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Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems

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

Abstract

This chapter is devoted to generic bifurcations of equilibria in two-parameter systems of differential equations. First, we make a complete list of such bifurcations. Then, we derive a parameter-dependent normal form for each bifurcation in the minimal possible phase dimension and specify relevant nondegeneracy conditions. Next, we truncate higher-order terms and present the bifurcation diagrams of the resulting system. The analysis is completed by a discussion of the effect of the higher-order terms. In those cases where the higher-order terms do not qualitatively alter the bifurcation diagram, the truncated systems provide topological normal forms for the relevant bifurcations. The results of this chapter can be applied to n-dimensional systems by means of the parameter-dependent version of the Center Manifold Theorem (see Chapter 5).

Keywords

Normal Form Hopf Bifurcation Phase Portrait Bifurcation Diagram Homoclinic Orbit 
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 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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