Generic super-exponential stability of invariant tori in Hamiltonian systems

TL;DR

This paper proves that solutions near certain stable invariant tori in Hamiltonian systems remain stable for super-exponentially long times under generic conditions, using a combination of classical and novel methods.

Contribution

It introduces a new approach to obtain generic Nekhoroshev estimates for stability near invariant tori in Hamiltonian systems.

Findings

Super-exponential stability times near invariant tori

Application of combined Birkhoff normal forms and new Nekhoroshev estimates

Focus on neighborhoods of elliptic fixed points

Abstract

In this article, we consider solutions starting close to some linearly stable invariant tori in an analytic Hamiltonian system and we prove results of stability for a super-exponentially long interval of time, under generic conditions. The proof combines classical Birkhoff normal forms and a new method to obtain generic Nekhoroshev estimates developed by the author and L. Niederman in another paper. We will mainly focus on the neighbourhood of elliptic fixed points, the other cases being completely similar.