Periodic orbits of a generalized Henon-Heiles system

In this paper, we apply the Lyapunov center theorem, Weinstein-Moser theorem and the averaging theory of second order to prove the existence of periodic orbits of a one-parameter generalized Henon-Heiles Hamiltonian system which includes the classical one. We show that this system has at least two-families of stable periodic orbits for energy level h > 0.