I. Stewart, M. Golubitsky and M. Pivato
Symmetry groupoids and patterns of synchrony in coupled cell networks
SIAM J. Appl. Dynam. Sys. 2 (4) (2003) 609-646.
The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information about the input sets of cells. (The input set of a cell consists of that cell and all cells connected to that cell.) The admissible vector fields for a given graph --- the dynamical systems with the corresponding internal dynamics and couplings --- are precisely those that are equivariant under the symmetry groupoid. A pattern of synchrony is `robust' if it arises for all admissible vector fields. The first main result shows that robust patterns of synchrony (invariance of `polydiagonal' subspaces under all admissible vector fields) is equivalent to the combinatorial condition that an equivalence relation on cells is `balanced'. The second main result shows that admissible vector fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled cell network, the `quotient network'. The existence of quotient networks has surprising implications for synchronous dynamics in coupled cell systems.