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

  • Zahra Aminzare
  • Biswadip Dey
  • Elizabeth N. Davison
  • Naomi Ehrich Leonard


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.


Cluster synchronization Contraction theory for stability Diffusively coupled nonlinear networks Neuronal oscillators 



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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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Zahra Aminzare
    • 1
  • Biswadip Dey
    • 2
  • Elizabeth N. Davison
    • 2
  • Naomi Ehrich Leonard
    • 2
  1. 1.The Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrincetonUSA

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