimemission lingain nonlingain state 0 tauscale Escale 0 5 parameter 5 lingainsol last state 1 Escale 0 5 stimemission 1 0 parameter 4 last state 0 Escale kappa first state 0 Escale state 1 tauscale Escale 0 5 parameter 5 lingainsol last state 0 Escale 0 5 stimemission 1 0 parameter 4 last state 1 Escale kappa first state 1 Escale state 2 tauscale Nscale parameter 7 parameter 8 last state 2 Nscale parameter 6 stimemission power The struct last contains the state variable at the present time and the struct first contains the state variable at the delay time 7 in the past The parameters are represented by the vector parameter with the values given in th
6. associated with the Floquet multiplier with modulus greater than fm see Fig 2 right gt gt fm 1 7 7 As the orbit intersects X a number of times in our case 20 times the intersection point for which we want to compute the unstable manifold gt gt intsn 19 The next step is to phase shift the periodic orbit A so that the end point of its profile lies in the plane X at the appropriate intersection point This is done using the function find spo m and the following user defined DDE BIFTOOL extra condition sys_cond m We denote the shifted orbit by C gt gt gt gt function resi condi sys_cond point xx AB NN par kt delta epsilon NO GN taup alpha taue II q Escale Nscale tauscale tau if point kind psol fix endpoint at NN 7 62 resi 1 point profile 3 end 7 62 condi 1 p_axpy 0 point condi 1 profile 3 end 1 else error SYS_COND point is not psol end return method df_mthod psol B C A0O find_spo A sec_coord sec_value tau nn intsn method From the periodic orbit C we extract the initial condition of length 7 with headpoint in the section X using the function convert_data m The resulting initial condition D1 shown in Fig 3 is written to the given file in this case data spokt2p5int19 dat gt gt gt gt gt gt gt gt gt gt gt gt D1 convert_data C sec_coord sec_value tau nn data spokt2p5int19 dat figure 3
7. clf hold on box on axis square plot C profile 1 C profile 3 k LineWidth 1 plot 3 5 3 5 7 62 7 62 k LineWidth 1 plot D1 1 D1 3 k LineWidth 4 plot D1 1 end D1 3 end k MarkerSize 24 7 66 V z NY AX 7 58 Figure 3 Phase shifted saddle periodic orbit C and initial condition D1 in bold shown in projection onto the Ez N plane The unstable eigenfunction associated with the shifted periodic orbit C is obtained with the function find_eigd m This function finds both branches of the local linear unstable manifold of C and manip ulates the data so that the end points of their profiles lie in 4 see Ref 4 for details Again we use convert_data m to extract two further initial conditions of length 7 along this local unstable manifold gt gt E F G1 G2 G2a H1 H2 newvec find_eigd C sec_coord sec_value tau nn fm pert method gt gt I1 convert_data H1 sec_coord sec_value tau nn data e1kt2p5int19 dat gt gt I2 convert_data H2 sec_coord sec_value tau nn data e2kt2p5int19 dat The small value of 6 is too small to see any difference between the saddle periodic orbit C and the unstable eigendirections H1 and H2 on the scale of Fig 3 However the user is encouraged to plot these orbits and use Matlab s zoom function to investigate them We now have the necessary starting data to compute the global 1D unstable manifold
