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.
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Communicated by E. Knobloch.
Research of M.F. supported in part by NSF Grants DMS-0600927 & DMS-0806321 and of P.A. by EPSRC Grant EP/C510771. Research of M.A. and A.D. supported in part by CMUP financed by FCT through the programmes POCTI and POSI, with Portuguese and European Community structural funds.
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Aguiar, M., Ashwin, P., Dias, A. et al. Dynamics of Coupled Cell Networks: Synchrony, Heteroclinic Cycles and Inflation. J Nonlinear Sci 21, 271–323 (2011). https://doi.org/10.1007/s00332-010-9083-9
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DOI: https://doi.org/10.1007/s00332-010-9083-9