Continuous symmetry reduction and return maps for high-dimensional flows

TL;DR

This paper introduces two methods for reducing high-dimensional dissipative flows with symmetry to simpler local return maps, demonstrating the effectiveness of the method of moving frames over polynomial basis approaches.

Contribution

The paper compares two continuous symmetry reduction methods, highlighting the practicality of the method of moving frames for high-dimensional flows.

Findings

Method of moving frames effectively reduces high-dimensional flows.

Hilbert polynomial basis approach is less feasible for high dimensions.

Reduction methods are demonstrated on the complex Lorenz system.

Abstract

We present two continuous symmetry reduction methods for reducing high-dimensional dissipative flows to local return maps. In the Hilbert polynomial basis approach, the equivariant dynamics is rewritten in terms of invariant coordinates. In the method of moving frames (or method of slices) the state space is sliced locally in such a way that each group orbit of symmetry-equivalent points is represented by a single point. In either approach, numerical computations can be performed in the original state-space representation, and the solutions are then projected onto the symmetry-reduced state space. The two methods are illustrated by reduction of the complex Lorenz system, a 5-dimensional dissipative flow with rotational symmetry. While the Hilbert polynomial basis approach appears unfeasible for high-dimensional flows, symmetry reduction by the method of moving frames offers hope.