Reduction of SO(2) symmetry for spatially extended dynamical systems

TL;DR

This paper introduces a simple symmetry reduction method for spatially extended systems with SO(2) symmetry, revealing hidden global structures in turbulence dynamics.

Contribution

The first Fourier mode slice provides an easy-to-implement reduction of SO(2) symmetry, uncovering new dynamical structures in turbulent flows.

Findings

Unveiled Kuramoto-Sivashinsky relative periodic orbits.

Regularized phase velocity near singularities.

Discovered unstable manifolds of travelling waves.

Abstract

Spatially extended systems, such as channel or pipe flows, are often equivariant under continuous symmetry transformations, with each state of the flow having an infinite number of equivalent solutions obtained from it by a translation or a rotation. This multitude of equivalent solutions tends to obscure the dynamics of turbulence. Here we describe the `first Fourier mode slice', a very simple, easy to implement reduction of SO(2) symmetry. While the method exhibits rapid variations in phase velocity whenever the magnitude of the first Fourier mode is nearly vanishing, these near singularities can be regularized by a time-scaling transformation. We show that after application of the method, hitherto unseen global structures, for example Kuramoto-Sivashinsky relative periodic orbits and unstable manifolds of travelling waves, are uncovered.