SPLITTING AND MELNIKOV POTENTIALS IN HAMILTONIAN SYSTEMS

Abstract

We consider a perturbation of an integrable Hamiltonian system possessing hyper bolic invariant tori with coincident whiskers Following an idea due to Eliasson we introduce a splitting potential whose gradient gives the splitting distance between the perturbed stable and unstable whiskers The homoclinic orbits to the perturbed whiskered tori are the critical points of the splitting potential and therefore their existence is ensured in both the regular or strongly hyperbolic or a priori unsta ble and the singular or weakly hyperbolic or a priori stable case The singular case is a model of a nearly integrable Hamiltonian near a single resonance In the regular case the Melnikov potential is a rst order approximation of the splitting potential and the standard Melnikov vector function is simply the gradient of the Melnikov potential Non degenerate critical points of the Melnikov potential give rise to transverse homoclinic orbits Explicit computations are carried out for some examples