Barriers to Transport and Mixing in Volume-Preserving Maps with Nonzero Flux

TL;DR

This paper investigates the geometric structures that limit transport and mixing in perturbed volume-preserving systems with nonzero flux, introducing an algorithm to analyze their properties and quantify their impact on dynamics.

Contribution

It identifies resonance-generated structures that restrict transport, and presents a simple algorithm for analyzing their distribution and properties in nonzero flux volume-preserving maps.

Findings

Distribution of escape times for unbounded orbits

Abundance and size of resonant regions

Quantitative properties of invariant tori

Abstract

In this paper we identify the geometric structures that restrict transport and mixing in perturbations of integrable volume-preserving systems with nonzero net flux. Unlike KAM tori, these objects cannot be continued to the tori present in the integrable system but are generated by resonance and have a contractible direction. We introduce a remarkably simple algorithm to analyze the behavior of these maps and obtain quantitative properties of the tori. In particular, we present assertions regarding the distribution of the escape times of the unbounded orbits, the abundance of tori, and the size of the resonant regions.