The boundedness of trajectories in the neighbourhood of the orbitally unstable periodic motion of a Hamiltonian system

Abstract

An autonomous Hamiltonian system with two degrees of freedom is considered. It is assumed that a periodic motion and a second-order resonance (parametric resonance) exist in the system. The unperturbed periodic motion is orbitally stable or unstable. However, even in the case of instability, the trajectories of the perturbed motion may remain in a bounded neighbourhood of the unperturbed trajectory for all values of the time. An asymptotic estimate of the size of this neighbourhood is given for the case when the Hamiltonian depends on a small parameter. The results are applied to the problem of the non-local stability of fast planar rotations of a heavy rigid body in the Kovalevskaya case, and to the problem of the stability of periodic Poincaré motions of the first kind in the restricted three-body problem, for one special case of second-order resonance.