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

We consider a nearly integrable, non-isochronous, a-priori unstable Hamiltonian system with a (trigonometric polynomial) O(μ)-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time Td=O((1/μ)log(1/μ)) 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 Td is optimal as a consequence of a general stability result proved via classical perturbation theory