Spiral Waves and Euclidean Symmetries

Spiral waves can be found in various chemical systems, for example in the Belousov–Zhabotinsky-reaction and in the catalysis on platinum surfaces. Such systems can be modelled by reaction-diffusion equations on the plane and have the symmetry of the Euclidean group of the plane. We present a center-manifold reduction ("slaving principle") near spiral waves which enables us to reduce the spiral wave dynamics to a small system of ordinary differential equations. Then we discuss the structure of the ordinary differential equations in detail. Our approach holds for any symmetry group G. As applications of our result we discuss the meandering transition of rotating spiral waves, periodic forcing of meandering spiral waves in the plane and dynamics of twisted scroll rings and helices in three-space.

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