Periodic orbits of a generalized Hénon–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 Hénon–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. (paper)