Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for estab- lishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connec- tions between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such trans- verse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary difierential equations.