Skip to main content
SpringerLink
Log in

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

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Winfree, A. T., Biolopgical Rhythms and the Behavior of Coupled Oscillators, J. Theoret. Biol., 1967, vol. 16, pp. 15–42.

    Article  Google Scholar 

  2. Kuramoto, Y., Chemical Oscillations, Waves and Turbulence, Berlin: Springer, 1984.

    Book  MATH  Google Scholar 

  3. Kuramoto, Y., Self-Entrainment of a Population of Coupled Non-Linear Oscillators, in Internat. Symp. on Mathematical Problems in Theoretical Physics (Kyoto University, Kyoto (Japan), January 23–29, 1975), H. Araki (Ed.), Lect. Notes Phys., vol. 39, New York: Springer, 1975, pp. 420–422.

    Chapter  Google Scholar 

  4. Michaels, D.C., Matyas, E.P., and Jalife, J., Mechanisms of Sinoatrial Pacemaker Synchronization: A New Hypothesis, Circ. Res., 1987, vol. 61, no. 5, pp. 704–714.

    Article  Google Scholar 

  5. Brown, E., Holmes, P., and Moehlis, J., Globally Coupled Oscillator Networks, in Perspectives and Problems in Nonlinear Science: A Celebratory Volume in Honor of Larry Sirovich, E. Kaplan, J. E. Marsden, K.R. Sreenivasan (Eds.), New York: Springer, 2003, pp. 183–215.

    Chapter  Google Scholar 

  6. Kopell, N. and Ermentrout, G. B., Coupled Oscillators and the Design of Central Pattern Generators, Math. Biosci., 1988, vol. 90, pp. 87–109.

    Article  MATH  MathSciNet  Google Scholar 

  7. Wiesenfeldt, K., Colet, P., and Strogatz, S., Frequency Locking in Josephson Junction Arrays: Connection with the Kuramoto Model, Phys. Rev. E, 1998, vol. 57, pp. 1563–1567.

    Article  Google Scholar 

  8. Neda, Z., Ravasz, E., Vicsek, T., Brecht, Y., Barabasi, A.-L., Physics of the Rhythmic Applause, Phys. Rev. E, 2000, vol. 61, pp. 6987–6992.

    Article  Google Scholar 

  9. Strogatz, S.H., Abrams, D.M., McRobie, A., Eckhardt, B., Ott, E., Theoretical Mechanics: Crowd Synchrony on the Millenium Bridge, Nature, 2005, vol. 438(70640), pp. 43–44.

    Article  Google Scholar 

  10. Kozyrev, G., Vladimirov, A. G., and Mandel, P., Global Coupling with the Time Delay in an Array of Semiconductor Lasers, Phys. Rev. Lett., 2000, vol. 85, no. 18, pp. 3809–3812.

    Article  Google Scholar 

  11. York, R.A. and Compton, R. C., Quasi-Optical Power Combining Using Mutually Synchronized Oscillator Arrays, IEEE Trans. Automat. Contr., 2012, vol. 57, no. 4, pp. 920–935.

    Article  Google Scholar 

  12. Dorfler, F. and Bullo, F., Synchronization in Complex Networks of Phase Oscillators: A Survey, Automatica, 2014, vol. 50, no. 6, pp. 1539–1564.

    Article  MathSciNet  Google Scholar 

  13. Acebron, J. A., Bonilla, L. L., Vicente, C. J.P., Ritort, F., and Spigler, R., The Kuramoto Model: A Simple Paradigm for Synchronization Phenomena, Rev. Modern Phys., 2005, vol. 77, no. 1, pp. 137–185.

    Article  Google Scholar 

  14. Gupta, Sh., Campa, A., and Ruffo, S., Kuramoto Model of Synchronization: Equilibrium and Nonequilibrium Aspects, J. Stat. Mech. Theory Exp., 2014, no. 8, R08001, 61 pp.

    Google Scholar 

  15. Aeyels, D. and Rogge, J.A., Existence of Partial Entrainment and Stability of Phase Locking Behavior of Coupled Oscillators, Progr. Theor. Exp. Phys., 2004, vol. 112, no. 6, pp. 921–942.

    Article  MATH  Google Scholar 

  16. Mirollo, R.E. and Strogatz, S.H., The Spectrum of the Partially Locked State of the Kuramoto Model of Coupled Oscillators, Phys. D, 2005, vol. 205, nos. 1–4, pp. 249–266.

    Article  MATH  MathSciNet  Google Scholar 

  17. Verwoerd, M. and Mason, O., Global Phase-Locking in Finite Populations of Phase-Coupled Oscillators, SIAM J. Appl. Dyn. Syst., 2008, vol. 7, no. 1, pp. 134–160.

    Article  MATH  MathSciNet  Google Scholar 

  18. Jadbabaie, A., Motee, N., and Barahona, N., On the Stability of the Kuramoto Model of Coupled Oscillators, in Proc. of the American Control Conference (Boston,Mass., June 30 2004–July 2, 2004): Vol. 5, pp. 4296–4301.

Download references

Author information

Authors and Affiliations

  1. Department of Control Theory, Nizhny Novgorod University, ul. Gagarina 23, Nizhny Novgorod, 603950, Russia

    Vladimir N. Belykh, Valentin S. Petrov & Grigory V. Osipov

Authors
  1. Vladimir N. Belykh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valentin S. Petrov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Grigory V. Osipov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir N. Belykh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belykh, V.N., Petrov, V.S. & Osipov, G.V. Dynamics of the finite-dimensional Kuramoto model: Global and cluster synchronization. Regul. Chaot. Dyn. 20, 37–48 (2015). https://doi.org/10.1134/S1560354715010037

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354715010037

MSC2010 numbers

Keywords

Search

Navigation