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Invariant Synchrony and Anti-synchrony Subspaces of Weighted Networks

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Abstract

The internal state of a cell in a coupled cell network is often described by an element of a vector space. Synchrony or anti-synchrony occurs when some of the cells are in the same or the opposite state. Subspaces of the state space containing cells in synchrony or anti-synchrony are called polydiagonal subspaces. We study the properties of several types of polydiagonal subspaces of weighted coupled cell networks. In particular, we count the number of such subspaces and study when they are dynamically invariant. Of special interest are the evenly tagged anti-synchrony subspaces in which the number of cells in a certain state is equal to the number of cells in the opposite state. Our main theorem shows that the dynamically invariant polydiagonal subspaces determined by certain types of couplings are either synchrony subspaces or evenly tagged anti-synchrony subspaces. A special case of this result confirms a conjecture about difference-coupled graph network systems.

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Data Availability

This paper’s data are computer-generated and available at Nijholt et al. (2022).

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Acknowledgements

E.N. acknowledges the support of the Center for Research in Mathematics Applied to Industry (FAPESP Cemeai grant 2013/07375-0) and the Serrapilheira Institute (Grant No. Serra-1709-16124)

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, London, UK

    Eddie Nijholt

  2. Department of Mathematics and Statistics, Northern Arizona University, PO Box 5717, Flagstaff, AZ, 86011-5717, USA

    Nándor Sieben & James W. Swift

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  1. Eddie Nijholt
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  2. Nándor Sieben
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  3. James W. Swift
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Correspondence to James W. Swift.

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Nijholt, E., Sieben, N. & Swift, J.W. Invariant Synchrony and Anti-synchrony Subspaces of Weighted Networks. J Nonlinear Sci 33, 63 (2023). https://doi.org/10.1007/s00332-023-09924-y

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