Generic perturbations of linear integrable Hamiltonian systems

TL;DR

This paper studies how generic small perturbations affect linear integrable Hamiltonian systems, showing that stability and instability results depend on whether the system's frequency is resonant or non-resonant, with implications for KAM and Nekhoroshev theorems.

Contribution

It proves generic instability in the resonant case and generic stability over doubly exponentially long times in the non-resonant case, advancing understanding of Hamiltonian perturbations.

Findings

Generic perturbations cause instability in resonant systems.

KAM theorem holds for generic perturbations in non-resonant systems.

Stability persists over doubly exponentially long times for generic perturbations.

Abstract

In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem which does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is non-resonant is more subtle. Our second result shows that for a generic perturbation, the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long interval of time, and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation, one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).