Relative Lyapunov Centre Bifurcations

Relative equilibria and relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur for example in celestial mechanics, molecular dynamics and rigid body motion. Relative equilibria are equilibria and RPOs are periodic orbits of the symmetry reduced system. Relative Lyapunov centre bifurcations are bifurcations of relative periodic orbits from relative equilibria corresponding to Lyapunov centre bifurcations of the symmetry reduced dynamics. In this paper we first prove a relative Lyapunov centre theorem by combining recent results on persistence of RPOs in Hamiltonian systems with a symmetric Lyapunov centre theorem of Montaldi et al. We then develop numerical methods for the detection of relative Lyapunov centre bifurcations along branches of RPOs and for their computation. We apply our methods to Lagrangian relative equilibria of the N-body problem. AMS subject classification. 37G15, 37J20, 37M20, 70H33