Global bifurcation of spiral wave solutions to the complex Ginzburg-Landau equation

We use the T² -equivariant degree to establish the existence of unbounded branches of rotating spiral wave solutions with any prescribed number of arms for the complex Ginzburg Landau equation (GLe) on the planar unit disc. By leveraging spatial symmetries inherent to the problem, our approach avoids the restrictive assumptions required in previous studies [5] that utilized the classical Leray-Schauder degree. Our results provide rigorous mathematical justification for the formation and persistence of these fundamental patterns, which are ubiquitous in physical, chemical, and biological systems but have, until now, eluded formal proof under general conditions.