Numerical Continuation of Symmetric Periodic Orbits

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. But there are hardly any results on the numerical computation of those bifurcations yet. In this paper we show how spatiotemporal 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 spatiotemporal 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 a tangential continuation method with implicit reparametrization. AMS subject classification. 37G15, 37G40, 37M20, 65P30