Splitting Potential and Poincar E{melnikov Theory for Whiskered Tori in Hamiltonian Systems

We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n + 1 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 that takes advantage of 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 + 1 homoclinic intersections between the perturbed whiskers. In the regular case, we develop Poincar e{Melnikov theory to obtain a rst order approximation for the splitting potential, and we call it the Melnikov potential. Its gradient, the (vector) Melnikov function, provides a rst order approximation for the splitting distance. Both the Melnikov potential and the Melnikov function are given by means of absolutely convergent integrals, which take into account the phase drift along the separatrix and the rst order deformation of the perturbed hyperbolic tori. In this regular case, 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 2 n critical points. Explicit computations showing that the Melnikov potential is a Morse function are carried out in diierent kinds of examples. 1 Setup and introduction 1.1 Perturbation of a hyperbolic integrable Hamiltonian It is well-known that the problem of giving conditions for the splitting of the whiskers of hyperbolic invariant tori is one of the main diiculties related with the Arnold diiusion, a phenomenon of instability in perturbations of integrable Hamiltonian systems with more than 2 degrees of freedom. The present paper is concerned with the study of the existence of homoclinic orbits and splitting in a wide class of Hamiltonians. The tools used are a geometric approach based on Eliasson's work Eli94], and Poincar e{ Melnikov theory. We start with a perturbation of a hyperbolic integrable Hamiltonian, with n + 1 3 degrees of freedom. In canonical variables z = (x; y; '; I) 2 D T R T n R n , with the symplectic form 1