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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 bifurcations of equilibria in generic 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 genericity 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 and Theorem 5.4 (see Chapter 5). We close this chapter with the derivation of the critical normal form coefficients for all codim 2 bifurcations using a combined reduction/normalization technique.

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 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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