Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps

TL;DR

This paper extends Greene's residue criterion to three-dimensional volume-preserving maps, analyzing the stability and destruction of invariant tori using periodic orbit residues, and identifies the most robust tori in a specific reversible map.

Contribution

It develops a residue-based method to predict torus destruction in 3D volume-preserving maps, expanding Greene's criterion beyond 2D systems.

Findings

Identified tori with Diophantine rotation vectors in a cubic algebraic field.

Computed the critical function for the map to predict torus breakdown.

Located the most robust invariant torus in the studied map.

Abstract

Invariant tori play a fundamental role in the dynamics of symplectic and volume-preserving maps. Codimension-one tori are particularly important as they form barriers to transport. Such tori foliate the phase space of integrable, volume-preserving maps with one action and $d$ angles. For the area-preserving case, Greene's residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits. Even though KAM theory applies to the three-dimensional case, the robustness of tori in such systems is still poorly understood. We study a three-dimensional, reversible, volume-preserving analogue of Chirikov's standard map with one action and two angles. We investigate the preservation and destruction of tori under perturbation by computing the "residue" of nearby periodic orbits. We find tori with Diophantine rotation vectors in the "spiral mean" cubic algebraic field. The residue is used to generate the critical function of the map and find a candidate for the most robust torus.