Homogeneous coupled cell networks with s_{3}-symmetric quotient

Pages: 1 - 9, Issue Special, September 2007

Abstract
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Manuela A. D. Aguiar - Centro de Matemática da Universidade do Porto, Faculdade de Economia, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal (email)

Ana Paula S. Dias - Centro de Matemática da Universidade do Porto (CMUP) and Dep. de Matemática Pur, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal (email)

Martin Golubitsky - Department of Mathematics, University of Houston, Houston TX 77204-3008, United States (email)

Maria Conceição A. Leite - Department of Mathematics, Purdue University, West Lafayette, IN 47906, United States (email)

Abstract:
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 S_{3}-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 S_{3}-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.

Keywords: Coupled cell network, S_3-symmetric quotient network, symmetry,synchrony-breaking bifurcation

Mathematics Subject Classification: Primary: 34A34, 34C15, 34C23, 37G40

Received: September 2006;
Revised:
February 2007;
Published: September 2007.