Local bifurcation in symmetric coupled cell networks: Linear theory

We consider a coupled cell network of differential equations with finite symmetry group Γ , where Γ 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. us on the analysis of the linearized vector field at a fully symmetric equilibrium and show that in the case of active cells, if Γ is Abelian the network structure does not influence the types of codimension one local bifurcations. We also show that beyond this context, when Γ is not Abelian, cells are passive, or when considering local bifurcations of higher codimensions, anomalies due to the network structure may arise.