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Effects of external global harmonic influence on chimera states

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Abstract

The effects of the global harmonic force on an ensemble of nonlocally coupled chaotic Rössler oscillators is studied. The autonomous ensemble without external force demonstrates a stable regime of chimera state. We have considered two types of the chimera states, namely phase chimera with a single incoherence cluster and combined chimera state with the incoherence clusters of the phase and amplitude chimeras. We have shown that even the sufficiently weak external influence leads to the qualitative changes in the spatiotemporal behavior of the ensemble under study. The diagram of regimes has been plotted in the plane of external signal parameters, where we detect regions with different dynamical regimes, induced by the external excitation. They are the spatiotemporal intermittence of structures, stable chimera states, stable regimes with a piecewise spatial profile, spatiotemporal chaos, complete chaotic synchronization and spatial incoherence. We have explored each of these regimes in detail and has found out that they are characterized by a certain distinctive range of the maximal Lyapunov exponent. We have found that the change of spatiotemporal behavior is not accompanied by the effects of locking the natural frequency or of suppression of selfoscillation by the external signal.

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References

  1. Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer Series in Synergetics, vol. 19. Springer, Berlin (1984)

    MATH  Google Scholar 

  2. Mikhailov, A.S., Loskutov, A.Y.: Foundations of Synergetics II: Complex Patterns, vol. 52. Springer, Berlin (2012)

    MATH  Google Scholar 

  3. Nekorkin, V.I., Velarde, M.G.: Synergetic Phenomena in Active Lattices. Springer Series in Synergetics. Springer, Berlin (2002)

    MATH  Google Scholar 

  4. Newman, M.E., Barabási, A.-L.E., Watts, D.J.: The Structure and Dynamics of Networks. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  5. Barrat, A., Barthelemy, M., Vespignani, A.: Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge (2008)

    MATH  Google Scholar 

  6. Cohen, R., Havlin, S.: Complex Networks: Structure, Robustness and Function. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  7. Boccaletti, S., Pisarchik, A., Genio, C., Amann, A.: Synchronization - From Coupled Systems to Complex Networks. Cambridge University Press (2018)

  8. Lindner, J.F., Meadows, B.K., Ditto, W.L., Inchiosa, M.E., Bulsara, A.R.: Array enhanced stochastic resonance and spatiotemporal synchronization. Phys. Rev. Lett. 75(1), 3 (1995)

    Google Scholar 

  9. Zhonghuai, H., Lingfa, Y., Zuo, X., Houwen, X.: Noise induced pattern transition and spatiotemporal stochastic resonance. Phys. Rev. Lett. 81(14), 2854 (1998)

    Google Scholar 

  10. Hu, B., Zhou, C.: Phase synchronization in coupled nonidentical excitable systems and array-enhanced coherence resonance. Phys. Rev. E 61(2), R1001 (2000)

    Google Scholar 

  11. Perc, M.: Stochastic resonance on excitable small-world networks via a pacemaker. Phys. Rev. E 76(6), 066203 (2007)

    MathSciNet  Google Scholar 

  12. Goldobin, D.S., Pikovsky, A.: Synchronization and desynchronization of self-sustained oscillators by common noise. Phys. Rev. E 71(4), 045201 (2005)

    MathSciNet  Google Scholar 

  13. Koukkari, W.L., Sothern, R.B.: Introducing Biological Rhythms: A Primer on the Temporal Organization of Life, with Implications for Health, Society, Reproduction, and the Natural Environment. Springer, Berlin (2007)

    Google Scholar 

  14. Acebrón, J.A., Lozano, S., Arenas, A.: Amplified signal response in scale-free networks by collaborative signaling. Phys. Rev. Lett. 99(12), 128701 (2007)

    Google Scholar 

  15. Freeman, D.K., Rizzo III, J.F., Fried, S.: Electric stimulation with sinusoids and white noise for neural prostheses. Front. Neurosci. 4, 1 (2010)

    Google Scholar 

  16. Liu, Z.: Organization network enhanced detection and transmission of phase-locking. EPL (Europhys. Lett.) 100(6), 60002 (2013)

    Google Scholar 

  17. Kori, H., Mikhailov, A.S.: Entrainment of randomly coupled oscillator networks by a pacemaker. Phys. Rev. Lett. 93(25), 254101 (2004)

    Google Scholar 

  18. Zhai, Y., Kiss, I.Z., Tass, P.A., Hudson, J.L.: Desynchronization of coupled electrochemical oscillators with pulse stimulations. Phys. Rev. E 71(6), 065202 (2005)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  21. Nomura, T., Glass, L.: Entrainment and termination of reentrant wave propagation in a periodically stimulated ring of excitable media. Phys. Rev. E 53(6), 6353 (1996)

