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Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems

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

Abstract

This chapter is devoted to the study of generic bifurcations of fixed points of two-parameter maps. First we derive a list of such bifurcations. As for the final two bifurcations in the previous chapter, the description of the majority of these bifurcations is incomplete in principle. For all but two cases, only approximate normal forms can be constructed. Some of these normal forms will be presented in terms of associated planar continuous-time systems whose evolution operator φ 1 approximates the map in question (or an appropriate iterate of the map). We present bifurcation diagrams of the approximate normal forms in minimal dimensions and discuss their relationships with the original maps. In general n-dimensional situation, these results should be applied to a map restricted to the center manifold. We give explicit computational formulas for the critical normal form coefficients of the restricted map in most of the codim 2 cases.

Keywords

Hopf Bifurcation Bifurcation Diagram Center Manifold Bifurcation Curve Heteroclinic Cycle 
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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Copyright information

© 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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