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Regular and Chaotic Dynamics

, Volume 16, Issue 1–2, pp 39–50 | Cite as

Dynamics and geometry near resonant bifurcations

  • Henk W. BroerEmail author
  • Sijbo J. Holtman
  • Gert Vegter
  • Renato Vitolo
Article

Abstract

This paper provides an overview of the universal study of families of dynamical systems undergoing a Hopf-Neĭmarck-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 Poincaré-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-Neĭmarck-Sacker dynamics in the form of planar Poincaré-Takens vector fields. To illustrate our methods a leading example is used: A periodically forced generalized Duffing-Van der Pol oscillator.

Keywords

periodically forced oscillator resonant Hopf-Neĭmarck-Sacker bifurcation geometric structure Lyapunov-Schmidt reduction equivariant singularity theory 

MSC2010 numbers

37G15 37G40 34C25 34C29 

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Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  • Henk W. Broer
    • 1
    Email author
  • Sijbo J. Holtman
    • 1
  • Gert Vegter
    • 1
  • Renato Vitolo
    • 2
  1. 1.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands
  2. 2.College of Engineering, Mathematics and Physical SciencesUniversity of ExeterExeterUK

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