M. Golubitsky, I. Stewart and A. Torok
Patterns of synchrony in coupled cell networks with multiple arrows
SIAM J. Appl. Dynam. Sys. 4 (1) (2005) 78-100.
A coupled cell system is a network of dynamical systems, or `cells', coupled together. The architecture of a coupled cell network is a graph that indicates how cells are coupled and which cells are equivalent. Stewart, Golubitsky, and Pivato presented a framework for coupled cell systems that permits a classification of robust synchrony in terms of network architecture. They also studied the existence of other robust dynamical patterns using a concept of quotient network. There are two difficulties with their approach. First, there are examples of networks with robust patterns of synchrony that are not included in their class of networks; and second, vector fields on the quotient do not in general lift to vector fields on the original network, thus complicating genericity arguments. We enlarge the class of coupled systems under consideration by allowing two cells to be coupled in more than one way, and we show that this approach resolves both difficulties. The theory that we develop, the `multiarrow formalism', parallels that of Stewart et al.