Double exponential stability of quasi-periodic motion in Hamiltonian systems

TL;DR

This paper proves that in Hamiltonian systems, invariant tori exhibit doubly exponential stability, meaning solutions stay close for extremely long times, with results applicable to real-analytic and Gevrey smooth systems.

Contribution

It establishes the doubly exponential stability of invariant tori in Hamiltonian systems, extending stability results to a broad class of smooth systems and generic perturbations.

Findings

Doubly exponential stability of invariant tori proven

Most KAM tori are doubly exponentially stable under small perturbations

Results apply to real-analytic and Gevrey smooth Hamiltonian systems

Abstract

We prove that generically, both in a topological and measure-theoretical sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is doubly exponentially stable in the sense that nearby solutions remain close to the torus for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the torus. We also prove that for an arbitrary small perturbation of a generic integrable Hamiltonian system, there is a set of almost full positive Lebesgue measure of KAM tori which are doubly exponentially stable. Our results hold true for real-analytic but more generally for Gevrey smooth systems.