Diffusion and Drift in Volume-Preserving Maps

TL;DR

This paper investigates how actions in nearly-integrable volume-preserving maps can experience significant drift along resonance channels, contrasting with the confinement seen in symplectic cases, using averaging theory and numerical comparisons.

Contribution

It extends the understanding of action drift from symplectic to volume-preserving maps, providing a theoretical framework and numerical validation for resonance-induced drift.

Findings

Actions can drift strongly along resonance channels in volume-preserving maps.

Averaging theory accurately predicts drift in rank-one resonance cases.

Numerical simulations confirm theoretical predictions for a 4D generalized Froeschlé map.

Abstract

A nearly-integrable dynamical system has a natural formulation in terms of actions, $y$ (nearly constant), and angles, $x$ (nearly rigidly rotating with frequency $\Omega(y)$). We study angle-action maps that are close to symplectic and have a positive-definite twist, the derivative of the frequency map, $D\Omega(y)$. When the map is symplectic, Nekhoroshev's theorem implies that the actions are confined for exponentially long times: the drift is exponentially small and numerically appears to be diffusive. We show that when the symplectic condition is relaxed, but the map is still volume-preserving, the actions can have a strong drift along resonance channels. Averaging theory is used to compute the drift for the case of rank-$r$ resonances. A comparison with computations for a generalized Froeschl\'e map in four-dimensions, shows that this theory gives accurate results for the rank-one case.