Reduction of continuous symmetries of chaotic flows by the method of slices

TL;DR

This paper explores symmetry reduction in chaotic dynamical systems using the method of slices, proposing a tessellation approach with multiple slices to effectively handle turbulence and avoid singularities.

Contribution

It introduces a set of multiple slices for symmetry reduction, improving upon the single-slice method by avoiding singularities and better capturing turbulent dynamics.

Findings

Single slice intersects all group orbits but is not ideal for turbulence.

Multiple slices form a tessellation that reduces singularities.

Application to complex Lorenz equations demonstrates effectiveness.

Abstract

We study continuous symmetry reduction of dynamical systems by the method of slices (method of moving frames) and show that a `slice' defined by minimizing the distance to a single generic `template' intersects the group orbit of every point in the full state space. Global symmetry reduction by a single slice is, however, not natural for a chaotic / turbulent flow; it is better to cover the reduced state space by a set of slices, one for each dynamically prominent unstable pattern. Judiciously chosen, such tessellation eliminates the singular traversals of the inflection hyperplane that comes along with each slice, an artifact of using the template's local group linearization globally. We compute the jump in the reduced state space induced by crossing the inflection hyperplane. As an illustration of the method, we reduce the SO(2) symmetry of the complex Lorenz equations.