An example of instability in high-dimensional Hamiltonian systems

TL;DR

This paper demonstrates that in high-dimensional Hamiltonian systems, certain perturbations can cause instability over polynomial timescales, challenging classical exponential stability results and refining stability thresholds.

Contribution

It introduces a new instability mechanism for high-dimensional Hamiltonian systems under moderate perturbations, improving bounds on stability thresholds.

Findings

Constructed an unstable solution with polynomial drifting time.

Showed classical exponential stability estimates do not hold under certain perturbations.

Provided a better upper bound on the validity of exponential stability estimates.

Abstract

In this article, we use a mechanism introduced by Herman, Marco and Sauzin to show that if a perturbation of a quasi-convex integrable Hamiltonian system is not too small with respect to the number of degrees of freedom, then the classical exponential stability estimates do not hold. Indeed, we construct an unstable solution whose drifting time is polynomial with respect to the inverse of the size of the perturbation. A different example was already given by Bourgain and Kaloshin, with a linear time of drift but with a perturbation which is larger than ours. As a consequence, we obtain a better upper bound on the threshold of validity of exponential stability estimates.