Skip to main content
SpringerLink
Log in

Collective chaos and period-doubling bifurcation in globally coupled phase oscillators

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Collective chaos has been intensively investigated in globally coupled map and oscillators in which the single unit is capable of producing chaos. In this work, we study collective chaos in globally coupled phase oscillators. We consider a model with a trimodal natural frequency distribution. We find that increasing coupling strength leads to series of bifurcations such as Hopf bifurcation, saddle–node bifurcation, and period-doubling bifurcation. We observe the collective chaotic partial synchronous state, which is developed from quasiperiodic partial synchronous ones through a cascade of period-doubling bifurcations. We also explore the impacts of model parameters on the conditions producing collective chaos.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Crawford, J.D.: Amplitude expansions for instabilities in populations of globally-coupled oscillators. J. Stat. Phys. 74, 1047–1084 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D 143, 1–20 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Acebrón, J.A., Bonilla, L.L., Pérez Vicente, C.J., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005)

    Article  Google Scholar 

  4. Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008)

    Article  MathSciNet  Google Scholar 

  5. Shibata, T., Kaneko, K.: Collective chaos. Phys. Rev. Lett. 81, 4116–4119 (1998)

    Article  Google Scholar 

  6. Cencini, M., Falcioni, M., Vergni, D., Vulpiani, A.: Macroscopic chaos in globally coupled maps. Phys. D 130, 58–72 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Matthews, P.C., Mirollo, R.E., Strogatz, S.H.: Dynamics of a large system of coupled nonlinear oscillators. Phys. D 52, 293–331 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Nakagawa, N., Kuramoto, Y.: Anomalous Lyapunov spectrum in globally coupled oscillators. Phys. D 80, 307–316 (1995)

  9. Hakim, V., Rappel, W.-J.: Dynamics of the globally coupled complex Ginzburg–Landau equation. Phys. Rev. A 46, R7347–7350 (1992)

  10. Olmi, S., Politi, A., Torcini, A.: Collective chaos in pulse-coupled neural networks. EPL 92, 60007 (2010)

    Article  Google Scholar 

  11. Pazó, D., Montbrió, E.: From quasiperiodic partial synchronization to collective chaos in populations of inhibitory neurons with delay. Phys. Rev. Lett. 116, 238101 (2016)

    Article  Google Scholar 

  12. So, P., Barreto, E.: Generating macroscopic chaos in a network of globally coupled phase oscillators. Chaos 21, 033127 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Komarov, M., Pikovsky, A.: Dynamics of multifrequency oscillator communities. Phys. Rev. Lett. 110, 134101 (2013)

    Article  Google Scholar 

  14. Alonso, L.M., Mindlin, G.B.: Average dynamics of a driven set of globally coupled excitable units. Chaos 21, 023102 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Skardal, P.S., Restrepo, J.G., Ott, E.: Frequency assortativity can induce chaos in oscillator networks. Phys. Rev. E 91, 060902(R) (2015)

    Article  MathSciNet  Google Scholar 

  16. Hemmen, J.L.V., Wreszinski, W.F.: Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators. J. Stat. Phys. 72, 145 (1993)

    Article  MATH  Google Scholar 

  17. Ott, E., Antonsen, T.M.: Low dimensional behavior of large systems of globally coupled oscillators. Chaos 18, 037113 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ott, E., Antonsen, T.M.: Long time evolution of phase oscillator systems. Chaos 19, 023117 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Martens, E.A., Barreto, E., Strogatz, S.H., Ott, E., So, P., Antonsen, T.M.: Exact results for the Kuramoto model with a bimodal frequency distribution. Phys. Rev. E 79, 026204 (2009)

    Article  MathSciNet  Google Scholar 

  20. Hong, H., Strogatz, S.H.: Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators. Phys. Rev. Lett. 106, 054102 (2011)

    Article  Google Scholar 

  21. Ju, P., Dai, Q., Cheng, H., Yang, J.: Dynamics in the Sakaguchi-Kuramoto model with two subpopulations. Phys. Rev. E 90, 012903 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by NSF of China under Grants Nos. 11505016, 11575036, and 71301012.

Author information

Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China

    Hongyan Cheng, Shuangjian Guo, Qionglin Dai, Haihong Li & Junzhong Yang

Authors
  1. Hongyan Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuangjian Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qionglin Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Haihong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Junzhong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junzhong Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, H., Guo, S., Dai, Q. et al. Collective chaos and period-doubling bifurcation in globally coupled phase oscillators. Nonlinear Dyn 89, 2273–2281 (2017). https://doi.org/10.1007/s11071-017-3585-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3585-z

Keywords

Search

Navigation