Ghosts of Order on the Frontier of Chaos
Mark Muldoon

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,…
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Taxonomy
TopicsQuantum chaos and dynamical systems · Mathematical Dynamics and Fractals
