Skip to main content
SpringerLink
Log in

Phase chaos and multistability in the discrete Kuramoto model

  • Published:
Nonlinear Oscillations

Abstract

The paper describes the appearance of a novel high-dimensional chaotic regime, called phase chaos, in the discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small and intermediate values of the coupling strength. It is caused by the nonlinear interaction of the oscillators, while the individual oscillators behave periodically when left uncoupled. For the four-dimensional discrete Kuramoto model, we outline the region of phase chaos in the parameter plane, distinguish the region where the phase chaos coexists with other periodic attractors, and demonstrate, in addition, that the transition to the phase chaos takes place through the torus destruction scenario.

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. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin (1984).

    MATH  Google Scholar 

  2. H. Haken, Principles of Brain Functioning, Springer, Berlin-Heidelberg (1996).

    MATH  Google Scholar 

  3. I. Blekhman, Synchronization in Science and Technology, Asme Press, New York (1988).

    Google Scholar 

  4. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization—A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge (2001).

    MATH  Google Scholar 

  5. E. Mosekilde, Yu. Maistrenko, and D. Postnov, Chaotic Synchronization. Application to Living Systems, World Scientific, New Jersey (2002).

    Google Scholar 

  6. A. Goldbeter and B. Wurster, “Regular oscillations in suspensions of a putatively chaotic mutant of Dictyostelium discoideum,” Experientia, 45, 363–365 (1989).

    Article  Google Scholar 

  7. E. Gylfe, E. Grapenqiesser, and B. Hillman, “Propagation of cytoplasmic Ca2+ oscillations in clusters of pancreatic β-cells exposed to glucose,” Cell Calcium, 12, 229–240 (1991).

    Article  Google Scholar 

  8. S. Danø, P. G. Sørensen, and F. Hynne, “Sustained oscillations in living cells,” Nature, 402, 320–322 (1999).

    Article  Google Scholar 

  9. N.-H. Holstein-Rathlou, K.-P. Yip, O. V. Sosnovtseva, and E. Mosekilde, “Synchronization phenomena in nephron-nephron interaction,” Chaos, 11, 417–426 (2001).

    Article  Google Scholar 

  10. P. Hadley, M. R. Beasley, and K. Wiesenfeld, “Phase locking of Josephson-junction series arrays,” Phys. Rev. B, 38, 8712–8719 (1988).

    Article  Google Scholar 

  11. I. Z. Kiss, Y. Zhai, and J. L. Hudson, “Emerging coherence in a population of chemical oscillators,” Science, 296, 1676–1678 (2002).

    Article  Google Scholar 

  12. N. F. Rulkov, “Images of synchronized chaos: experiments with circuits,” Chaos, 6, 262–279 (1996).

    Article  MathSciNet  Google Scholar 

  13. D. E. Postnov, A. G. Balanov, O. V. Sosnovtseva, and E. Mosekilde, “Chaotic hierarchy in high dimensions,” Int. J. Mod. Phys. B, 14, 2511–2527 (2000).

    Google Scholar 

  14. S. De Monte, F. d'Ovidio, and E. Mosekilde, “Coherent regimes of globally coupled dynamical systems,” Phys. Rev. Lett., 90 (2003).

  15. P. A. Tass, Phase Resetting in Medicine and Biology, Springer, Berlin (1999).

    MATH  Google Scholar 

  16. Yu. L. Maistrenko, O. V. Popovych, and P. A. Tass, “Desynchronization and chaos in the Kuramoto model,” in: Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Springer, Berlin (2005), pp. 285–306.

    Chapter  Google Scholar 

  17. O. V. Popovych, Yu. L. Maistrenko, and P. A. Tass, “Phase chaos in coupled oscillators,” Phys. Rev. E, 71 (2005).

  18. Yu. Maistrenko, O. Popovych, O. Burylko, and P. Tass, “Mechanism of desynchronization in the finite-dimensional Kuramoto model,” Phys. Rev. Lett., 93 (2004).

  19. A. Vasylenko, Yu. Maistrenko, and M. Hasler, “Modelling the phase synchronization in systems of two and three coupled oscillators,” Nonlin. Oscillations, 7, No. 3, 311–327 (2004).

    MATH  MathSciNet  Google Scholar 

  20. V. I. Arnol'd, V. S. Afraimovich, Yu. S. Il'yashenko, and L. P. Shilnikov, Dynamical Systems, Springer, Berlin (1977).

    Google Scholar 

  21. Z. Liu, Y.-C. Lai, and M. A. Matias, “Universal scaling of Lyapunov exponents in coupled chaotic oscillators,” Phys. Rev. E, 67 (2003).

  22. N. Nakagawa and Y. Kuramoto, “Anomalous Lyapunov spectrum in globally coupled oscillators,” Physica D, 80, 307–316 (1995).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev, Ukraine

    V. Maistrenko, A. Vasylenko & Yu. Maistrenko

  2. Institute of Medicine, Research Centre Jülich, Jülich, Germany

    Yu. Maistrenko

  3. Department of Physics, Technical University of Denmark, Kongens Lyngby, Denmark

    E. Mosekilde

Authors
  1. V. Maistrenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Vasylenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu. Maistrenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. Mosekilde
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Published in Neliniini Kolyvannya, Vol. 11, No. 2, pp. 217–229, April–June, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maistrenko, V., Vasylenko, A., Maistrenko, Y. et al. Phase chaos and multistability in the discrete Kuramoto model. Nonlinear Oscill 11, 229–241 (2008). https://doi.org/10.1007/s11072-008-0026-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11072-008-0026-4

Keywords

Search

Navigation