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Zbl 1210.37034
Wulff, Claudia; Schebesch, Andreas
Numerical continuation of symmetric periodic orbits. (English)
[J] SIAM J. Appl. Dyn. Syst. 5, No. 3, 435-475, electronic only (2006). ISSN 1536-0040/e

Summary: The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for symmetric dynamical systems. However, there are hardly any results on the numerical computation of those bifurcations yet. In this paper we show how spatio-temporal symmetries of periodic orbits can be exploited numerically. We describe methods for the computation of symmetry breaking bifurcations of periodic orbits for free group actions and show how bifurcations increasing the spatio-temporal symmetry of periodic orbits (including period halving bifurcations and equivariant Hopf bifurcations) can be detected and computed numerically. Our pathfollowing algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization.

MSC 2000:: *37G15 Bifurcations of limit cycles and periodic orbits
37G40 Symmetries, equivariant bifurcation theory
37M20 Computational methods for bifurcation problems
65P30 Bifurcation problems

Keywords: Numerical continuation; symmetry breaking bifurcations; symmetric periodic orbits

Cited in: Zbl 1168.37014

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