Dynamics and geometry near resonant bifurcations

Hendrik Broer, Sijbo J. Holtman, Gert Vegter, Renato Vitolo

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Abstract

This paper provides an overview of the universal study of families of dynamical systems undergoing a Hopf-NeAmarck-Sacker bifurcation as developed in [1-4]. The focus is on the local resonance set, i.e., regions in parameter space for which periodic dynamics occurs. A classification of the corresponding geometry is obtained by applying Poincar,-Takens reduction, Lyapunov-Schmidt reduction and contact-equivalence singularity theory, equivariant under an appropriate cyclic group. It is a classical result that the local geometry of these sets in the nondegenerate case is given by an Arnol'd resonance tongue. In a mildly degenerate situation a more complicated geometry given by a singular perturbation of a Whitney umbrella is encountered. Our approach also provides a skeleton for the local resonant Hopf-NeAmarck-Sacker dynamics in the form of planar Poincar,-Takens vector fields. To illustrate our methods a leading example is used: A periodically forced generalized Duffing-Van der Pol oscillator.

Original languageEnglish
Pages (from-to)39-50
Number of pages12
JournalRegular & chaotic dynamics
Volume16
Issue number1-2
DOIs
Publication statusPublished - Feb-2011

Keywords

  • periodically forced oscillator
  • resonant Hopf-Neimarck-Sacker bifurcation
  • geometric structure
  • Lyapunov-Schmidt reduction
  • equivariant singularity theory
  • SACKER FAMILIES
  • RECOGNITION

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