Skip to main content
SpringerLink
Log in

Periodic synchronization in a system of coupled phase oscillators with attractive and repulsive interactions

  • Research Article
  • Published:
Frontiers of Physics Aims and scope Submit manuscript

Abstract

In this study we investigate the collective behavior of the generalized Kuramoto model with an external pinning force in which oscillators with positive and negative coupling strengths are conformists and contrarians, respectively. We focus on a situation in which the natural frequencies of the oscillators follow a uniform probability density. By numerically simulating the model, it is shown that the model supports multistable synchronized states such as a traveling wave state, π state and periodic synchronous state: an oscillating π state. The oscillating π state may be characterized by the phase distribution oscillating in a confined region and the phase difference between conformists and contrarians oscillating around π periodically. In addition, we present the parameter space of the oscillating π state and traveling wave state of the model.

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, International symposium on mathematical problems in theoretical physics, in: H. Araki (Editor), Lecture Notes in Physics 39 (420–422), New York: Springer 1975

    Google Scholar 

  2. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80(4), 1275 (2008)

    Article  ADS  Google Scholar 

  3. C. von Cube, S. Slama, D. Kruse, C. Zimmermann, P. W. Courteille, G. R. M. Robb, N. Piovella, and R. Bonifacio, Self-synchronization and dissipation-induced threshold in collective atomic recoil lasing, Phys. Rev. Lett. 93(8), 083601 (2004)

    Article  ADS  Google Scholar 

  4. J. Javaloyes, M. Perrin, and A. Politi, Collective atomic recoil laser as a synchronization transition, Phys. Rev. E 78(1), 011108 (2008)

    Article  ADS  Google Scholar 

  5. M. Wickramasinghe and I. Z. Kiss, Phase synchronization of three locally coupled chaotic electrochemical oscillators: Enhanced phase diffusion and identification of indirect coupling, Phys. Rev. E 83(1), 016210 (2011)

    Article  ADS  Google Scholar 

  6. I. Z. Kiss, W. Wang, and J. L. Hudson, Populations of coupled electrochemical oscillators, Chaos 12(1), 252 (2002)

    Article  ADS  Google Scholar 

  7. J. W. Swift, S. H. Strogatz, and K. Wiesenfeld, Averaging of globally coupled oscillators, Physica D 55(3–4), 239 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. K. Wiesenfeld, P. Colet, and S. H. Strogatz, Synchronization transitions in a disordered Josephson series array, Phys. Rev. Lett. 76(3), 404 (1996)

    Article  ADS  Google Scholar 

  9. K. Wiesenfeld, P. Colet, and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E 57(2), 1563 (1998)

    Article  ADS  Google Scholar 

  10. G. Grüner, The dynamics of charge-density waves, Rev. Mod. Phys. 60(4), 1129 (1988)

    Article  ADS  Google Scholar 

  11. C. M. Marcus, S. H. Strogatz, and R. M. Westervelt, Delayed switching in a phase-slip model of charge-densitywave transport, Phys. Rev. B 40(8), 5588 (1989)

    Article  ADS  Google Scholar 

  12. J. Buck and E. Buck, Synchronous fireflies, Sci. Am. 234(5), 74 (1976)

    Article  Google Scholar 

  13. C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Science Publication, 268–278, New York: Springer, 1975

    Google Scholar 

  14. I. Z. Kiss, et al., Emerging coherence in a population of chemical oscillators, Science 296(5573), 1676 (2002)

    Article  ADS  Google Scholar 

  15. G. Filatrella, A. H. Nielsen, and N. F. Pedersen, Analysis of a power grid using a Kuramoto-like model, Eur. Phys. J. B 61(4), 485 (2008)

    Article  ADS  Google Scholar 

  16. M. Rohden, A. Sorge, M. Timme, and D. Witthaut, Selforganized synchronization in decentralized power grids, Phys. Rev. Lett. 109(6), 064101 (2012)

    Article  ADS  Google Scholar 

  17. F. Dorfler, M. Chertkov, and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proc. Natl. Acad. Sci. USA 110(6), 2005 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Z. Néda, E. Ravasz, T. Vicsek, Y. Brechet, and A. L. Barabási, Physics of the rhythmic applause, Phys. Rev. E 61(6), 6987 (2000)

