Bifurcations from Synchrony in Homogeneous Networks: Linear Theory

A regular network is a network with one kind of node and one kind of coupling. We show that a codimension one bifurcation from a synchronous equilibrium in a regular network is at linear level isomorphic to a generalized eigenspace of the adjacency matrix of the network, at least when the dimension of the internal dynamics of each node is greater than 1. We also introduce the notion of a product network—a network where the nodes of one network are replaced by copies of another network. We show that generically the center subspace of a bifurcation in product networks is the tensor product of eigenspaces of the adjacency matrices of the two networks.