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Numerical Analysis of Bifurcations

  • Yuri A. Kuznetsov
Part of the Applied Mathematical Sciences book series (AMS, volume 112)

Abstract

In this chapter we shall describe some of the basic techniques used in the numerical analysis of dynamical systems. We assume that low-level numerical routines like those for solving linear systems, finding eigenvectors and eigenvalues, and performing numerical integration of ODEs are known to the reader. Instead we focus on algorithms that are more specific to bifurcation analysis, specifically those for the location of equilibria (fixed points) and their continuation with respect to parameters, and for the detection, analysis, and continuation of bifurcations. Special attention is given to location and continuation of limit cycles and their associated bifurcations, as well as to continuation of homoclinic orbits. We deal mainly with the continuous-time case and give only brief remarks on discrete-time systems. Appendix A summarizes simple estimates of convergence of Newton-like methods. Appendix B gives some background information on the bialternate matrix product used to detect Hopf and Neimark-Sacker bifurcations. Appendix C presents numerical methods for detection of higher-order homoclinic bifurcations. The bibliographical notes in Appendix D include references to standard noninteractive software packages and interactive programs available for continuation and bifurcation analysis of dynamical systems. Actually, the main goal of this chapter is to provide the reader with an understanding of the methods implemented in widely used software for dynamical systems analysis.

Keywords

Jacobian Matrix Hopf Bifurcation Homoclinic Orbit Bifurcation Curve Homoclinic Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Yuri A. Kuznetsov
    • 1
    • 2
  1. 1.Department of MathematicsUtrecht UniversityUtrechtThe Netherlands
  2. 2.Institute of Mathematical Problems of BiologyRussian Academy of SciencesPushchino, Moscow RegionRussia

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