Orbits homoclinic to resonances: the Hamiltonian case

In this paper we develop methods to show the existence of orbits homoclinic or heteroclinic to periodic orbits, hyperbolic fixed points or combinations of hyperbolic fixed points and/or periodic orbits in a class of two-degree-offreedom, integrable Hamiltonian systems subject to arbitrary Hamiltonian perturbations. Our methods differ from previous methods in that the invariant sets (periodic orbits, fixed points) are created, and become hyperbolic, as a result of the interaction of the perturbation with a resonance in the unperturbed system. This results in a very degenerate situation that requires a combination of geometric singular perturbation theory, higher-dimensional Melnikov-type methods, and transversality theory. We establish a simple energy-phase criterion which gives a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonance. We apply our methods to a two-mode truncation of the driven nonlinear Schrodinger equation first studied by Bishop et al. In this example we show that as the energy is increased at resonance, orbits homoclinic to hyperbolic periodic orbits are created in pairs in a global bifurcation that is best described as a saddle-node bifurcation of homoclinic orbits.