Spiral waves on spherical domains: A dynamical systems approach

Abstract

This thesis is concerned with the dynamics and bifurcations of spiral waves in excitable media with spherical or approximately spherical geometry (e.g. cardiac tissue). First, we study parameter-dependent systems of reaction-diffusion partial differential equations on a sphere which are equivariant under the group SO(3) of all rigid rotations on a sphere. Two main types of spatial-temporal patterns that can appear in such systems are rotating waves (equilibrium in a co-rotating frame) and modulated rotating waves (periodic solution in a co-rotating frame). The transition from rotating waves to modulated rotating waves on spherical domains is explained via a supercritical Hopf bifurcation from a rotating wave and SO(3) symmetry. The Baker-Campbell-Haussdorff formula in the Lie algebra so(3) is used to get a formula for a primary frequency vector, as well as a formula for the periodic part associated to any modulated rotating wave obtained by a supercritical Hopf bifurcation from a rotating wave. In the resonant case, the primary frequency vector of the modulated rotating wave is generically orthogonal to the frequency vector of the initial rotating wave. In the second part of the thesis, we study the effects of forced symmetry-breaking from SO(3) to SO(2) for a normally hyperbolic relative equilibrium. This is done by introducing a small SO(2)-equivariant perturbation into the above reaction-diffusion system. The relative equilibrium persists to a normally hyperbolic SO(2)-invariant manifold that is SO(2)-equivariant diffeomorphic to SO(3). The perturbed SO(2)-equivariant dynamics on this manifold are studied by using the projection onto the orbit space SO(3)/ SO(2). Depending on the frequency vectors of the rotating waves that form the relative equilibrium, these rotating waves (up to SO(2)) will give either SO(2)-orbits of rotating waves or SO(2)-orbits of modulated rotating waves (if some transversality conditions hold). The orbital stability of these solutions is established as well.