Nekhoroshev's estimates for quasi-periodic time-dependent perturbations

TL;DR

This paper extends Nekhoroshev's stability estimates to quasi-periodic time-dependent perturbations of Hamiltonian systems, providing long-term stability results and improving understanding near resonances.

Contribution

It generalizes Nekhoroshev's estimates from periodic to quasi-periodic perturbations and removes the Diophantine condition in some cases, advancing long-standing conjectures.

Findings

Proves exponential stability of action variables under quasi-periodic perturbations

Extends Nekhoroshev's estimates to more general perturbations

Provides improved stability bounds near and far from resonances

Abstract

In this paper, we consider a Diophantine quasi-periodic time-dependent analytic perturbation of a convex integrable Hamiltonian system, and we prove a result of stability of the action variables for an exponentially long interval of time. This extends known results for periodic time-dependent perturbations, and partly solves a long standing conjecture of Chirikov and Lochak. We also obtain improved stability estimates close to resonances or far away from resonances, and a more general result without any Diophantine condition.