Homogeneous three-cell networks

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.