8. dimensional maps can be generalised to the setting of DDEs However with methods such as fundamental domain iteration the distribution of points may be poor 7 Our approach has the advantage that by adapting the point distribution according to the curvature of the trace the amount of points computed are minimised while an accurate representation of the manifold is obtained This report is organised as follows In Section 2 we define the first example system that of a semi conductor laser with phase conjugate feedback PCF and demonstrate how to compute 1D unstable manifolds of saddle steady states In Section 3 we demonstrate how to compute 1D unstable manifolds of saddle periodic orbits in the PCF laser In Section 4 we consider a special case of computing 1D unstable manifolds in a system with rotational symmetry namely in a semiconductor laser with conventional optical feedback COF A list of the files needed for this demonstration are given in Appendix A If you use the algorithm presented here in a publication please give credit with a note such as Figure 1 was produced using the techniques described in Refs 4 7 that is refer specifically to Ref 4 and Ref 7 of the reference list Installation The gzipped tar file ManDde tgz can be downloaded from http www cs kuleuven ac be cwis research twr research public software shtml and unpacked using the command tar xvzf ManDde tgz A directory ManDde will be created c
9. m cofdemo sys_rhs m cofdemo sys_tau m C files AdamsIntegrator cc Bisect cc ClassDefinitions h CopyCDList cc CreateListFromFile cc FindInterval cc FindingCandidate cc GlobalVariables cc GlobalVariables h GrowingManifold cc GrowingManifoldData cc GrowingManifoldHeader cc InterpolateToSection cc Iteration0fPoincarePoint cc ListManipulation cc Main cc Makef ile Parameters cc PhasePortrait cc PoincareStep cc Prototypes h RHS cc SectionCheck cc SuperPoincare cc SystemPar cc 16 Matlab files convert_data m find _eigd m find_spo m pefdemo psol_plots m pefdemo psol_script m pefdemo stst_plots m pefdemo stst_script m cofdemo stst_plots m cofdemo stst_script m References 1 K Engelborghs T Luzyanina and G Samaey DDE BIFTOOL v2 00 a Matlab package for bifurc ation analysis of delay differential equations Technical Report TW 330 Department of Computer Science K U Leuven Belgium 2001 http www cs kuleuven ac be koen delay ddebiftool shtml K Green and B Krauskopf Global bifurcations at the locking boundaries of a semiconductor laser with phase conjugate feedback Phys Rev E 66 016220 2002 K Green B Krauskopf and K Engelborghs Bistability and torus break up in a semiconductor laser with phase conjugate feedback Phys D 173 114 129 2002 K Green B Krauskopf and K Engelborghs One dimensional unstable eigenfunction and manifold computations in delay differenti
10. state 1 parameter 0 cos parameter 1 parameter 2 parameter 10 first state 0 parameter 0 sin parameter 1 parameter 2 parameter 10 first state 1 state 1 0 5 parameter 4 last state 2 parameter 5 last state 1 parameter 3 0 5 parameter 4 last state 2 parameter 5 parameter 2 last state 0 parameter 0 cos parameter 1 parameter 2 parameter 10 first state 1 parameter 0 sin parameter 1 parameter 2 parameter 10 first state 0 state 2 parameter 6 parameter 7 last state 2 parameter 8 parameter 9 last state 2 parameter 5 last state 0 last state 0 last state 1 last state 1 and parameters 25 0 31 5 51 85278770791984 3 5 50 0 5 0 8 0 1 0 0 55 0 05 0 22 After writing the new file RHS cc recompile using make As in Section 2 we use DDE BIFTOOL to compute the unstable eigenvector and starting data We therefore change the current directory to cofdemo and start Matlab gt gt sys_init gt gt load cp3ip5stst mat gt gt lbda cp31pbstst stability 11 1 gt gt AO sys_derilcp3ip5stst x cp3ipbstst parameter 0 gt gt Al sys_deri cp3ip5stst x cp3ipbstst parameter 1 gt gt tau cp31p5stst parameter 11 gt gt Delta A0 A1 exp lbda tau gt gt v el eig Delta gt gt v v 3 13 3 5 3 5 0 0 Figure 5 Two branches of a 1D unstab
