M. Golubitsky and R. Lauterbach
Bifurcations from synchrony in homogeneous networks: linear theory
SIAM J. Appl. Dynam. Sys.
8 (1) (2009) 40-75.
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 wh
en 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 generalized eigenspaces of the adjacency matrices
of the two networks.