M. Golubitsky and I. Stewart
Symmetry and Pattern Formation in Coupled Cell Networks
In: Pattern Formation in Continuous and Coupled
Systems, (M. Golubitsky, D. Luss and S.H. Strogatz, eds.) IMA Volumes
in Mathematics and its Applications 115, Springer, New York, 1999, 65-82.
We describe some basic concepts and techniques from symmetric bifurcation
theory in the context of coupled systems of cells (`oscillator networks').
These include criteria for the existence of symmetry-breaking branches of
steady and periodic states. We emphasize the role of symmetry as a
general framework for such analyses. As well as overt symmetries of
the network we discuss internal symmetries of the cells, `hidden'
symmetries related to Neumann boundary conditions, and spatio-temporal
symmetries of periodic states. The methods are applied to a
model central pattern generator for legged animal locomotion.