Hopf Bifurcation in Coupled Cell Networks with Interior Symmetries

We consider an important class of non-symetric networks that lies between the class of general networks and the class of symmetric networks, where group theoretic methods still apply – namely, networks admitting “interior symmetries”. The main result of this paper is the full analogue of the Equivariant Hopf Theorem for networks with symmetries. We extend the result of Golubitsky, Pivato and Stewart (Interior symmetry and local bifurcation in coupled cell networks, Dynamical Systems 19 (4) (2004) 389–407) to obtain states whose linearizations on certain subsets of cells, near bifurcation, are superpositions of synchronous states with states having spatio-temporal symmetries. AMS Classification Codes: 34C15, 34C25, 37G40