Claudia Wulff; Andreas Schebesch.

Numerical continuation of Hamiltonian relative periodic orbits.

he bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been a rapid progress in the development of a bifurcation theory for dynamical systems with structure, like symmetry or symplecticity. But there are few results on the numerical computation of those bifurcations yet. In this paper we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Moreover we numerically compute relative periodic orbits persisting from periodic orbits in a symmetry breaking bifurcation and present a method for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. We apply our methods to continue the famous Figure Eight choreography of the three-body system. We find a new less symmetric Figure Eight and a new family of non-reversible rotating choreographies bifurcating from it. 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.