11. 01 2 6336061e 00 7 6200000e 00 Algm 5 2 1646162e 02 8 4930729e 01 2 6338749e 00 7 6200000e 00 Algm 6 2 5646162e 02 8 4542921e 01 2 6344525e 00 7 6200000e 00 Algm 7 3 3646162e 02 giving the value of the intersection of the manifold with X also written to pcefdemo data man1kt2p5int19 dat the number of points computed along the manifold and the arclength The algorithm exits after 75 points have been calculated with an arclength of 1 818 after convergence to a fixed point is detected with the message Attractor reached through convergence To compute the other longer branch of the manifold we change the input and output files in GrowingManifoldData dat to pcfdemo data spokt2p5int19 dat InputFile1 pcfdemo data e2kt2p5int19 dat InputFile2 pcfdemo data man2kt2p5int19 dat OutputFile1 and run GMRUN again This time the program breaks after an arclength of 5 5 has been reached with the message 10 2 9 2 3 Figure 4 Both branches of the 1D unstable manifold of the saddle fixed point Prescribed arclength reached Exit y or n If y is selected the program will exit If n is selected the following message is displayed and the maximum arclength can be increased to for example 6 0 n Increase the arclength to 6 We now use Matlab to plot the computed manifold gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt figure 4 clf hold
12. 3 2 Globalising the unstable manifold We now use the algorithm in Ref 7 to compute the global unstable manifold of the saddle periodic orbit C With SystemPar dat and RHS cc defined as in Section 2 the new parameter values are given in the file Parameters dat as 2 5 3 57e 8 1 64e8 1190 0 1 4e 12 3 0 2 0e 9 65 1e 3 1 6e 19 The next step is to edit the file GrowingManifoldData dat containing input and output information and the values of the accuracy parameters outlined in Section 3 These are 1 first_delta initial A 2 epsilon bisection tolerance alpha max Qmax alpha min Qmin delta alpha max Aa max delta alpha min Aa min delta min Amin convergence convergence to fixed point re GOED AO en Ga sec _coord state in which lies 10 sec value value of 11 arclength maximum length of manifold 12 intersections number of intersections of C with X 13 time fudge integration time tolerance see Ref 7 for further details pcfdemo data spokt2p5int19 dat InputFile1 pcfdemo data e1kt2p5int19 dat InputFile2 pcfdemo data manikt2p5int19 dat OutputFile1 5 0e 4 first_delta 0 2 epsilon 0 3 alpha_max 0 2 alpha_min 1 0e 3 delta_alpha_max 1 0e 4 delta_alpha_min 1 0e 2 delta_min 1 0e 4 convergence 2 sec_coord 7 62 sec_value 5 5 arclength 20 intersections 0 3 time_fudge The program can now be executed by typing at the command line while in the directory ManDde
13. al equations J Comput Phys 2004 to appear B Haegeman K Engelborghs D Roose D Pieroux and T Erneux Stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback Phys Rev E 66 046216 2002 T Heil I Fischer W Els fer B Krauskopf K Green and A Gavrielides Delay dynamics of semiconductor lasers with short external cavities Bifurcation scenarios and mechanisms Phys Rev E 67 066214 2003 B Krauskopf and K Green Computing unstable manifolds of periodic orbits in delay differential equations J Comput Phys 186 230 249 2003 B Krauskopf and H M Osinga Growing 1d and quasi 2d unstable manifolds of maps J Comput Phys 146 404 1998 B Krauskopf G H M Van Tartwijk and G R Gray Symmetry properties of lasers subject to optical feedback Opt Commun 177 347 353 2000 17
14. anch of the unstable manifold is clearly seen to spiral down and into a fixed point while the other branch accumulates on a chaotic attractor gt gt figure 5 clf gt gt load data COFici dat gt gt load data COFic2 dat gt gt AC 1 sqrt COFici 1 COFic1 1 COFic1 2 COFic1 2 gt gt B 1 sqrt COFic2 1 COFic2 1 COFic2 2 COFic2 2 gt gt subplot 1 2 1 box on hold on axis square gt gt plot A 1 COFic1 3 b gt gt plot A 1 1 COFic1 1 3 kx MarkerSize 12 LineWidth 2 gt gt axis 0 8 3 5 6 gt gt subplot 1 2 2 box on hold on axis square gt gt plot B 1 COFic2 3 r gt gt plot B 1 1 COFic2 1 3 kx MarkerSize 12 LineWidth 2 gt gt axis 0 8 3 5 6 Acknowledgements My thanks