Birkhoff Averages and the Breakdown of Invariant Tori in Volume-Preserving Maps

TL;DR

This paper introduces numerical methods based on weighted Birkhoff averages to study invariant tori in volume-preserving maps, distinguishing chaotic from regular dynamics and identifying torus destruction points.

Contribution

The authors develop a novel, symmetry-independent numerical approach for analyzing invariant tori and their breakdown in volume-preserving maps, including a three-dimensional generalization of Chirikov's standard map.

Findings

Sharp distinction between chaotic and regular dynamics via convergence rates.

Accurate computation of rotation vectors for regular orbits.

Identification of critical parameters where tori are destroyed.

Abstract

In this paper, we develop numerical methods based on the weighted Birkhoff average for studying two-dimensional invariant tori for volume-preserving maps. The methods do not rely on symmetries, such as time-reversal symmetry, nor on approximating tori by periodic orbits. The rate of convergence of the average gives a sharp distinction between chaotic and regular dynamics and allows accurate computation of rotation vectors for regular orbits. Resonant and rotational tori are distinguished by computing the resonance order of the rotation vector to a given precision. Critical parameter values, where tori are destroyed, are computed by a sharp decrease in convergence rate of the Birkhoff average. We apply these methods for a three-dimensional generalization of Chirikov's standard map: an angle-action map with two angle variables. Computations on grids in frequency and perturbation amplitude allow estimates of the critical set. We also use continuation to follow tori with fixed rotation vectors. We test three conjectures for cubic fields that have been proposed to give locally robust invariant tori.