Periodic orbit analysis of a system with continuous symmetry - a tutorial

TL;DR

This paper demonstrates how to analyze chaotic systems with continuous symmetry by using symmetry reduction techniques, Poincaré sections, and cycle averaging formulas, enabling systematic identification of relative periodic orbits and their dynamics.

Contribution

It introduces and compares symmetry reduction methods for chaotic systems with continuous symmetry, illustrating their application with a 4D model and providing tools for orbit analysis.

Findings

Symmetry reduction simplifies the analysis of chaotic symmetric systems.

A Poincaré section on the symmetry slice reduces the flow to a unimodal map.

Cycle averaging formulas effectively compute dynamical averages from relative periodic orbits.

Abstract

Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains. In these cases, it is natural to express the state of the fluid in terms of a Fourier series truncated to a finite number of modes. Here, we study a 4-dimensional model with chaotic dynamics and SO(2) symmetry similar to those that appear in fluid dynamics problems. A crucial step in the analysis of such a system is symmetry reduction. We use the model to illustrate different symmetry-reduction techniques. Its relative equilibria are conveniently determined by rewriting the dynamics in terms of a symmetry-invariant polynomial basis. However, for the analysis of its chaotic dynamics, the `method of slices', which is applicable to very high-dimensional problems, is preferable. We show that a Poincar\'e section taken on the `slice' can be used to further reduce this flow to what is for all practical purposes a unimodal map. This enables us to systematically determine all relative periodic orbits and their symbolic dynamics up to any desired period. We then present cycle averaging formulas adequate for systems with continuous symmetry and use them to compute dynamical averages using relative periodic orbits. The convergence of such computations is discussed.