KAM Theory Near Multiplicity One Resonant Surfaces in Perturbations of A-Priori Stable Hamiltonian Systems

Abstract

We consider a near-integrable Hamiltonian system in the action-angle variables with analytic Hamiltonian. For a given resonant surface of multiplicity one we show that near a Cantor set of points on this surface, whose remaining frequencies enjoy the usual diophantine condition, the Hamiltonian may be written in a simple normal form which, under certain assumptions, may be related to the class which, following Chierchia and Gallavotti [1994], we call a-priori unstable. For the a-priori unstable Hamiltonian we prove a KAM-type result for the survival of whiskered tori under the perturbation as an infinitely differentiable family, in the sense of Whitney, which can then be applied to the above normal form in the neighborhood of the resonant surface.