On nonlinear motions of a hamiltonian system in the case of external resonance

We consider a nonlinear conservative Hamiltonian system with one degree of freedom subject to a small periodic perturbation such that the external resonance takes place. We deal with the case of bifurcation of 2π-periodic solution emanating from the equilibrium position of the unperturbed system. By using the method of normal forms and KAM theory we study the behaviour of the system near its 2π-periodic solutions in detail. We show that one of the 2π-periodic solutions is stable whereas the other 2π-periodic solution is unstable. The instability is soft, i.e. trajectories starting in a vicinity of the unstable periodic solution remain in its small neighbourhood forever. We apply our results to the problem of planar motions of a satellite in an elliptic orbit.