Skip to main content
SpringerLink
Log in

Cluster synchronization in complex network of coupled chaotic circuits: An experimental study

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

Abstract

By a small-size complex network of coupled chaotic Hindmarsh-Rose circuits, we study experimentally the stability of network synchronization to the removal of shortcut links. It is shown that the removal of a single shortcut link may destroy either completely or partially the network synchronization. Interestingly, when the network is partially desynchronized, it is found that the oscillators can be organized into different groups, with oscillators within each group being highly synchronized but are not for oscillators from different groups, showing the intriguing phenomenon of cluster synchronization. The experimental results are analyzed by the method of eigenvalue analysis, which implies that the formation of cluster synchronization is crucially dependent on the network symmetries. Our study demonstrates the observability of cluster synchronization in realistic systems, and indicates the feasibility of controlling network synchronization by adjusting network topology.

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, Berlin: Springer, 1984

    Book  MATH  Google Scholar 

  2. A. S. Pikovsky, M. G. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Science, Cambridge: Cambridge University Press, 2001

    Book  MATH  Google Scholar 

  3. S. Strogatz, Sync: The Emerging Science of Spontaneous Order, New York: Hyperion, 2003

    Google Scholar 

  4. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, Complex networks: Structure and dynamics, Phys. Rep. 424(4–5), 175 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, and C. S. Zhou, Synchronization in complex networks, Phys. Rep. 469(3), 93 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  6. L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64(8), 821 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett. 80(10), 2109 (1998)

    Article  ADS  Google Scholar 

  8. G. Hu, J. Z. Yang, and W. Liu, Instability and controllability of linearly coupled oscillators: Eigenvalue analysis, Phys. Rev. E 58(4), 4440 (1998)

    Article  ADS  Google Scholar 

  9. L. Huang, Q. Chen, Y. C. Lai, and L. M. Pecora, Generic behavior of master-stability functions in coupled nonlinear dynamical systems, Phys. Rev. E 80(3), 036204 (2009)

    Article  ADS  Google Scholar 

  10. L. M. Pecora, Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems, Phys. Rev. E 58(1), 347 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  11. 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 

  12. V. N. Belykh and E. Mosekilde, One-dimensional map lattices: Synchronization, bifurcations, and chaotic structures, Phys. Rev. E 54(4), 3196 (1996)

    Google Scholar 

  13. M. Hasler, Yu. Maistrenko, and O. Popovych, Simple example of partial synchronizaiton of chaotic systems, Phys. Rev. E 58(5), 6843 (1998)

    Article  ADS  Google Scholar 

  14. Y. Zhang, G. Hu, H. A. Cerdeira, S. Chen, T. Braun, and Y. Yao, Partial synchronization and spontaneous spatial ordering in coupled chaotic systems, Phys. Rev. E 63(2), 026211 (2001)

    Article  ADS  Google Scholar 

  15. A. Pikovsky, O. Popovych, and Yu. Maistrenko, Resolving clusters in chaotic ensembles of globally coupled identical oscillators, Phys. Rev. Lett. 87(4), 044102 (2001)

    Article  ADS  Google Scholar 

  16. I. A. Heisler, T. Braun, Y. Zhang, G. Hu, and H. A. Cerdeira, Experimental investigation of partial synchronization in coupled chaotic oscillators, Chaos 13(1), 185 (2003)

    Article  ADS  Google Scholar 

  17. B. Ao and Z. G. Zheng, Partial synchronization on complex networks, Europhys. Lett. 74(2), 229 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  18. C. S. Zhou and J. Kurths, Hierarchical synchronization in complex networks with heterogeneous degrees, Chaos 16(1), 015104 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. J. Zhang, Y. Yu, and X. G. Wang, Synchronization of coupled metronomes on two layers, Front. Phys. 12(6), 120508 (2017)

    Article  Google Scholar 

  20. M. Golubitsky, I. Stewart, and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Springer-Verlag, 1985

    Book  MATH  Google Scholar 

  21. E. Basar, Brain Function and Oscillation, New York: Springer, 1998

    Book  MATH  Google Scholar 

  22. D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small world networks, Nature 393(6684), 440 (1998)

    Article  ADS  MATH  Google Scholar 

  23. A. L. Barabási and R. Albert, Emergence of scaling in random networks, Science 286(5439), 509 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. P. M. Gade, Synchronization in coupled map lattices with random nonlocal connectivity, Phys. Rev. E 54(1), 64 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  25. P. M. Gade and C. K. Hu, Synchronous chaos in coupled map lattices with small-world interactions, Phys. Rev. E 62(5), 6409 (2000)

    Article  ADS  Google Scholar 

  26. M. Barahona and L. M. Pecora, Synchronization in Small-World Systems, Phys. Rev. Lett. 89(5), 054101 (2002)

    Article  ADS  Google Scholar 

  27. T. Nishikawa, A. E. Motter, Y. C. Lai, and F. C. Hoppensteadt, Heterogeneity in oscillator networks: Are smaller worlds easier to synchronize? Phys. Rev. Lett. 91(1), 014101 (2003)

    Article  ADS  Google Scholar 

  28. A. E. Motter, C. Zhou, and J. Kurths, Weighted networks are more synchronizable: How and why, AIP Conf. Proc. 776, 201 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  29. X. G. Wang, Y. C. Lai, and C. H. Lai, Enhancing synchronization based on complex gradient networks, Phys. Rev. E 75(5), 056205 (2007)

    Article  ADS  Google Scholar 

  30. C. Fu, H. Zhang, M. Zhan, and X. G. Wang, Synchronous patterns in complex systems? Phys. Rev. E 85(6), 066208 (2012)

    Article  ADS  Google Scholar 

  31. Z. He, X. G. Wang, G. Y. Zhang, and M. Zhan, Control for a synchronization-desynchronization switch, Phys. Rev. E 90(1), 012909 (2014)

