Synchrony and elementary operations on coupled cell networks (Q2790861)

The authors investigate how elementary operations on a network, such as removing/adding an edge or a node, rewiring an edge, etc., effect the synchrony subspaces of the network. Synchrony subspaces are defined by equalities of cell coordinates and left invariant under every coupled cell system respecting the network structure. Moreover, it is known that synchrony subspaces are in one-to-one correspondence with the so called balanced equivalence relations. As a main result, sufficient and sometimes necessary and sufficient conditions for the persistence of synchrony subspaces (balanced equivalence relations) under elementary operations are obtained. For instance, by removing and/or adding edges, a relatively simple criterum is obtained: ``an equivalence relation is simultaneously balanced on two networks related by deletion and addition of edges if and only if it is balanced for the two networks, \(G_{F}\) and \(G_{E}\), having the same set of cells as the two original networks but only with edges corresponding, one to the deleted edges and the other one to the added edges, respectively''. This criterion allows reducing the problem to the study of much simpler networks \(G_{F}\) and \(G_{E}\) than the original ones. For the case of rewiring, the paper, in particular, extends the result of \textit{M. Field} [Dyn. Syst. 19, No. 3, 217--243 (2004; Zbl 1058.37008)].