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2013
Impact Factor

Sijbo Holtman

P.O. Box 407, 9700 AK Groningen, The Netherlands
Johann Bernoulli Institute for Mathematics and Computer Science University of Groningen

Publications:

Broer H. W., Holtman S. J., Vegter G., Vitolo R.
Dynamics and geometry near resonant bifurcations
2011, vol. 16, no. 1-2, pp.  39-50
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
Citation: Broer H. W., Holtman S. J., Vegter G., Vitolo R.,  Dynamics and geometry near resonant bifurcations, Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 39-50
DOI:10.1134/S1560354710520023

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