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This article is cited in 9 scientific papers (total in 9 papers)
Dynamics and geometry near resonant bifurcations
Henk W. Broera, Sijbo J. Holtmana, Gert Vegtera, Renato Vitolob a 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
Citation:
Henk W. Broer, Sijbo J. Holtman, Gert Vegter, Renato Vitolo, “Dynamics and geometry near resonant bifurcations”, Regul. Chaotic Dyn., 16:1-2 (2011), 39–50
Linking options:
https://www.mathnet.ru/eng/rcd425 https://www.mathnet.ru/eng/rcd/v16/i1/p39
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