Abstract
Coupled cell systems are networks of dynamical systems (the cells), where the links between the cells are described through the network structure, the coupled cell network. Synchrony subspaces are spaces defined in terms of equalities of certain cell coordinates that are flow-invariant for all coupled cell systems associated with a given network structure. The intersection of synchrony subspaces of a network is also a synchrony subspace of the network. It follows, then, that, given a coupled cell network, its set of synchrony subspaces, taking the inclusion partial order relation, forms a lattice. In this paper we show how to obtain the lattice of synchrony subspaces for a general network and present an algorithm that generates that lattice. We prove that this problem is reduced to obtain the lattice of synchrony subspaces for regular networks. For a regular network we obtain the lattice of synchrony subspaces based on the eigenvalue structure of the network adjacency matrix.
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Notes
\(V_k\) is a subspace of \(\ker \left( A- \lambda _i \mathrm{Id}_n\right) ^k\).
\(<B_C> \subseteq \ker \left( A- \lambda _i \mathrm{Id}_n\right) ^{k-1} \subseteq V_k\).
If row \(r_j-r_k\) of \(\overline{M}\) is zero, then \(x_j=x_k\) for all vectors in \(V_k\).
Equivalently, \(C_N\) is the set of equality conditions satisfied by the vectors in \(<\overline{\overline{B}}_k>\).
See Remark 6.7.
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Acknowledgments
The authors thank the referees for their comments, which improved the paper’s presentation. Research was partially funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT under Projects PEst-C/MAT/UI0144/2011 and PTDC/MAT/100055/2008.
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Communicated by P. Newton.
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Aguiar, M.A.D., Dias, A.P.S. The Lattice of Synchrony Subspaces of a Coupled Cell Network: Characterization and Computation Algorithm. J Nonlinear Sci 24, 949–996 (2014). https://doi.org/10.1007/s00332-014-9209-6
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DOI: https://doi.org/10.1007/s00332-014-9209-6