Lyapunov's center theorem for resonant equilibrium

Lyapunov's center theorem relative to the existence of families of periodic orbits emanating from an equilibrium is generalized to cases where a resonance occurs between two basic frequencies. Analytical Hamiltonian systems are considered and the theorems depend on the nonannulation of an invariant of the system.The proof is performed in two steps. In a first step the theorems are shown to be valid for some approximation of the Hamiltonian system. These results are described in a previous paper (Henrard, 1970) and are only summarized here. In a second step Poincaré's perturbation theorem is generalized in order to transfer to the original system the conclusions relatives to its approximations.In the conclusion, our results are compared with similar results published recently.