Synchrony versus Symmetry in Coupled Cells

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’.