Hopf Bifurcation In Coupled Cell Networks With Interior Symmetries

We consider an important class of nonsymmetric 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 [Dyn. Syst., 19 (2004), pp. 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.