The Destruction of Tori in Volume-Preserving Maps

TL;DR

This paper investigates how invariant tori in volume-preserving maps are destroyed as perturbations increase, revealing critical phenomena and resonance structures similar to those in lower-dimensional systems.

Contribution

It provides numerical evidence of a critical perturbation size where all tori are destroyed in a 3D volume-preserving map, extending understanding of invariant tori breakdown.

Findings

Existence of a critical perturbation size $_c$ for torus destruction

Power law singularity in crossing time near $_c$

Presence of high-order resonances near the critical torus

Abstract

Invariant tori are prominent features of symplectic and volume preserving maps. From the point of view of chaotic transport the most relevant tori are those that are barriers, and thus have codimension one. For an $n$-dimensional volume-preserving map, such tori are prevalent when the map is nearly "integrable," in the sense of having one action and $n-1$ angle variables. As the map is perturbed, numerical studies show that the originally connected image of the frequency map acquires gaps due to resonances and domains of nonconvergence due to chaos. We present examples of a three-dimensional, generalized standard map for which there is a critical perturbation size, $\epsilon_c$, above which there are no tori. Numerical investigations to find the "last invariant torus" reveal some similarities to the behavior found by Greene near a critical invariant circle for area preserving maps: the crossing time through the newly destroyed torus appears to have a power law singularity at $\epsilon_c$, and the local phase space near the critical torus contains many high-order resonances.