Numerical Bifurcation Analysis

Global analysis of dynamical systems. Festschrift dedicated to floris takens for his 60th birthday Lei-den. (2007) Small divisors and large multi-pliers (Petits diviseurs et grands multiplicateurs). Ann l'institut Fourier 57(2):603–628 Broer HW, Levi M (1995) Geometrical aspects of stability theory for Hill's equations. Problèmes des modules pour les èqua-tions différentielles non linéaires du premier ordre. Publ IHES 5563–164 Martinet J Ramis JP (1983) Classification analytique des équations différentielles non linéaires résonnantes du premier ordre. Glossary Dynamical system A rule for time evolution on a state space. The term system will be used interchangeably. Here a system is a family given by an ordinary differential equation (ODE) depending on parameters. Equilibrium A constant solution of the system, for given parameter values. Limit cycle An isolated periodic solution of the system, for given parameter values. Bifurcation A qualitative change in the dynamics of a dy-namical system produced by changing its parameters. Bifurcation points are the critical parameter combinations at which this happens for arbitrarily small parameter perturbations. Normal form A simplified model system for the analysis of a certain type of bifurcation. Codimension The minimal number of parameters needed to perturb a family of systems in a generic manner. Defining system A set of suitable equations so that the zero set corresponds to a bifurcation of a certain type or to a particular solution of the system. Also called defining function or equation. Continuation A numerical method suited for tracing one-dimensional manifolds, curves (here called branches) of solutions for a defining system while one or more parameters are varied. Test function A function designed to have a regular zero at a bifurcation. During continuation a test function can be monitored to detect bifurcations. Branch switching Several branches of different codimen-sion can emanate from a bifurcation point. Switching from the computation of one branch to an other requires appropriate procedures. Definition of the Subject The theory of dynamical systems studies the behavior of solutions of systems, like nonlinear ordinary differential equations (ODEs), depending upon parameters. Using qualitative methods of bifurcation theory, the behavior of the system is characterized for various parameter combinations. In particular, the catalog of system behaviors showing qualitative differences can be identified, together with the regions in parameter space where the different behaviors occur. Bifurcations delimit such regions. Symbolic and analytical approaches are in general infeasible, but numerical bifurcation analysis is a powerful tool that aids in the understanding of …

[1]  Z. Mei.  Numerical Bifurcation Analysis for Reaction-Diffusion Equations , 2000 .

[2]  Willy Govaerts,et al.  Computation of Periodic Solution Bifurcations in ODEs Using Bordered Systems , 2003, SIAM J. Numer. Anal..

[3]  Yu. A. Kuznetsov,et al.  Numerical Normalization Techniques for All Codim 2 Bifurcations of Equilibria in ODE's , 1999 .

[4]  Yuri A. Kuznetsov,et al.  Continuation of Connecting orbits in 3D-ODES (I): Point-to-Cycle Connections , 2007, Int. J. Bifurc. Chaos.

[5]  Wolf-Jürgen Beyn,et al.  The Numerical Computation of Connecting Orbits in Dynamical Systems , 1990 .

[6]  Willy Govaerts,et al.  Numerical Continuation of Bifurcations of Limit Cycles in MATLAB , 2005, SIAM J. Sci. Comput..

[7]  Dirk Roose,et al.  Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL , 2002, TOMS.

[8]  Alan R. Champneys,et al.  Homoclinic Branch Switching: a Numerical Implementation of Lin's Method , 2003, Int. J. Bifurc. Chaos.

[9]  A. I. Khibnik,et al.  LINLBF: A Program for Continuation and Bifurcation Analysis of Equilibria Up to Codimension Three , 1990 .

[10]  Yu. A. Kuznetsov,et al.  Remarks on interacting Neimark–Sacker bifurcations , 2006 .

[11]  John Guckenheimer,et al.  Numerical Analysis of Dynamical Systems , 1999 .

[12]  Wolf-Jürgen Beyn,et al.  The Numerical Computation of Homoclinic Orbits for Maps , 1997 .

[13]  Willy Govaerts,et al.  Continuation of Homoclinic Orbits in Matlab , 2005, International Conference on Computational Science.

[14]  G. Iooss,et al.  Global Characterization of the Normal Form for a Vector Field Near a Closed Orbit , 1988 .

[15]  Willy Govaerts,et al.  Numerical Periodic Normalization for Codim 1 Bifurcations of Limit Cycles , 2005, SIAM J. Numer. Anal..

[16]  Y. Kuznetsov.  Elements of applied bifurcation theory (2nd ed.) , 1998 .

[17]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and an Introduction to Chaos , 2003 .

[18]  C. Stéphanos.  Sur une extension du calcul des substitutions linéaires , 1900 .

[19]  W. Beyn.  Numerical methods for dynamical systems , 1991 .

[20]  Bernd Krauskopf,et al.  A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits , 2008 .

[21]  Y. Kuznetsov,et al.  New features of the software MatCont for bifurcation analysis of dynamical systems , 2008 .

[22]  R. Russell,et al.  Adaptive Mesh Selection Strategies for Solving Boundary Value Problems , 1978 .