go to Bernd Krauskopf for collaboration in developing the manifold algorithm Jan Sieber for testing the code and Pieter De Ceuninck for helpful discussions Appendix A List of files System definitions GrowingManifoldData dat Parameters dat SystemPar dat 15 pcfdemo Parameters_PCF dat pcfdemo RHS_PCF cc pcfdemo SystemPar_PCF dat pefdemo ktOp6stst mat pefdemo kt2p480psol mat pefdemo sys_cond m pefdemo sys_deri m pefdemo sys_init m pefdemo sys_rhs m pefdemo sys_tau m cofdemo Parameters_COF dat cofdemo RHS_COF cc cofdemo SystemPar_COF dat cofdemo cp31p5stst mat cofdemo sys_cond m cofdemo sys_deri m cofdemo sys_init
15. e file Parameters dat 0 6 3 57e 8 1 64e8 1190 0 1 4e 12 3 0 2 0e 9 65 1e 3 1 6e 19 Copies of SystemPar dat RHS cc and Parameters dat for the PCF laser can be found in the subdir ectory pcfdemo After editing the file RHS cc it is necessary to recompile using the command make Before computing the starting data needed for a manifold computation we first need to define the PCF laser system for use inside DDE BIFTOOL with the following files again given in pcfdemo sys init m sys_rhs m sys _deri m sys_tau m The user must edit the file sys init m in order to add the DDE BIFTOOL routines to their Matlab file space for example ours reads function name dim sys_init name phase conjugate feedback dim 3 path path home kirk ddebiftool path path home kirk ManDde return 2 1 Computing the manifold In order to compute a 1D unstable manifold of a steady state we first need starting data in the direction of the unstable eigenvector along the linear unstable manifold of the steady state This starting data is computed using DDE BIF TOOL Therefore start Matlab in the subdirectory pcfdemo We will assume that we have computed a branch of steady state solutions from which we have extracted a saddle steady state namely ktOp6stst mat A bifurcation analysis involving this steady state can be found in Ref 2 DDE BIFTOOL does not automatically provide the unstable eigenvector of thi
16. et multiplier using the techniques developed by Krauskopf and Green described in detail in Ref 7 Before attempting a manifold computation the user should be familiar with the continuation package DDE BIFTOOL 1 which is used to compute the necessary starting data Specifically we use DDE BIFTOOL to compute saddle steady states and saddle periodic orbits together with their one dimensional unstable eigenfunctions 4 The 1D unstable manifold of a steady state can then be found by time integration However this is not the case for a saddle periodic orbit and hence the need to employ more sophisticated techniques The algorithm for computing 1D unstable manifolds of saddle periodic orbits is generalised from one for finite dimensional maps 8 Using the idea of a Poincar map we grow the unstable manifold of a saddle periodic point of this map corresponding to a saddle periodic orbit of the DDE 7 Note that the unstable manifold of a periodic orbit of a DDE is a two dimensional object in an infinite dimensional phase space whose intersection with the Poincar plane its trace is a one dimensional curve Due to projection from the infinite dimensional phase space this trace may have self intersections The algorithm computes the manifold as a sequence of points where the distance between successive points is governed by the curvature of the trace see Section 3 and Ref 7 We remark that any method for computing unstable manifolds of finite
