V.N.Biktashev 
, A.V.Holden 
and E.V.Nikolaev 
July 26, 1996
Institute for Mathematical Problems in Biology,
Pushchino, Moscow Region, 142292, Russia
Department of Physiology, University of Leeds, Leeds LS2 9JT, UK
Author to whom correspondence should be addressed
We present a general group-theoretic approach that explains the main qualitative features of the meander of spiral wave solutions on the plane. The approach is based on the well known space reduction method and is to separate the motions in the system into superposition of those `along' orbits of the Euclidean symmetry group, the group of all isometric transformations of the plane, and `across' the group orbits. It has the visual interpretation as passing to a reference frame attached to the spiral wave's tip. The system of ODEs governing the tip movement is obtained. It is the system that describes the movements along the group orbits. The motions across the group orbits are described by a PDE which lacks the Euclidean symmetry. Consequences of the Euclidean symmetry on the spiral wave dynamics are discussed. In particular, we explicitly derive the model system for bifurcation from rigid to biperiodic rotation, suggested earlier by Barkley (1994) from a priori symmetry considerations.