Journal of Nonlinear Science

, Volume 21, Issue 2, pp 271–323 | Cite as

Dynamics of Coupled Cell Networks: Synchrony, Heteroclinic Cycles and Inflation

Article

Abstract

We consider the dynamics of small networks of coupled cells. We usually assume asymmetric inputs and no global or local symmetries in the network and consider equivalence of networks in this setting; that is, when two networks with different architectures give rise to the same set of possible dynamics. Focussing on transitive (strongly connected) networks that have only one type of cell (identical cell networks) we address three questions relating the network structure to dynamics. The first question is how the structure of the network may force the existence of invariant subspaces (synchrony subspaces). The second question is how these invariant subspaces can support robust heteroclinic attractors. Finally, we investigate how the dynamics of coupled cell networks with different structures and numbers of cells can be related; in particular we consider the sets of possible “inflations” of a coupled cell network that are obtained by replacing one cell by many of the same type, in such a way that the original network dynamics is still present within a synchrony subspace. We illustrate the results with a number of examples of networks of up to six cells.

Keywords

Coupled cell network Synchrony Heteroclinic cycle Inflation 

Mathematics Subject Classification (2000)

37A250 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.CMUP and Faculdade de EconomiaUniversidade do PortoPortoPortugal
  2. 2.Mathematics Research Institute, SECaMUniversity of ExeterExeterUK
  3. 3.CMUP and Dep. de Matemática, Faculdade de CiênciasUniversidade do PortoPortoPortugal
  4. 4.Department of MathematicsUniversity of HoustonHoustonUSA

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