    Google Scholar 

  22. Sakaguchi, H., Fujimoto, T.: Forced entrainment and elimination of spiral waves for the Fitzhugh–Nagumo equation. Prog. Theor. Phys. 108(2), 241–252 (2002)

    MathSciNet  MATH  Google Scholar 

  23. Chen, J.-X., Zhang, H., Qiao, L.-Y., Liang, H., Sun, W.-G.: Interaction of excitable waves emitted from two defects by pulsed electric fields. Commun. Nonlinear Sci. Numer. Simul. 54, 202–209 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Cui, R.-F., Chen, Q.-H., Chen, J.-X.: Separation of nanoparticles via surfing on chemical wavefronts. Nanoscale 12(23), 12275–12280 (2020)

    Google Scholar 

  25. Good, L.B., Sabesan, S., Marsh, S.T., Tsakalis, K., Treiman, D., Iasemidis, L.: Control of synchronization of brain dynamics leads to control of epileptic seizures in rodents. Int. J. Neural Syst. 19(03), 173–196 (2009)

    Google Scholar 

  26. Boon, P., Raedt, R., De Herdt, V., Wyckhuys, T., Vonck, K.: Electrical stimulation for the treatment of epilepsy. Neurotherapeutics 6(2), 218–227 (2009)

    Google Scholar 

  27. Eusebio, A., Thevathasan, W., Gaynor, L.D., Pogosyan, A., Bye, E., Foltynie, T., Zrinzo, L., Ashkan, K., Aziz, T., Brown, P.: Deep brain stimulation can suppress pathological synchronisation in parkinsonian patients. J. Neurol. Neurosurg. Psychiatry 82(5), 569–573 (2011)

    Google Scholar 

  28. Dayan, E., Censor, N., Buch, E.R., Sandrini, M., Cohen, L.G.: Noninvasive brain stimulation: from physiology to network dynamics and back. Nat. Neurosci. 16(7), 838 (2013)

    Google Scholar 

  29. González, H., Nagai, Y., Bub, G., Glass, L., Shrier, A.: Resetting and annihilating reentrant waves in a ring of cardiac tissue: theory and experiment. Progr. Theor. Phys. Suppl. 139, 83–89 (2000)

    Google Scholar 

  30. Luther, S., Fenton, F.H., Kornreich, B.G., Squires, A., Bittihn, P., Hornung, D., Zabel, M., Flanders, J., Gladuli, A., Campoy, L., et al.: Low-energy control of electrical turbulence in the heart. Nature 475(7355), 235–239 (2011)

    Google Scholar 

  31. Feng, X., Gao, X., Pan, D.-B., Li, B.-W., Zhang, H.: Unpinning of rotating spiral waves in cardiac tissues by circularly polarized electric fields. Sci. Rep. 4(1), 1–5 (2014)

    Google Scholar 

  32. Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom. Complex Syst. 5(4), 380–385 (2002)

    Google Scholar 

  33. Abrams, D.M., Strogatz, S.H.: Chimera states for coupled oscillators. Phys. Rev. Lett. 93(17), 174102 (2004)

    Google Scholar 

  34. Zakharova, A.: Chimera Patterns in Networks: Interplay between Dynamics, Structure, Noise, and Delay. Springer, Berlin (2020)

    Google Scholar 

  35. Omelchenko, I., Maistrenko, Y., Hövel, P., Schöll, E.: Loss of coherence in dynamical networks: spatial chaos and chimera states. Phys. Rev. Lett. 106, 234102 (2011)

    Google Scholar 

  36. Hagerstrom, A.M., Murphy, T.E., Roy, R., Hövel, P., Omelchenko, I., Schöll, E.: Experimental observation of chimeras in coupled-map lattices. Nat. Phys. 8, 658–661 (2012)

    Google Scholar 

  37. Martens, E.A., Thutupalli, S., Fourrière, A., Hallatschek, O.: Chimera states in mechanical oscillator networks. Proc. Nat. Acad. Sci. 110(26), 10563–10567 (2013)

    Google Scholar 

  38. Zakharova, A., Kapeller, M., Schöll, E.: Chimera death: symmetry breaking in dynamical networks. Phys. Rev. Lett. 112(15), 154101 (2014)

    Google Scholar 

  39. Kapitaniak, T., Kuzma, P., Wojewoda, J., Czolczynski, K., Maistrenko, Y.: Imperfect chimera states for coupled pendula. Sci. Rep. 4, 6379 (2014)

    Google Scholar 

  40. Panaggio, M.J., Abrams, D.M.: Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 28(3), R67–R87 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Maistrenko, Y., Sudakov, O., Osiv, O., Maistrenko, V.: Chimera states in three dimensions. New J. Phys. 17(7), 073037 (2015)

