Spiral anchoring in anisotropic media with multiple inhomogeneities: a dynamical system approach

Abstract

Various PDE models have been suggested in order to explain and predict the dynamics of spiral waves in excitable media. In two landmark papers, Barkley noticed that some of the behaviour could be explained by the inherent Euclidean symmetry of these models. LeBlanc and Wulff then introduced forced Euclidean symmetry-breaking (FESB) to the analysis, in the form of individual translational symmetry-breaking (TSB) perturbations and rotational symmetry-breaking (RSB) perturbations; in either case, it is shown that spiral anchoring is a direct consequence of the FESB. In this article, we provide a characterization of spiral anchoring when two perturbations, a TSB term and a RSB term, are combined, where the TSB term is centered at the origin and the RSB term preserves rotations by multiples of $\frac{2π}{\jmath^*}$, where $\jmath^*\geq 1$ is an integer. When $\jmath^*>1$ (such as in a modified bidomain model), it is shown that spirals anchor at the origin, but when $\jmath^* =1$ (such as in a planar reaction-diffusion-advection system), spirals generically anchor away from the origin.