    Article  ADS  Google Scholar 

  32. W. Yang, W. Lin, X. G. Wang, and L. Huang, Synchronization of networked chaotic oscillators under external periodic driving, Phys. Rev. E 91(3), 032912 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  33. T. Nishikawa and A. E. Motter, Symmetric states requiring system asymmetry? Phys. Rev. Lett. 117(11), 114101 (2016)

    Article  ADS  Google Scholar 

  34. K. Park, L. Huang, and Y. C. Lai, Desynchronization waves in small-world networks, Phys. Rev. E 75(2), 026211 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  35. F. Sorrentino and E. Ott, Network synchronization of groups, Phys. Rev. E 76(5), 056114 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  36. V. N. Belykh, G. V. Osipov, V. S. Petrov, J. A. K. Suykens, and J. Vandewalle, Cluster synchronization in oscillatory networks, Chaos 18(3), 037106 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. G. Russo and J. J. E. Slotine, Symmetries, stability, and control in nonlinear systems and networks, Phys. Rev. E 84(4), 041929 (2011)

    Google Scholar 

  38. T. Dahms, J. Lehnert, and E. Schöll, Cluster and group synchronization in delay-coupled networks, Phys. Rev. E 86(1), 016202 (2012)

    Article  ADS  Google Scholar 

  39. V. Nicosia, M. Valencia, M. Chavez, A. Diaz-Guilera, and V. Latora, Remote synchronization reveals network symmetries and functional modules, Phys. Rev. Lett. 110(17), 174102 (2013)

    Article  ADS  Google Scholar 

  40. C. Fu, Z. Deng, L. Huang, and X. G. Wang, Topological control of synchronous patterns in systems of networked chaotic oscillators, Phys. Rev. E 87(3), 032909 (2013)

    Article  ADS  Google Scholar 

  41. C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators, Phys. Rev. Lett. 110(6), 064104 (2013)

    Article  ADS  Google Scholar 

  42. C. Fu, W. Lin, L. Huang, and X. G. Wang, Synchronization transition in networked chaotic oscillators: The viewpoint from partial synchronization, Phys. Rev. E 89(5), 052908 (2014)

    Article  ADS  Google Scholar 

  43. L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Cluster synchronization and isolated desynchronization in complex networks with symmetries, Nat. Commun. 5, 4079 (2014)

    Article  ADS  Google Scholar 

  44. F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Complete characterization of stability of cluster synchronization in complex dynamical networks, Sci. Adv. 2(4), e1501737 (2016)

    Article  ADS  Google Scholar 

  45. D. Hart, K. Bansal, T. E. Murphy, and R. Roy, Experimental observation of chimera and cluster states in a minimal globally coupled network, Chaos 26(9), 094801 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  46. T. Nishikawa and A. E. Motter, Network-complement transitions, symmetries, and cluster synchronization, Chaos 26(9), 094818 (2016)

    MATH  Google Scholar 

  47. M. T. Schaub, N. O’Clery, Y. N. Billeh, J. C. Delvenne, R. Lambiotte, and M. Barahona, Graph partitions and cluster synchronization in networks of oscillators, Chaos 26(9), 094821 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. F. Sorrentino and L. Pecora, Approximate cluster synchronization in networks with symmetries and parameter mismatches, Chaos 26(9), 094823 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Y. S. Cho, T. Nishikawa, and A. E. Motter, Stable chimeras and independently synchronizable clusters, Phys. Rev. Lett. 119(8), 084101 (2017)

    Article  ADS  Google Scholar 

  50. W. Stein, http://www.sagemath.org/sage/ and http:// sage.scipy.org/ for SAGE: Software for Algebra and Geometry Experimentation, 2013

  51. W. Lin, H. Fan, Y. Wang, H. Ying, and X. G. Wang, Controlling synchronous patterns in complex networks, Phys. Rev. E 93(4), 042209 (2016)

    Article  ADS  Google Scholar 

  52. W. Lin, H. Li, H. Ying, and X. G. Wang, Inducing isolated-desynchronization states in complex network of coupled chaotic oscillators, Phys. Rev. E 94(6), 062303 (2016)

    Article  ADS  Google Scholar 

  53. J. Sun, E. M. Bollt, and T. Nishikawa, Master stability functions for coupled nearly identical dynamical systems, EPL 85(6), 60011 (2009)

    Article  ADS  Google Scholar 

  54. J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. Lond. B Biol. Sci. 221(1222), 87 (1984)

    Article  ADS  Google Scholar 

  55. G. Ren, J. Tang, J. Ma, and Y. Xu, Detection of noise effect on coupled neuronal circuits, Commun. Nonlinear Sci. Numer. Simul. 29(1–3), 170 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 11375109 and the Fundamental Research Funds for the Central Universities under Grant No. GK201601001. Y.Z. Yu and X.G. Wang thank the support from the National Demonstration Center for Experimental X-physics Education (Shaanxi Normal University).

Author information

Authors and Affiliations

  1. School of Physics and Information Technology, Shaanxi Normal University, Xi’an, 710062, China

    Ben Cao, Ya-Feng Wang, Liang Wang, Yi-Zhen Yu & Xin-Gang Wang

  2. School of Physical Education, Shaanxi Normal University, Xi’an, 710062, China

    Ben Cao

Authors
  1. Ben Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ya-Feng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yi-Zhen Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xin-Gang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xin-Gang Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, B., Wang, YF., Wang, L. et al. Cluster synchronization in complex network of coupled chaotic circuits: An experimental study. Front. Phys. 13, 130505 (2018). https://doi.org/10.1007/s11467-018-0775-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11467-018-0775-1

Keywords

Search

Navigation