M. Golubitsky and I. Stewart
Synchrony versus symmetry in coupled cells
In: Equadiff 2003: Proceedings of the International Conference on Differential Equations.
(F. Dumortier, H.W. Broer, J. Mawhin, A. Vanderbauwhede and S.M. Verduyn Lunel, eds.)
World Scientific Publ. Co., Singapore, 2005, 13-24.
A coupled cell system is a network of dynamical systems, or `cells',
coupled together. Such systems are represented schematically
by a directed graph whose nodes correspond to cells and whose
edges represent couplings. Symmetry of coupled cell systems can
lead to synchronized cells. We show that symmetry is not the only
mechanism that can create such states in a coupled cell
system. The first main result shows that robust synchrony
is equivalent to the condition that an equivalence
relation on cells is `balanced'. The second main result shows that
admissible vector fields restricted to synchrony subspaces are
themselves admissible vector fields for a new coupled cell network,
the `quotient network'.