On the Existence of Drifting Orbits for Non-Convex Hamiltonian Systems

TL;DR

This paper proves the existence of drifting orbits in certain perturbed non-convex Hamiltonian systems, showing that action variables can change significantly over time despite small perturbations, with numerical simulations supporting the theoretical results.

Contribution

It demonstrates the existence of drifting orbits in non-convex Hamiltonian systems under small perturbations, a phenomenon previously not well-understood.

Findings

Drifting orbits can occur near resonant surfaces with small perturbations.

Action variables can change by order one infinitely often.

Numerical simulations compare instability conditions in specific examples.

Abstract

We show the existence of drifting orbits for certain perturbations of non-convex Hamiltonian systems with several degrees of freedom. These orbits remain in the vicinity of resonant surfaces where the action variables can undergo changes $O(1)$ infinitely often although the size of perturbations $O(\epsilon)$ can be arbitrarily small. The first drifts occur in a period of time $O(1/\epsilon)$ and then reoccur with frequencies independent of $\epsilon$. We also perform numerical simulations to compare the effects of two conditions for instability in two four-dimensional examples with random parameters.