M.C.A. Leite and M. Golubitsky
Homogeneous three-cell networks
Nonlinearity.
19 (2006) 2313-2363.
A cell is a system of differential equations. Coupled cell systems
are networks of cells. The architecture of a coupled cell network
is a graph indicating which cells are identical and which cells are
coupled to which. In this paper we continue the work of Stewart,
Golubitsky, Pivato, and Torok by classifying all homogeneous
three-cell networks (where each cell has at most two inputs), and
classifying all generic codimension-one steady-state and Hopf bifurcations
from a synchronous equilibrium. We use combinatorial arguments to show
that there are 34 distinct homogeneous three-cell networks as opposed
to only three such two-cell networks.
We show that codimension one bifurcations in homogeneous three-cell
networks can exhibit interesting features that are due to network
architecture. Indeed, network architecture determines, even at
linear level, the kind of generic transitions from a synchronous
equilibrium that can occur as we vary one parameter, and plays a
crucial role in establishing how the solutions on the bifurcating
branches manifest themselves in each cell.