17. le manifold showing bistability between a fixed point and a chaotic attractor in the COF laser 5 546346744963866e 01 8 230244811552463e 01 1 225197178637979e 01 gt gt dlta 0 01 gt gt ici cp3ipbstst xtdlta v gt gt ic2 cp3ip5stst x dlta v ici 2 488933150761937e 00 8 230244811552464e 03 4 679107519514766e 00 2 500025844251865e 00 8 230244811552464e 03 4 676657125157490e 00 We execute the program GMRUN or in Matlab unix GMRUN as before and output the unstable manifold data to the files cofdemo data COFic1 dat and cofdemo data COFic2 dat For example parameter 0 25 parameter 1 31 5 parameter 2 51 8528 parameter 3 3 5 parameter 4 50 parameter 5 5 parameter 6 8 parameter 7 1 parameter 8 0 55 parameter 9 0 05 parameter 10 0 22 14 1 Unstable manifold of a fixed point 2 Unstable manifold of a saddle periodic orbit Select option from menu 1 Calculating a phase portrait Please enter a file name cofdemo data COFici dat Enter transient time in tau 0 Enter calculation time in tau 50 How many points to skip in output in tau n intervals 5 Please give initial value for state 0 2 488933150761937e 00 Please give initial value for state 1 8 230244811552464e 03 Please give initial value for state 2 4 679107519514766e 00 Taking advantage of the rotational symmetry of the LK equations we plot the intensity A t against N see Fig 5 and Refs 6 9 One br
18. on box on axis square load data manikt2p5int19 dat plot manikt2p5int19 1 manikt2p5int19 2 k LineWidth 1 plot manikt2p5int19 1 manikt2p5int19 2 b MarkerSize 6 plot manikt2p5int19 1 1 manikt2p5int19 1 2 k MarkerSize 15 LineWidth 2 plot manikt2p5int19 end 1 manikt2p5int19 end 2 kx MarkerSize 15 LineWidth 2 load data man2kt2p5int19 dat plot man2kt2p5int19 1 man2kt2p5int19 2 k LineWidth 1 plot man2kt2p5int19 1 man2kt2p5int19 2 r MarkerSize 6 plot man2kt2p5int19 1 1 man2kt2p5int19 1 2 k MarkerSize 15 LineWidth 2 plot man2kt2p5int19 end 1 man2kt2p5int19 end 2 kx MarkerSize 15 LineWidth 2 11 To check the accuracy 7 of these computations change the accuracy parameters to 1 0e 4 first_delta 0 2 epsilon 0 3 alpha_max 0 2 alpha_min 5 0e 4 delta_alpha_max 5 0e 5 delta_alpha_min 5 0e 3 delta_min 5 0e 5 convergence and recompute the manifolds 4 Unstable manifolds of external cavity modes in the COF laser We now demonstrate a special case of computing 1D unstable manifolds in systems with rotational St symmetry 9 Namely the Lang Kobayashi LK equations describing a semiconductor laser subject to conventional optical feedback COF These equations can be written as 6 SE ZM 1 ia NE Ny El KEE 1 exp iC 2 IX
19. ontaining the C subroutines for a manifold computation and in which the algorithm can be compiled using the command make 2 Unstable manifolds of steady states in the PCF laser As the first illustrative example we will consider the following system describing the PCF laser 2 3 EO L Paar Noa 4 cw 2 Bomber 1 dN t I N t 5 Ea COR for the evolution of the complex electric field E t E t iE t and the population inversion N t Nonlinear gain is included as G t Gn N t No 1 P t where P t E t is the intensity of the electric field In what follows we rescale time by the value of the delay 7 and obtain order 1 values of E and N by rescaling E t by 1 0 x 10 and N t by 1 0 x 108 Note that this does not change the dynamics As it requires the storage of large amounts of data the algorithm for the computation of global 1D unstable manifolds has been written in C 7 To set up the PCF system in C we first edit the file SystemPar dat containing the system parameters that is the dimension of the system the size of the delay interval 7 and the number of subintervals used in the integration scheme 3 dimension 1 0 tau 2500 subintervals Secondly the right hand side of equation 1 must be coded into RHS cc as follows include ClassDefinitions h include GlobalVariables h include Prototypes h include lt cmath gt include lt vector gt void Datapt RHS
