Local bifurcation in symmetric coupled cell networks: Linear theory

We consider a coupled cell network of differential equations with finite symmetry group Gamma, where Gamma permutes cells transitively. We show how the structure of the coupled cell network, represented by a directed graph whose vertices represent individual cells and edges represent couplings, can be taken into account in the bifurcation analysis of a fully symmetric steady-state solution. We focus on the analysis of the linearized vector field at a fully symmetric equilibrium and show that in the case of active cells, if Gamma is Abelian the network structure does not influence the types of codimension one local bifurcations. We also show that beyond this context, when Gamma is not Abelian, cells are passive, or when considering local bifurcations of higher codimensions, anomalies due to the network structure may arise. (c) 2006 Elsevier B.V. All rights reserved.