On the continuation of degenerate periodic orbits in Hamiltonian systems

The continuation of non-isolated periodic orbits lying on the resonant invariant tori of an integrable Hamiltonian system with respect to a small perturbative parameter cannot be proved by a direct application of the continuation theorem, since their monodromy matrix possesses more than a single pair of unit eigenvalues. In this case one may use Poincaré's theorem which proves that, if the integrable part of the Hamiltonian is non-degenerate and the average value of the perturbing function, evaluated along the unperturbed periodic orbits, possesses a simple extremum on such an orbit, then this orbit can be analytically continued with respect to the perturbation. In the present paper we prove a criterion for the continuation of the non-isolated periodic orbits, for which this average value is constant along the periodic orbits of the resonant torus and Poincaré's theorem is not applicable. We apply the results in two such systems of two degrees of freedom.