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

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

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)2235-2257
Number of pages23
JournalJournal of Nonlinear Science
Issue number5
StatePublished - Oct 1 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • General Engineering
  • Applied Mathematics


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


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