Ginzburg-Landau patterns in circular and spherical geometries: vortices, spirals and attractors

TL;DR

This paper analyzes pattern formation in Ginzburg-Landau equations on circular and spherical geometries, providing a comprehensive description of vortex solutions, their bifurcations, and attractors, with new methods for proving hyperbolicity.

Contribution

It offers a complete bifurcation diagram, proves persistence of bifurcation curves under perturbations, and explicitly constructs the global attractor for vortex solutions using a novel shooting method.

Findings

Complete bifurcation diagram of vortex equilibria

Existence of spiral waves in perturbed systems

Explicit construction of the global attractor

Abstract

This paper consists of three results on pattern formation of Ginzburg-Landau $m$-armed vortex solutions and spiral waves in circular and spherical geometries. First, we completely describe the global bifurcation diagram of vortex equilibria. Second, we prove persistence of all bifurcation curves under perturbations of parameters, which yields the existence of spiral waves for the complex Ginzburg-Landau equation. Third, we explicitly construct the global attractor of $m$-armed vortex solutions. Our main tool is a new shooting method that allows us to prove hyperbolicity of vortex equilibria in the invariant subspace of vortex solutions.