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

Massimiliano Berti, Luca Biasco, Philippe Bolle

Published 2002-03-19Version 1

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.