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Synchrony patterns in Laplacian networks

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Abstract

A network of coupled dynamical systems is represented by a graph whose vertices represent individual cells and whose edges represent couplings between cells. Motivated by the impact of synchronization results of the Kuramoto networks, we introduce the generalized class of Laplacian networks, governed by mappings whose Jacobian at any point is a symmetric matrix with row entries summing to zero. By recognizing this matrix with a weighted Laplacian of the associated graph, we derive the optimal estimates of its positive, null and negative eigenvalues directly from the graph topology. Furthermore, we provide a characterization of the mappings that define Laplacian networks. Lastly, we discuss stability of equilibria inside synchrony subspaces for two types of Laplacian network on a ring with some extra couplings.

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Data Availability

The data that support the synchrony computation code are openly available at the links below: https://drive.google.com/file/d/1-ogTHwezqxY4PCLgXQ12ZbBBqISQTERi/view?usp=sharing and https://drive.google.com/file/d/1RUUYSppXsWaUT-VnhbHZPZepqtmUjopl/view?usp=sharing

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Acknowledgements

TAA acknowledges financial support by FAPESP grant 2019/2130-0, and MM acknowledges financial support by FAPESP grant 2019/21181-0.

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  1. Mathematics Department, ICMC, University of São Paulo, 13560-970, P.O. box 668, São Carlos, SP, Brazil

    Tiago de Albuquerque Amorim & Miriam Manoel

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  1. Tiago de Albuquerque Amorim
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  2. Miriam Manoel
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Correspondence to Miriam Manoel.

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de Albuquerque Amorim, T., Manoel, M. Synchrony patterns in Laplacian networks. Res Math Sci 11, 23 (2024). https://doi.org/10.1007/s40687-024-00428-z

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