An Extension of the Subharmonic Melnikov Method for Resonant Periodic Orbits

We extend the subharmonic Melnikov method to study the persistence of resonant periodic orbits in nonlinear ‐dimensional systems formed by an integrable zero‐order part plus a generic perturbation. The unperturbed system is assumed to be a decoupled reunion of Hamiltonian periodic annuli, and we treat both non‐degenerate and degenerate resonances. The main novelty is that we allow the subharmonic method to be applied in the presence of a continuum of periodic orbits. We illustrate the scope of the theory with model applications that display degenerate and ‐degenerate resonances, and we collect stability conditions via Floquet theory.