[23]  Willy Govaerts,et al.  Numerical Methods for Two-Parameter Local Bifurcation Analysis of Maps , 2007, SIAM J. Sci. Comput..

[24]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[25]  Eugene L. Allgower,et al.  Numerical continuation methods - an introduction , 1990, Springer series in computational mathematics.

[26]  W. Govaerts,et al.  MIXED BLOCK ELIMINATION FOR LINEAR-SYSTEMS WITH WIDER BORDERS. , 1993 .

[27]  Eusebius J. Doedel,et al.  Lecture Notes on Numerical Analysis of Nonlinear Equations , 2007 .

[28]  J. Guckenheimer,et al.  Defining Functions for Multiple Hopf Bifurcations , 1997 .

[29]  Pierre Gaspard,et al.  Local birth of homoclinic chaos , 1993 .

[30]  E. A. Spiegel,et al.  Amplitude Equations for Systems with Competing Instabilities , 1983 .

[31]  K. Lusty,et al.  Direct Computation of Period Doubling Bifurcation Points of Large-scale Systems of Odes Using a Newton-picard Method , 1998 .

[32]  Gábor Stépán,et al.  Continuation of Bifurcations in Periodic Delay-Differential Equations Using Characteristic Matrices , 2006, SIAM J. Sci. Comput..

[33]  H. Osinga,et al.  Continuation of quasiperiodic invariant tori , 2006 .

[34]  A. Fuller,et al.  Conditions for a matrix to have only characteristic roots with negative real parts , 1968 .

[35]  Willy Govaerts,et al.  Bifurcations of Maps in the Software Package CONTENT , 1999, CASC.

[36]  Àngel Jorba,et al.  Numerical computation of the normal behaviour of invariant curves of n-dimensional maps , 2001 .

[37]  Willy Govaerts,et al.  Numerical methods for bifurcations of dynamical equilibria , 1987 .

[38]  Randy C. Paffenroth,et al.  Elemental Periodic orbits Associated with the libration Points in the Circular Restricted 3-Body Problem , 2007, Int. J. Bifurc. Chaos.

[39]  Yuri A. Kuznetsov,et al.  Continuation of Connecting orbits in 3D-ODES (II) : Cycle-to-Cycle Connections , 2008, Int. J. Bifurc. Chaos.

[40]  J. Guckenheimer,et al.  Computing Hopf Bifurcations I , 1997 .

[41]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[42]  Wolf-Jürgen Beyn,et al.  Numerical analysis of homoclinic orbits emanating from a Takens-Bogdanov point , 1994 .

[43]  P. Deuflhard,et al.  Efficient numerical path following beyond critical points , 1987 .

[44]  Louis A. Romero,et al.  Bifurcation Tracking Algorithms and Software for Large Scale Applications , 2005, Int. J. Bifurc. Chaos.

[45]  P. Coullet,et al.  A simple global characterization for normal forms of singular vector fields , 1987 .

[46]  John Guckenheimer,et al.  A Survey of Methods for Computing (un)Stable Manifolds of Vector Fields , 2005, Int. J. Bifurc. Chaos.

[47]  Frank Schilder,et al.  Fourier methods for quasi‐periodic oscillations , 2006, International Journal for Numerical Methods in Engineering.

[48]  Bard Ermentrout,et al.  Simulating, analyzing, and animating dynamical systems - a guide to XPPAUT for researchers and students , 2002, Software, environments, tools.

[49]  Willy Govaerts,et al.  Numerical Continuation of Branch Points of Equilibria and Periodic orbits , 2005, Int. J. Bifurc. Chaos.

[50]  Michael E. Henderson,et al.  Higher-Dimensional Continuation , 2007 .

[51]  Peter Deuflhard,et al.  Scientific Computing with Ordinary Differential Equations , 2002 .

[52]  Dirk Roose,et al.  A Direct Method for the Computation of Hopf Bifurcation Points , 1985 .

[53]  Björn Sandstede,et al.  A numerical toolbox for homoclinic bifurcation analysis , 1996 .

[54]  W. Govaerts,et al.  Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs , 2008 .

[55]  F. Takens.  Chapter 3 – Local Invariant Manifolds and Normal Forms , 2010 .

[56]  Meiyuan Zhen,et al.  A numerical approximation for the simple bifurcation problems , 1989 .

[57]  Bernd Krauskopf,et al.  Homoclinic bifurcations in a neutral delay model of a transmission line oscillator , 2007 .

[58]  Dirk Roose,et al.  An Adaptive Newton-Picard Algorithm with Subspace Iteration for Computing Periodic Solutions , 1998, SIAM J. Sci. Comput..

[59]  Fabio Dercole.  BPCONT: An Auto Driver for the Continuation of Branch Points of Algebraic and Boundary-Value Problems , 2008, SIAM J. Sci. Comput..

[60]  John Guckenheimer,et al.  Dstool: Computer assisted exploration of dynamical systems , 1992 .

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

[62]  Willy Govaerts,et al.  MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs , 2003, TOMS.