This article is cited in 9 papers

Dynamics and geometry near resonant bifurcations

Henk W. Broer^a, Sijbo J. Holtman^a, Gert Vegter^a, Renato Vitolo<sup>b

a</sup> Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands
^b College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter EX4 4QF, UK

Abstract: This paper provides an overview of the universal study of families of dynamical systems undergoing a Hopf–Neimarck–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–Neimarck–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–Neimarck–Sacker bifurcation, geometric structure, Lyapunov–Schmidt reduction, equivariant singularity theory.

Received: 04.04.2010
Accepted: 21.06.2010

Language: English

DOI: 10.1134/S1560354710520023

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