    Article  ADS  Google Scholar 

  19. J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77(1), 137 (2005)

    Article  ADS  Google Scholar 

  20. A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16(1), 15 (1967)

    Article  Google Scholar 

  21. H. Daido, Population dynamics of randomly interacting self-oscillators, Prog. Theor. Phys. 77(3), 622 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  22. C. Börgers, S. Epstein, and N. J. Kopell, Background gamma rhythmicity and attention in cortical local circuits: A computational study, Proc. Natl. Acad. Sci. USA 102(19), 7002 (2005)

    Article  ADS  Google Scholar 

  23. C. Börgers and N. Kopell, Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity, Neural Comput. 15(3), 509 (2003)

    Article  MATH  Google Scholar 

  24. H. Hong and S. H. Strogatz, Kuramoto model of coupled oscillators with positive and negative coupling parameters: An example of conformist and contrarian oscillators, Phys. Rev. Lett. 106(5), 054102 (2011)

    Article  ADS  Google Scholar 

  25. H. Hong and S. H. Strogatz, Conformists and contrarians in a Kuramoto model with identical natural frequencies, Phys. Rev. E 84(4), 046202 (2011)

    Article  ADS  Google Scholar 

  26. C. Freitas, E. Macau, and A. Pikovsky, Partial synchronization in networks of non-linearly coupled oscillators: The Deserter Hubs Model, Chaos 25(4), 043119 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. H. Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields, Prog. Theor. Phys. 79(1), 39 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  28. H. Kori and A. S. Mikhailov, Strong effects of network architecture in the entrainment of coupled oscillator systems, Phys. Rev. E 74(6), 066115 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  29. T. M. Jr Antonsen, R. T. Faghih, M. Girvan, E. Ott, and J. Platig, External periodic driving of large systems of globally coupled phase oscillators, Chaos 18(3), 037112 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos 18(3), 037113 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. S. H. Park and S. Kim, Noise-induced phase transitions in globally coupled active rotators, Phys. Rev. E 53(4), 3425 (1996)

    Article  ADS  Google Scholar 

  32. S. Shinomoto and Y. Kuramoto, Phase transitions in active rotator systems, Prog. Theor. Phys. 75(5), 1105 (1986)

    Article  ADS  Google Scholar 

  33. H. Hong, Periodic synchronization and chimera in conformist and contrarian oscillators, Phys. Rev. E 89(6), 062924 (2014)

    Article  ADS  Google Scholar 

  34. D. Hansel, G. Mato, and C. Meunier, Clustering and slow switching in globally coupled phase oscillators, Phys. Rev. E 48(5), 3470 (1993)

    Article  ADS  Google Scholar 

  35. O. Burylko, Y. Kazanovich, and R. Borisyuk, Bifurcation study of phase oscillator systems with attractive and repulsive interaction, Phys. Rev. E 90(2), 022911 (2014)

    Article  ADS  Google Scholar 

  36. C. Bick, M. Timme, D. Paulikat, D. Rathlev, and P. Ashwin, Chaos in symmetric phase oscillator networks, Phys. Rev. Lett. 107(24), 244101 (2011)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant Nos. 11447001, 11475004, and U1504108), the Key Project of Scientific and Technological Research of the Education Department of Henan Province (Grant Nos. 16A140002, 18A140012, and 18B140001), and the Innovation Foundation for Students of Anyang Normal University (Grant No. ASCX/2017-Z59).

Author information

Authors and Affiliations

  1. School of Physics and Electrical Engineering, Anyang Normal University, Anyang, 455000, China

    Di Yuan, Jun-Long Tian, Fang Lin, Dong-Wei Ma, Jing Zhang & Hai-Tao Cui

  2. School of Physics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Yi Xiao

Authors
  1. Di Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun-Long Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fang Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dong-Wei Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Hai-Tao Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Yi Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Di Yuan or Jun-Long Tian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuan, D., Tian, JL., Lin, F. et al. Periodic synchronization in a system of coupled phase oscillators with attractive and repulsive interactions. Front. Phys. 13, 130504 (2018). https://doi.org/10.1007/s11467-018-0748-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11467-018-0748-4

Keywords

Search

Navigation