Non-integrability and chaos for natural Hamiltonian systems with a random potential

TL;DR

This paper proves that high-dimensional Hamiltonian systems with Gaussian random potentials on a torus typically display both chaotic regions and invariant tori, indicating non-integrability and complex dynamics.

Contribution

It establishes that, with high probability, such random Hamiltonian systems are neither integrable nor ergodic, extending results to general Riemannian manifolds.

Findings

Presence of chaotic regions in most systems as degree increases

Existence of positive-volume invariant tori coexist with chaos

Systems are typically non-integrable and non-ergodic

Abstract

Consider the ensemble of Gaussian random potentials $\{V^L(q)\}_{L=1}^\infty$ on the $d$-dimensional torus where, essentially, $V^L(q)$ is a real-valued trigonometric polynomial of degree $L$ whose coefficients are independent standard normal variables. Our main result ensures that, with a probability tending to 1 as $L\to\infty$, the dynamical system associated with the natural Hamiltonian function defined by this random potential, $H^L:=\frac12|p|^2+ V^L(q)$, exhibits a number of chaotic regions which coexist with a positive-volume set of invariant tori. In particular, these systems are typically neither integrable with non-degenerate first integrals nor ergodic. An analogous result for random natural Hamiltonian systems defined on the cotangent bundle of an arbitrary compact Riemannian manifold is presented too.