Stable amplitude chimera states and chimera death in repulsively coupled chaotic oscillators

Original Paper
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Abstract

Amplitude chimera states, representing a spontaneous symmetry breaking of a population of coupled identical oscillators into two distinct clusters with one oscillating in spatial coherent amplitude, while the other displaying oscillations in a spatially incoherent manner, have been observed as a kind of transient dynamics in the process of transition to the in-phase synchronization in coupled limit-cycle oscillators. Here, we obtain a kind of stable amplitude chimera state in the chaotic regime of a system of repulsively coupled Lorenz oscillators. With the increment of the coupling strength, the coupled oscillators transit from spatiotemporal chaos to amplitude chimera states then to coherent oscillation death or chimera death states. Moreover, the number of clusters in amplitude chimera patterns has a power-law dependence on the number of coupled neighbors. The amplitude chimera and the chimera death states coexist at certain coupling strength. Moreover, the amplitude chimera and the amplitude death patterns are related to the initial condition for given coupling strength. Our findings of amplitude chimera states and chimera death states in coupled chaotic system may enrich the knowledge of the symmetry-breaking-induced pattern formation.

Keywords

Amplitude chimera states Chimera death Coupled oscillators 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NSFC) (Grants Nos. 61377067, 11775034, 11765008), and Weiqing Liu is supported by the Qingjiang Program for Excellent Young Talents of Jiangxi University of Science and Technology.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Pikvosky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2003)MATHGoogle Scholar
  2. 2.
    Ujjwal, S.R., Punetha, N., Prasad, A., Ramaswamy, R.: Emergence of chimeras through induced multistability. Phys. Rev. E 95, 032203 (2017)CrossRefGoogle Scholar
  3. 3.
    Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillations. Nonlinear phenom. Complex Syst. 5, 380 (2002)Google Scholar
  4. 4.
    Abrams, D.M., Strogatz, S.H.: Chimera states for coupled oscillators. Phys. Rev. Lett. 93, 174102 (2004)CrossRefGoogle Scholar
  5. 5.
    Sakaguchi, H.: Instability of synchronized motion in nonlocally coupled neural oscillators. Phys. Rev. E 73, 031907 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Olmi, S., Politi, A., Torcini, A.: Collective chaos in pulse-coupled neural networks. Europhys. Lett. 92, 60007 (2010)CrossRefGoogle Scholar
  7. 7.
    Nkomo, S., Tinsley, M.R., Showalter, K.: Chimera states in populations of nonlocally coupled chemical oscillators. Phys. Rev. Lett. 110, 244102 (2013)CrossRefGoogle Scholar
  8. 8.
    Martens, E.A., Thutupalli, S., Fourrire, A., Hallatschek, O.: Chimera states in mechanical oscillator networks. Proc. Nat. Acad. Sci. USA 110, 10563–10567 (2013)CrossRefGoogle Scholar
  9. 9.
    Hagerstrom, A.M., et al.: Experimental observation of chimeras in coupled-map lattices. Nat. Phys. 8, 658 (2012)CrossRefGoogle Scholar
  10. 10.
    Kuramoto, Y., Shima, S.: Rotating spirals without phase singularity in reaction–diffusion systems. Progr. Theor. Phys. Suppl. 150, 115 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Schmidt, L., Schonleber, K., Krischer, K., Garcia-Morales, V.: Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling. Chaos 24, 013102 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sethia, G.C., Sen, A.: Chimera states: the existence criteria revisited. Phys. Rev. Lett. 112, 144101 (2014)CrossRefGoogle Scholar
  13. 13.
    Yeldesbay, A., Pikovsky, A., Rosenblum, M.: Chimeralike states in an ensemble of globally coupled oscillators. Phys. Rev. Lett. 112, 144103 (2014)CrossRefGoogle Scholar
  14. 14.
    Laing, C.R.: Chimeras in networks with purely local coupling. Phys. Rev. E 92, 050904 (2015)CrossRefGoogle Scholar
  15. 15.
    Wolfrum, M., Omelchenko, O.E.: Chimera states are chaotic transients. Phys. Rev. E 84, 015201 (2011)CrossRefGoogle Scholar
  16. 16.
    Rosin, D.P., Rontani, D., Haynes, N.D., Schőll, E., Gauthier, D.J.: Transient scaling and resurgence of chimera states in coupled Boolean phase oscillators. Phys. Rev. E 90, 030902(R) (2014)CrossRefGoogle Scholar
