Ghosts of Order on the Frontier of Chaos

TL;DR

This paper investigates the persistence and destruction of invariant tori in higher-dimensional Hamiltonian systems, extending Aubry-Mather theory, through numerical and rigorous methods, revealing regimes where chaos dominates.

Contribution

It extends Aubry-Mather theory to four-dimensional systems by combining numerical approximations of Birkhoff orbits with rigorous proofs of torus destruction.

Findings

Most invariant tori can be approximated by Birkhoff orbits.

Large perturbations can destroy all KAM tori in the studied systems.

Existence of regimes with no invariant tori, indicating chaotic behavior.

Abstract

What kinds of motion can occur in classical mechanics? We address this question by looking at the structures traced out by trajectories in phase space; the most orderely, completely integrable systems, are charactrized by phase trajectories confined to low-dimensional, invariant tori. The KAM theory examines what happens to the tori when an integrable system is subjected to a small perturbation and finds that, for small enough perturbations, most of them survive. The KAM theory is mute on the suject of the disrupted tori, but, for two dimensional systems, Aubry and Mather discovered an astonishing picture: the broken tori are replaced by "cantori", tattered, Cantor-set remnants of the original invariant curves. We seek to extend Aubry and Mather's picture to higher dimensional systems and report two kinds of studies; both concern perturbations of a completely integrable, four-dimensional symplectic map. In the first study we compute some numerical approximations to Birkhoff periodic orbits; sequences of such orbits should approximate any higher dimensional analogs of the cantori. In the second study we prove converse KAM theorems; that is, we use a combination of analytic arguments and rigorous, machine-assisted computations to find perturbations so large that no KAM tori survive. We are able to show that the last few of our Birkhoff orbits exist in a regime where there are no tori.