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Dynamics of meandering spiral waves with weak lattice perturbations

Re-entrant spiral waves are observed in many different situations in nature, perhaps most importantly in excitable electrophysiological tissue where they are believed to be responsible for pathological conditions such as cardiac arrhythmias, epileptic seizures and hallucinations. Mathematically, spiral waves occur as solutions to systems of reaction-diffusion partial differential equations (RDPDEs) which are frequently used as models for electrophysiological phenomena. Because of the invariance of these RDPDEs with respect to the Euclidean group SE(2) of planar translations and rotations, much progress has been made in understanding the dynamics and bifurcations of spiral waves using the theory of group-equivariant dynamical systems. In reality however, Euclidean symmetry is at best an approximation. Inhomogeneities and anisotropy in the medium of propagation of the waves break the Euclidean symmetry, and can lead to such phenomena as anchoring and drifting. In this paper, we study the effects on quasi-periodic meandering spiral waves of a small perturbation which breaks the continuous SE(2) symmetry, but preserves the symmetry of a regular square lattice.