Regular and Chaotic Dynamics

, Volume 20, Issue 1, pp 37–48 | Cite as

Dynamics of the finite-dimensional Kuramoto model: Global and cluster synchronization

  • Vladimir N. BelykhEmail author
  • Valentin S. Petrov
  • Grigory V. Osipov


Synchronization phenomena in networks of globally coupled non-identical oscillators have been one of the key problems in nonlinear dynamics over the years. The main model used within this framework is the Kuramoto model. This model shows three main types of behavior: global synchronization, cluster synchronization including chimera states and totally incoherent behavior. We present new sufficient conditions for phase synchronization and conditions for an asynchronous mode in the finite-size Kuramoto model. In order to find these conditions for constant and time varying frequency mismatch, we propose a simple method of comparison which allows one to obtain an explicit estimate of the phase synchronization range. Theoretical results are supported by numerical simulations.


phase oscillators Kuramoto model global synchronization existence and stability conditions asynchronous mode 

MSC2010 numbers

34C25 34C28 34C46 37C75 


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Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • Vladimir N. Belykh
    • 1
    Email author
  • Valentin S. Petrov
    • 1
  • Grigory V. Osipov
    • 1
  1. 1.Department of Control TheoryNizhny Novgorod UniversityNizhny NovgorodRussia

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