Numerical Bifurcation of Hamiltonian Relative Periodic Orbits

Relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur for example in celestial mechanices, molecular dynamics and rigid body motion. RPOs are solutions which are periodic orbits of the symmetry-reduced system. In this paper we analyze certain symmetry-breaking bifurcations of Hamiltonian relative periodic orbits and show how they can be detected and computed numerically. These are turning points of RPOs, relative period-doubling and relative period-halving bifurcations along branches of RPOs. In a comoving frame the latter correspond to symmetry-breaking/symmetryincreasing pitchfork bifurcations or to period doubling/period-halving bifurcations. We apply our methods to the family of rotating choreographies which bifurcate from the famous Figure Eight solution of the three body problem as angular momentum is varied. We find that the Eights rotating around the e2-axis bifurcate to the family of rotating Eights that connect to the Lagrange relative equilibrium. Moreover, we find several relative period-doubling bifurcations and a turning point of the planar rotating choreography which bifurcates from the Figure Eight solution when the third component of the angular momentum vector is varied.