  17. 17.
    Wacker, A., Bose, S., Scholl, E.: Transient spatiotemporal chaos in a reaction–diffusion model. Europhys. Lett. 31, 257 (1995)CrossRefGoogle Scholar
  18. 18.
    Sethia, G.C., Sen, A., Johnston, G.L.: Amplitude-mediated chimera states. Phys. Rev. E 88(4), 042917 (2013)CrossRefGoogle Scholar
  19. 19.
    Zakharova, A., Kapeller, M., Scholl, E.: Chimera death: symmetry breaking in dynamical networks. Phys. Rev. Lett. 112, 154101 (2014)CrossRefGoogle Scholar
  20. 20.
    Gjurchinovski, A., Scholl, E., Zakharova, A.: Control of amplitude chimeras by time delay in oscillator networks. Phys. Rev. E 95, 042218 (2017)CrossRefGoogle Scholar
  21. 21.
    Zakharova, A., Loos, S.A.M., Siebert, J., et al.: Controlling chimera patterns in networks: interplay of structure, noise, and delay. Control of self-organizing nonlinear systems, Springer International Publishing, pp 1–21 (2016)Google Scholar
  22. 22.
    Tinsley, M.R., Nkomo, S., Showalter, K.: Chimera and phase cluster states in populations of coupled chemical oscillators. Nat. Phys. 8, 662–665 (2012)CrossRefGoogle Scholar
  23. 23.
    Gambuzza, L.V., Buscarino, A., Chessari, S., Fortuna, L., Meucci, R., Frasca, M.: Experimental investigation of chimera states with quiescent and synchronous domains in coupled electronic oscillators. Phys. Rev. E 90, 032905 (2014)CrossRefGoogle Scholar
  24. 24.
    Davidenko, J.M., Pertsov, A.M., Salomonsz, R., Baxter, W., Jalife, J.: Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 355, 349 (1992)CrossRefGoogle Scholar
  25. 25.
    Motter, A.E., Myers, S.A., Anghel, M., Nishikawa, T.: Spontaneous synchrony in power-grid networks. Nat. Phys. 9, 191–197 (2013)CrossRefGoogle Scholar
  26. 26.
    Rattenborg, N.C., Amlaner, C.J., Lima, S.L.: Behavioral, neurophysiological and evolutionary perspectives on unihemispheric sleep. Neurosci. Biobehav. Rev. 24, 817–842 (2000)CrossRefGoogle Scholar
  27. 27.
    Liu, W., Xiao, J., Qian, X., Yang, J.: Antiphase synchronization in coupled chaotic oscillators. Phys. Rev. E 73(5), 057203 (2006)CrossRefGoogle Scholar
  28. 28.
    Liu, W., Volkov, E., Xiao, J., Zou, W., Zhan, M., et al.: Inhomogeneous stationary and oscillatory regimes in coupled chaotic oscillators. Chaos 22, 033144 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Qu, Z., Shiferaw, Y., Weiss, J.N.: Nonlinear dynamics of cardiac excitation-contraction coupling: an iterated map study. Phys. Rev. E 75, 011927 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Song, Z., Ko, C.Y., Nivala, M., Weiss, J.N., Qu, Z.: Complex early and delayed afterdepolarization dynamics caused by voltage-calcium coupling in cardiac myocytes. Biophys. J. 108(2), 261–262 (2015)CrossRefGoogle Scholar
  31. 31.
    Song, Z., Karma, A., Karagueuzian, H.S., Weiss, J.N., Qu, Z.: Voltage and calcium coupling in the genesis of cardiac afterdepolarizations. Biophys. J. 106(2), 631 (2014)CrossRefGoogle Scholar
  32. 32.
    Dziubak, V., Maistrenko, Y., Scholl, E.: Coherent traveling waves in nonlocally coupled chaotic systems. Phys. Rev. E 87, 032907 (2013)CrossRefGoogle Scholar
  33. 33.
    Jaros, P., Maistrenko, Y., Kapitaniak, T.: Chimera states on the route from coherence to rotating waves. Phys. Rev. E 91, 022907 (2015)CrossRefGoogle Scholar
  34. 34.
    Dudkowski, D., Maistrenko, Y., Kapitaniak, T.: Different types of chimera states: an interplay between spatial and dynamical chaos. Phys. Rev. E 90(3–1), 032920 (2014)CrossRefGoogle Scholar
  35. 35.
    Dudkowski, D., Maistrenko, Y., Kapitaniak, T.: Occurrence and stability of chimera states in coupled externally excited oscillators. Chaos 26, 116306 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Gopal, R., Chandrasekar, V.K., Venkatesan, A., et al.: Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling. Phys. Rev. E 89(5), 052914 (2014)CrossRefGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.School of ScienceJiangxi University of Science and TechnologyGanzhouChina
  3. 3.State Key Lab of Information Photonics and Optical CommunicationsBeijing University of Posts and TelecommunicationsBeijingChina

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