Translational symmetry-breaking for spiral waves
This paper concerns small SO(2)-equivariant perturbations of SE(2)-equivariant reaction-diffusion systems of the type \[ \partial_t u(t,x)=d \Delta u(t,x) + f(u(t,x)) + \varepsilon g(|x|,u(t,0)),\quad x \in \mathbb R^2. \] In fact, the authors study a three dimensional and a five dimensional system of ordinary differential equations, which describe the dynamics (after a center manifold reduction) near normally hyperbolic relative equilibria and relative periodic orbits, respectively. The perturbation behavior of rigid spiral rotations, of epicycle-like motions of the spiral tip (quasi-periodic meandering) and of circular motions of the spiral tip superimposed with a linear drift (linear meandering) is investigated. The generic effects are: Stabilization of certain SO(2)-orbits of rotating waves or of modulated waves (which are determined by Melnikov-like functions) and drifting along these SO(2)-orbits (thereby introducing another frequency).
breaking of the translation symmetry while keeping the rotation symmetry
scientific article; zbMATH DE number 1563278