20. r 6 2e 09 parameter 7 0 0651 parameter 8 1 6e 19 1 Unstable manifold of a fixed point 2 Unstable manifold of a saddle periodic orbit Select option from menu 1 Calculating a phase portrait Please enter a file name pcfdemo data PCFic2 dat Enter transient time in tau 0 Enter calculation time in tau 300 How many points to skip in output in tau n intervals 50 Please give initial value for state 0 1 402293685085163e 00 Please give initial value for state 1 1 209517588934237e 00 Please give initial value for state 2 7 647454996518048e 00 After similarly computing the unstable manifold corresponding to the other initial condition ic1 we plot the results see Fig 1 and stst_plots m gt gt load PCFici dat gt gt load PCFic2 dat gt gt figure 1 clf hold on box on axis square view 50 15 gt gt plot3 PCFic1 1 PCFic1 2 PCFic1 3 r gt gt plot3 PCFici 1 1 PCFici 1 2 PCFici 1 3 r gt gt plot3 PCFic2 1 PCFic2 2 PCFic2 3 b gt gt plot3 PCFic2 1 1 PCFic2 1 2 PCFic2 1 3 b 3 Unstable manifolds of saddle periodic orbits in the PCF laser We now demonstrate how to compute 1D unstable manifolds of a saddle periodic orbit with one unstable Floquet multiplier of the DDE describing the PCF laser introduced in Section 2 To compute a 1D unstable manifold of a saddle periodic orbit we first need as starting data initial conditions of length 7
21. s solution but we can compute it as follows see also stst_script m gt gt sys_init ans Phase conjugate feedback laser gt gt load ktOp6stst mat gt gt lbda ktOp6stst stability 11 1 gt gt AO sys_deri ktOp6stst x ktOp6stst parameter 0 gt gt Al sys_deri ktOp6stst x ktOp6stst parameter 1 gt gt tau ktOp6stst parameter 13 gt gt Delta A0 A1 exp lbda tau gt gt v e eig Delta gt gt vev 1 7 073726928039000e 01 7 068010414291569e 01 7 494085002089528e 03 z 65 Figure 1 Both branches of the 1D unstable manifold of a saddle steady state Two initial conditions along the linear unstable eigenvector are then found by perturbing the saddle steady state along this eigenvector by a distance dlta gt gt dlta 0 01 gt gt ici ktOp6stst x tdlta v gt gt ic2 ktOp6stst x dlta v ici 1 416441138941241e 00 1 195381568105654e 00 7 647305114818007e 00 ic2 1 402293685085163e 00 1 209517588934237e 00 7 647454996518048e 00 The program can now be executed at the command line by typing while in the directory ManDde GMRUN in Matlab unix GMRUN resulting in the following output including as a check the parameter values and required input where we compute the unstable manifold corresponding to the initial condition ic2 parameter 0 0 6 parameter 1 3 57e 08 parameter 2 1 64e 08 parameter 3 1190 parameter 4 1 4e 12 parameter 5 3 paramete
22. t with a single unstable Floquet multiplier namely kt2p500psol mat This file can be found in the subdirectory pcfdemo in which we start Matlab A bifurcation analysis involving this saddle periodic orbit can be found in Ref 3 We first initialise the system and load this orbit into Matlab We call this orbit A it is shown along with its stability information in Fig 2 see psol script m and psol plots m gt gt sys_init gt gt load kt2p500psol mat gt gt A kt2p500psol gt gt figure 2 clf gt gt subplot 1 2 1 box on hold on axis square gt gt plot A profile 1 A profile 3 k LineWidth 1 gt gt plot A profile 1 end A profile 3 end k MarkerSize 15 gt gt plot 3 5 3 5 7 62 7 62 k LineWidth 1 gt gt subplot 1 2 2 box on hold on axis square gt gt p_splot A Now that we have the saddle periodic orbit A we use Matlab to compute the starting data For this we need to define some parameters 1 The state variable as first defined in sys_rhs m this defines the plane of intersection gt gt sec_coord 3 2 The value of X see Fig 2 left gt gt sec_value 7 62 3 The size of the delay 7 gt gt tau 1 0 4 The number of subintervals in a delay interval gt gt nn 2500 5 The perturbation 6 applied to the eigenfunction gt gt pert 0 02 6 A real value fm in the complex plane for which we want to compute the eigenfunction
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