## 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 language | English |
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Pages (from-to) | 39-50 |

Number of pages | 12 |

Journal | Regular & chaotic dynamics |

Volume | 16 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Feb-2011 |

## Keywords

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