Homogeneous Coupled Cell Networks with S3-symmetric Quotient

A coupled cell network represents dynamical systems (the coupled cell systems) that can be seen as a set of individual dynamical systems (the cells) with interactions between them. Every coupled cell system associated to a network, when restricted to a flow-invariant subspace defined by the equality of certain cell coordinates, corresponds to a coupled cell system associated to a smaller network, called quotient network. In this paper we consider homogeneous networks admitting a S3-symmetric quotient network. We assume that a codimension-one synchrony-breaking bifurcation from a synchronous equilibrium occurs for that quotient network. We aim to investigate, for different networks admitting that S3-symmetric quotient, if the degeneracy condition leading to that bifurcation gives rise to branches of steady-state solutions outside the flow-invariant subspace associated with the quotient network. We illustrate that the existence of new solutions can be justified directly or not by the symmetry of the original network. The bifurcation analysis of a six-cell asymmetric network suggests that the existence of new solutions outside the flow-invariant subspace associated with the quotient is ‘forced’ by the symmetry of a five-cell quotient network. 2000 Mathematics Subject Classification. Primary: 34A34, 34C15, 34C23, 37G40.