Dynamics and geometry near resonant bifurcations |
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Authors: | Henk W Broer Sijbo J Holtman Gert Vegter Renato Vitolo |
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Institution: | 1. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK, Groningen, The Netherlands 2. College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter, EX4 4QF, UK
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Abstract: | This paper provides an overview of the universal study of families of dynamical systems undergoing a Hopf-Ne?marck-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-Ne?marck-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. |
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