On the continuation of degenerate periodic orbits via normal form : full dimensional resonant tori

We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamiltonian systems. In particular we focus on periodic orbits that arise from the breaking of a completely resonant maximal torus. We here propose a suitable normal form construction that allows to identify and approximate the periodic orbits which survive to the breaking of the resonant torus. Our algorithm allows to treat the continuation of approximate orbits which are at leading order degenerate, hence not covered by classical averaging methods. We discuss possible future extensions and applications to localized periodic orbits in chains of weakly coupled oscillators.