Symmetry and synchrony in coupled cell networks 1 : Fixed-point spaces

Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetry-breaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Gamma of permutations of the nodes ('' cells ''). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals - ones that are not fixed-point spaces - can occur for such networks. We also prove the '' folk theorem '' that in any Gamma-equivariant dynamical system on R-k the only flow-invariant subspaces are the fixed-point spaces of subgroups of Gamma