Pere GUTIERREZ

Departament de Matematica Aplicada II
Universitat Politecnica de Catalunya
Pau Gargallo 5, E-08028 Barcelona
Spain
gutierrez@ma2.upc.es
 

On Melnikov theory for the splitting potential of whiskered tori in regular and singular hyperbolic Hamiltonians

Abstract : A perturbation of an integrable Hamiltonian system, having hyperbolic invariant tori with coincident whiskers, is considered. Our goal is to detect homoclinic intersections between the perturbed whiskers, and to measure the splitting distance. Using a geometric approach (based on a paper by Eliasson), the splitting distance can be shown to be the gradient of a (scalar and periodic) splitting potential. For a regular (or strongly hyperbolic) system, a first order approximation for this potential is given through Melnikov theory. This approach is designed also in the singular (or weakly hyperbolic) case, which is more involved and constitutes a model for a nearly-integrable Hamiltonian near a single resonance. The Melnikov potential in this singular case is studied, and is expected to be useful in order to give predictions of the splitting under some weak hypotheses on the perturbation. Explicit computations in concrete examples are also given.