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Cluster Synchronization of Diffusively Coupled Nonlinear Systems: A Contraction-Based Approach

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Abstract

Finding the conditions that foster synchronization in networked nonlinear systems is critical to understanding a wide range of biological and mechanical systems. However, the conditions proved in the literature for synchronization in nonlinear systems with linear coupling, such as has been used to model neuronal networks, are in general not strict enough to accurately determine the system behavior. We leverage contraction theory to derive new sufficient conditions for cluster synchronization in terms of the network structure, for a network where the intrinsic nonlinear dynamics of each node may differ. Our result requires that network connections satisfy a cluster-input-equivalence condition, and we explore the influence of this requirement on network dynamics. For application to networks of nodes with FitzHugh–Nagumo dynamics, we show that our new sufficient condition is tighter than those found in previous analyses that used smooth or nonsmooth Lyapunov functions. Improving the analytical conditions for when cluster synchronization will occur based on network configuration is a significant step toward facilitating understanding and control of complex networked systems.

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Notes

  1. If D is not diagonal, an appropriate change of coordinate can render it diagonal.

  2. The statement of Theorem 3 in Russo and Slotine (2010) is correct; however, the proof needs revision to be complete.

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Acknowledgements

This work was jointly supported by the National Science Foundation under NSF-CRCNS grant DMS-1430077 and the Office of Naval Research under ONR Grant N00014-14-1-0635. This material is also based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant DGE-1656466. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors thank the anonymous reviewers for their thoughtful and detailed comments.

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Authors and Affiliations

  1. The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Zahra Aminzare

  2. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 08540, USA

    Biswadip Dey, Elizabeth N. Davison & Naomi Ehrich Leonard

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  1. Zahra Aminzare
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  2. Biswadip Dey
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  4. Naomi Ehrich Leonard
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Correspondence to Zahra Aminzare.

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Communicated by Danielle S. Bassett.

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Aminzare, Z., Dey, B., Davison, E.N. et al. Cluster Synchronization of Diffusively Coupled Nonlinear Systems: A Contraction-Based Approach. J Nonlinear Sci 30, 2235–2257 (2020). https://doi.org/10.1007/s00332-018-9457-y

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