Stability calculations for Piecewise-Smooth Delay Equations

This paper describes a new method for computing the stability of nonsmooth periodic orbits of piecewise-smooth dynamical systems with delay. Stability computations for piecewise-smooth dynamical systems without delay have previously been performed using discontinuity mappings to "correct" the linearized period map. However, this approach is less convenient for systems with delays due to the infinite dimensional nature of the problem. Additional problems arise due to the discontinuity propagation properties of delay differential equations. The method proposed is based around a multi-point boundary value solver, which allows the correct linearized period map to be constructed directly. We present numerical examples showing the rapid convergence of the method and also illustrate its use as part of a numerical bifurcation study.

[1]  Fatihcan M. Atay,et al.  VAN DER POL'S OSCILLATOR UNDER DELAYED FEEDBACK , 1998 .

[2]  Yuri A. Kuznetsov,et al.  SlideCont: An Auto97 driver for bifurcation analysis of Filippov systems , 2005, TOMS.

[3]  Dirk Roose,et al.  COMPUTATION, CONTINUATION AND BIFURCATION ANALYSIS OF PERIODIC SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS , 1997 .

[4]  S. Coombes,et al.  Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  Marian Wiercigroch,et al.  Applied nonlinear dynamics and chaos of mechanical systems with discontinuities , 2000 .

[6]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[7]  R. Seydel.  Practical Bifurcation and Stability Analysis , 1994 .

[8]  Harry Dankowicz,et al.  TC-HAT: A Novel Toolbox for the Continuation of Periodic Trajectories in Hybrid Dynamical Systems , 2008, SIAM J. Appl. Dyn. Syst..

[9]  Harry Dankowicz,et al.  Bifurcation Analysis of a Microactuator Using a New Toolbox for Continuation of Hybrid System Trajectories , 2007 .

[10]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[11]  Michael C. Mackey,et al.  Bifurcation and Bistability in a Model of Hematopoietic Regulation , 2007, SIAM J. Appl. Dyn. Syst..

[12]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[13]  M. Mackey,et al.  Oscillatory modes in a nonlinear second-order differential equation with delay , 1990 .

[14]  Peter Whittle.  Bifurcations and Chaos , 2010 .

[15]  Yuri A. Kuznetsov,et al.  One-Parameter bifurcations in Planar Filippov Systems , 2003, Int. J. Bifurc. Chaos.

[16]  Takashi Matsumoto,et al.  A chaotic attractor from Chua's circuit , 1984 .

[17]  P. Piiroinen.  Recurrent dynamics of nonsmooth systems with application to human gait , 2002 .

[18]  Christopher Edwards,et al.  Sliding mode control : theory and applications , 1998 .

[19]  Willy Govaerts,et al.  Cl_matcont: a continuation toolbox in Matlab , 2003, SAC '03.

[20]  Erik Mosekilde,et al.  Bifurcations and chaos in piecewise-smooth dynamical systems , 2003 .

[21]  Gábor Stépán,et al.  Dynamics of delayed piecewise linear systems , 2003 .

[22]  Tamás Kalmár-Nagy,et al.  Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations , 2001 .

[23]  C. Budd,et al.  Review of ”Piecewise-Smooth Dynamical Systems: Theory and Applications by M. di Bernardo, C. Budd, A. Champneys and P. 2008” , 2020 .

[24]  Koen Engelborghs,et al.  Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations , 2002, Numerische Mathematik.

[25]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[26]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[27]  Tatyana Luzyanina,et al.  Computing Floquet Multipliers for Functional Differential Equations , 2002, Int. J. Bifurc. Chaos.

[28]  G. Samaey,et al.  DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations , 2001 .

[29]  Enric Fossas,et al.  Study of chaos in the buck converter , 1996 .

[30]  Mario di Bernardo,et al.  Complex Dynamics in a Hysteretic Relay Feedback System with Delay , 2007, J. Nonlinear Sci..

[31]  J. Tyson,et al.  Computational Cell Biology , 2010 .

[32]  Vadim I. Utkin,et al.  A control engineer's guide to sliding mode control , 1999, IEEE Trans. Control. Syst. Technol..

[33]  B. Krauskopf,et al.  Control based bifurcation analysis for experiments , 2008 .

[34]  K Pyragas,et al.  Analytical properties and optimization of time-delayed feedback control. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  H. B. Keller,et al.  NUMERICAL ANALYSIS AND CONTROL OF BIFURCATION PROBLEMS (II): BIFURCATION IN INFINITE DIMENSIONS , 1991 .

[36]  Guido Herrmann,et al.  A robust delay adaptation scheme for Pyragas' chaos control method , 2001 .

[37]  Bernd Krauskopf,et al.  Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation , 2006 .

[38]  Lawrence F. Shampine,et al.  A friendly Fortran DDE solver , 2006 .

[39]  S. Doole,et al.  A piece wise linear suspension bridge model: nonlinear dynamics and orbit continuation , 1996 .

[40]  Gábor Stépán,et al.  Modelling nonlinear regenerative effects in metal cutting , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[41]  Dirk Roose,et al.  Numerical bifurcation Analysis of differential equations with State-Dependent Delay , 2001, Int. J. Bifurc. Chaos.

[42]  Wolf Bayer,et al.  Oscillation Types and Bifurcations of a Nonlinear Second-Order Differential-Difference Equation , 1998 .

[43]  Arne Nordmark,et al.  Non-periodic motion caused by grazing incidence in an impact oscillator , 1991 .

[44]  Dirk Roose,et al.  Collocation Methods for the Computation of Periodic Solutions of Delay Differential Equations , 2000, SIAM J. Sci. Comput..

[45]  John R. Terry,et al.  A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. , 2006, Cerebral cortex.