Topological Dynamics of Volume-Preserving Maps Without an Equatorial Heteroclinic Curve

TL;DR

This paper extends homotopic lobe dynamics to analyze the topological structure of volume-preserving maps in three dimensions without an equatorial heteroclinic curve, focusing on invariant manifolds of fixed points and circles.

Contribution

It generalizes the homotopic lobe dynamics method to cases lacking an equatorial heteroclinic curve by shifting focus to invariant manifolds of invariant circles.

Findings

Successfully applied the extended HLD to example maps.

Provided a symbolic description of the topology of invariant manifolds.

Enhanced understanding of phase space structures in higher-dimensional systems.

Abstract

Understanding the topological structure of phase space for dynamical systems in higher dimensions is critical for numerous applications, including the computation of chemical reaction rates and transport of objects in the solar system. Many topological techniques have been developed to study maps of two-dimensional (2D) phase spaces, but extending these techniques to higher dimensions is often a major challenge or even impossible. Previously, one such technique, homotopic lobe dynamics (HLD), was generalized to analyze the stable and unstable manifolds of hyperbolic fixed points for volume-preserving maps in three dimensions. This prior work assumed the existence of an equatorial heteroclinic intersection curve, which was the natural generalization of the 2D case. The present work extends the previous analysis to the case where no such equatorial curve exists, but where intersection curves, connecting fixed points may exist. In order to extend HLD to this case, we shift our perspective from the invariant manifolds of the fixed points to the invariant manifolds of the invariant circle formed by the fixed-point-to-fixed-point intersections. The output of the HLD technique is a symbolic description of the minimal underlying topology of the invariant manifolds. We demonstrate this approach through a series of examples.