Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems

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

Abstract

In the previous two chapters we studied bifurcations of equilibria and fixed points in generic one-parameter dynamical systems having the minimum possible phase dimensions. Indeed, the systems we analyzed were either one- or two-dimensional. This chapter shows that these bifurcations occur in “essentially” the same way for generic n-dimensional systems. As we shall see, there are certain parameter-dependent one- or two-dimensional invariant manifolds on which the system exhibits the corresponding bifurcations, while the behavior off the manifolds is somehow “trivial,” for example, the manifolds may be exponentially attractive. Moreover, such manifolds (called center manifolds) exist for many dissipative infinite-dimensional dynamical systems. Below we derive quadratic approximations to the center manifolds in finite dimensions and for systems restricted to them at bifurcation parameter values. Using these results, we derive explicit invariant formulas for the critical normal form coefficients at all studied codimension 1 bifurcations of equilibria and fixed points. In Appendix A we consider a reaction-diffusion system on an interval to illustrate the necessary modifications of the technique to handle infinite-dimensional systems.

Keywords

Hopf Bifurcation Invariant Manifold Center Manifold Fold Bifurcation Reduction Principle 
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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