Splitting potential and Poincar e Melnikov method for whiskered tori in Hamiltonian systems

We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n degrees of freedom The integrable system is assumed to have n dimensional hyperbolic invariant tori with coincident whiskers separatrices Following Eliasson we use a geometric approach closely related to the Lagrangian properties of the whiskers to show that the splitting distance between the perturbed stable and unstable whiskers is the gradient of a periodic scalar function of n phases which we call splitting potential This geometric approach works for both the singular or weakly hyperbolic case and the regular or strongly hyperbolic case and provides the existence of at least n homoclinic intersections between the perturbed whiskers In the regular case we also obtain a rst order approximation for the splitting potential that we call Melnikov potential Its gradient the vector Melnikov function provides a rst order approximation for the splitting distance Then the nondegenerate critical points of the Melnikov potential give rise to transverse homoclinic intersections between the whiskers Generically when the Melnikov potential is a Morse function there exist at least n critical points The rst order approximation relies on the n dimensional Poincar e Melnikov method to which an important part of the paper is devoted We develop the method in a general setting giving the Melnikov potential and the Melnikov function in terms of absolutely convergent integrals which take into account the phase drift along the separatrix and the rst order deformation of the perturbed hyperbolic tori We provide formulas useful in several cases and carry out explicit computations that show that the Melnikov potential is a Morse function in di erent kinds of examples