Synchrony Branching Lemma for Regular Networks.

Coupled cell systems are dynamical systems associated to a network and synchrony subspaces, given by balanced colorings of the network, are invariant subspaces for every coupled cell systems associated to that network. Golubitsky and Lauterbach [SIAM J. Appl. Dyn. Syst., 8 (2009), pp. 40-75] prove an analogue of the equivariant branching lemma in the context of regular networks. We generalize this result proving the generic existence of steady-state bifurcation branches for regular networks with maximal synchrony. We also give necessary and sucient conditions for the existence of steady- state bifurcation branches with some submaximal synchrony. Those conditions only depend on the network structure, but the lattice structure of the balanced colorings is not sufficient to decide which synchrony subspaces support a steady-state bifurcation branch. [ABSTRACT FROM AUTHOR]