Numerical Continuation of Symmetric Periodic Orbits

Abstract

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.

Numerical Continuation of Hamiltonian Relative Periodic Orbits

“…Moreover we have the following convergence theorem which generalizes corresponding results in [15,29,35] to periodic orbits with spatio-temporal symmetry:…”

“…A symmetric periodic orbit of a Γ-equivariant dissipative systemẋ = f (x) with drift symmetry α ∈ Γ of order can be computed as solution of the following underdetermined system (see [10,38])…”

“…If DF (x,T ) has full rank in the solution (x,T ) then the equation F (y) = 0 where y = (x, T ) can be solved by a Gauss-Newton method [10,38]:…”

“…In this paper we are concerned with the numerical continuation of symmetric Hamiltonian periodic orbits and RPOs. We show how to numerically exploit spatio-temporal symmetries in the computation of Hamiltonian periodic orbits by extending corresponding methods for dissipative systems [38] to Hamiltonian systems. Moreover, in the case of continuous symmetries, we compute symmetry breaking bifurcations of relative periodic orbits from periodic orbits and show how to continue non-degenerate RPOs of compact group actions in the conserved quantities momentum and energy, building on the persistence results from [34].…”

“…The paper is organized as follows: In Section 2 we extend the numerical continuation techniques for symmetric periodic orbits of general systems which we developed in [38] to Hamiltonian systems. Then, in Section 3, we consider systems with continuous symmetry groups and continuation of relative periodic orbits.…”