Bifurcations from regular quotient networks: A first insight

• Published in: Physica. D Vol. 238; no. 2; pp. 137 - 155
• Main Authors: Aguiar, Manuela A.D., Dias, Ana Paula S., Golubitsky, Martin, Leite, Maria da Conceição A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.01.2009; Elsevier
• Subjects: Bidirectional; Bifurcation; Bifurcations; Coupled cell network; Dynamical systems; Dynamics; Exact sciences and technology; Networks; Nonlinear dynamics; Physics; Quotient network; Quotients; Subspaces; Bifurcation; Coupled cell network; Quotient network; Equivalence relation; Necessary condition; Invariant subspace; Dynamics; Existence of solution; Non linear phenomenon
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2008.10.006

We consider regular (identical-edge identical-node) networks whose cells can be grouped into classes by an equivalence relation. The identification of cells in the same class determines a new network — the quotient network. In terms of the dynamics, this corresponds to restricting the coupled cell systems associated with a network to flow-invariant subspaces given by equality of certain cell coordinates. Assuming a bifurcation occurs for a coupled cell system restricted to the quotient network, we ask how that bifurcation lifts to the overall space. Surprisingly, for certain networks, new branches of solutions occur besides the ones that occur in the quotient network. To investigate this phenomenon we develop a systematic method that enumerates all networks with a given quotient. We also prove necessary conditions for the existence of solution branches not predicted by the quotient. We then apply our method to two particular quotient networks; namely, two- and three-cell bidirectional rings. We show there are no additional bifurcating solution branches when the quotient network is a two-cell bidirectional ring. However, two of the 12 five-cell networks that have the three-cell bidirectional ring as a quotient network exhibit bifurcating solutions that do not occur in the quotient itself. Thus, network architecture sometimes forces the existence of bifurcating branches in addition to the ones determined by the quotient.