Pattern-Selective Feedback Stabilization of Ginzburg--Landau Spiral Waves

TL;DR

This paper introduces a novel symmetry-breaking control method to stabilize previously unstable multi-armed spiral waves in the complex Ginzburg--Landau equation, enabling the observation of stable spirals with arbitrary arms.

Contribution

It develops the first method to stabilize unstable multi-armed spiral waves in PDEs using an equivariant Pyragas control generalization.

Findings

Successfully stabilizes multi-armed spiral waves in simulations.

Achieves stable spirals with any number of arms.

Extends control techniques to complex pattern-forming PDEs.

Abstract

The complex Ginzburg--Landau equation serves as a paradigm of pattern formation and the existence and stability properties of Ginzburg--Landau $m$-armed spiral waves have been investigated extensively. However, many multi-armed spiral waves are unstable and thereby rarely visible in experiments and numerical simulations. In this article we selectively stabilize certain significant classes of unstable spiral waves within circular and spherical geometries. As a result, stable spiral waves with an arbitrary number of arms are obtained for the first time. Our tool for stabilization is the symmetry-breaking control triple method, which is an equivariant generalization of the widely applied Pyragas control to the setting of PDEs.