Using invariant manifolds to construct symbolic dynamics for three-dimensional volume-preserving maps

TL;DR

This paper extends topological techniques, specifically homotopic lobe dynamics, from 2D to 3D volume-preserving maps, enabling symbolic dynamics construction and entropy computation in higher-dimensional systems relevant to fluid and magnetic field line transport.

Contribution

It introduces a method to apply homotopic lobe dynamics to 3D volume-preserving maps using invariant manifolds, advancing topological analysis in higher-dimensional dynamical systems.

Findings

Successfully extended HLD to 3D maps

Constructed symbolic dynamics from invariant manifolds

Detected differences in 2D and 3D stretching rates

Abstract

Topological techniques are powerful tools for characterizing the complexity of many dynamical systems, including the commonly studied area-preserving maps of the plane. However, the extension of many topological techniques to higher dimensions is filled with roadblocks preventing their application. This article shows how to extend the homotopic lobe dynamics (HLD) technique, previously developed for 2D maps, to volume-preserving maps of a three-dimensional phase space. Such maps are physically relevant to particle transport by incompressible fluid flows or by magnetic field lines. Specifically, this manuscript shows how to utilize two-dimensional stable and unstable invariant manifolds, intersecting in a heteroclinic tangle, to construct a symbolic representation of the topological dynamics of the map. This symbolic representation can be used to classify system trajectories and to compute topological entropy. We illustrate the salient ideas through a series of examples with increasing complexity. These examples highlight new features of the HLD technique in 3D. Ultimately, in the final example, our technique detects a difference between the 2D stretching rate of surfaces and the 1D stretching rate of curves, illustrating the truly 3D nature of our approach.