Numerical Analysis of Bifurcations

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


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 performing numerical differentiation of functions or 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. We deal mainly with the continuous-time case and give only brief remarks on discrete-time systems. Appendix 1 summarizes estimates of convergence of Newton-like methods. Appendix 2 presents numerical methods for the continuation and analysis of homoclinic bifurcations. The bibliographical notes in Appendix 3 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.


Hopf Bifurcation Bifurcation Point Homoclinic Orbit Newton Iteration Bifurcation Curve 
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Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Yuri A. Kuznetsov
    • 1
    • 2
  1. 1.Centrum voor Wiskunde en InformaticaAmsterdamThe Netherlands
  2. 2.Institute of Mathematical Problems of BiologyRussian Academy of SciencesPushchino, Moscow RegionRussia

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