Optimal stability and instability results for a class of nearly integrable Hamiltonian systems

Abstract

We consider a nearly integrable, non-isochronous, a-priori unstable Hamiltonian system with a (trigonometric polynomial) $O(\mu )$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $T_d=O((1/\mu )\log (1/\mu ))$ by a variational method which does not require the existence of ``transition chains of tori'' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d$ is optimal as a consequence of a general stability result proved via classical perturbation theory.

6 pages