| Coupled cell systems are networks of differential equations. The architecture of a coupled cell network is a graph indicating which cells are identical and which cells are coupled to which. The aim of this work is (1) classify all different homogeneous three-cell networks with each cell having at most two inputs; (2) classify generic local 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.; Bifurcation theory, either in general systems or in symmetric systems, does not apply directly to coupled cell systems. Therefore, to classify the generic local codimension-one bifurcations we use standard techniques from bifurcation analysis, but we do not use the standard results. We also use the coupled cell system theory developed by Stewart, Golubitsky, Pivato, Torok and co-workers.; This work shows that homogeneous three-cell networks can exhibit interesting features due to network architecture. Surprisingly, the network architecture (a) determines, even at the linear level, the kind of generic local transitions from a synchronous equilibrium that can occur in the 34 homogeneous three-cell networks as we vary one parameter; (b) plays a crucial role in establishing how the solutions on the bifurcating branches manifest themselves in each cell. |