    Google Scholar 

  42. Bogomolov, S.A., Slepnev, A.V., Strelkova, G.I., Schöll, E., Anishchenko, V.S.: Mechanisms of appearance of amplitude and phase chimera states in ensembles of nonlocally coupled chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 43, 25–36 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Andrzejak, R.G., Ruzzene, G., Malvestio, I.: Generalized synchronization between chimera states. Chaos 27, 053114 (2017)

    MathSciNet  MATH  Google Scholar 

  44. Bukh, A., Rybalova, E., Semenova, N., Strelkova, G., Anishchenko, V.: New type of chimera and mutual synchronization of spatiotemporal structures in two coupled ensembles of nonlocally interacting chaotic maps. Chaos 27, 111102 (2017)

    MathSciNet  MATH  Google Scholar 

  45. Rybalova, E., Vadivasova, T., Strelkova, G., Anishchenko, V., Zakharova, A.: Forced synchronization of a multilayer heterogeneous network of chaotic maps in the chimera state mode. Chaos Interdiscip. J. Nonlinear Sci. 29(3), 033134 (2019)

    MathSciNet  MATH  Google Scholar 

  46. Rakshit, S., Faghani, Z., Parastesh, F., Panahi, S., Jafari, S., Ghosh, D., Perc, M.: Transitions from chimeras to coherence: an analytical approach by means of the coherent stability function. Phys. Rev. E 100(1), 012315 (2019)

    Google Scholar 

  47. Majhi, S., Perc, M., Ghosh, D.: Chimera states in a multilayer network of coupled and uncoupled neurons. Chaos Interdiscip. J. Nonlinear Sci. 27(7), 073109 (2017)

    MathSciNet  Google Scholar 

  48. Majhi, S., Bera, B.K., Ghosh, D., Perc, M.: Chimera states in neuronal networks: a review. Phys. Life Rev. 28, 100–121 (2019)

    Google Scholar 

  49. Semenova, N., Strelkova, G., Anishchenko, V., Zakharova, A.: Temporal intermittency and the lifetime of chimera states in ensembles of nonlocally coupled chaotic oscillators. Chaos Interdiscip. J. Nonlinear Sci. 27(6), 061102 (2017)

    MathSciNet  MATH  Google Scholar 

  50. Loos, S.A., Claussen, J.C., Schöll, E., Zakharova, A.: Chimera patterns under the impact of noise. Phys. Rev. E 93(1), 012209 (2016)

    Google Scholar 

  51. Semenov, V., Zakharova, A., Maistrenko, Y., Schöll, E.: Delayed-feedback chimera states: forced multiclusters and stochastic resonance. EPL (Europhys. Lett.) 115(1), 10005 (2016)

    Google Scholar 

  52. Bukh, A.V., Slepnev, A.V., Anishchenko, V.S., Vadivasova, T.E.: Stability and noise-induced transitions in an ensemble of nonlocally coupled chaotic maps. Regul. Chaotic Dyn. 23(3), 325–338 (2018)

    MathSciNet  MATH  Google Scholar 

  53. Rybalova, E.V., Klyushina, D.Y., Anishchenko, V.S., Strelkova, G.I.: Impact of noise on the amplitude chimera lifetime in an ensemble of nonlocally coupled chaotic maps. Regul. Chaotic Dyn. 24(4), 432–445 (2019)

    MathSciNet  Google Scholar 

  54. Tang, J., Zhang, J., Ma, J., Luo, J.: Noise and delay sustained chimera state in small world neuronal network. Sci. China Technol. Sci. 62(7), 1134–1140 (2019)

    Google Scholar 

  55. Semenova, N., Zakharova, A., Anishchenko, V., Schöll, E.: Coherence-resonance chimeras in a network of excitable elements. Phys. Rev. Lett. 117, 014102 (2016)

    Google Scholar 

  56. Shepelev, I., Vadivasova, T.: Inducing and destruction of chimeras and chimera-like states by an external harmonic force. Phys. Lett. A 382(10), 690–696 (2018)

    MathSciNet  Google Scholar 

  57. Shepelev, I., Vadivasova, T.: External localized harmonic influence on an incoherence cluster of chimera states. Chaos Solitons Fract. 133, 109642 (2020)

    MathSciNet  Google Scholar 

  58. Clerc, M., Coulibaly, S., Ferré, M., Rojas, R.: Chimera states in a duffing oscillators chain coupled to nearest neighbors. Chaos Interdiscip. J. Nonlinear Sci. 28(8), 083126 (2018)

    MathSciNet  MATH  Google Scholar 

  59. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)

    MATH  Google Scholar 

  60. Bordyugov, G., Pikovsky, A., Rosenblum, M.: Self-emerging and turbulent chimeras in oscillator chains. Phys. Rev. E 82(3), 035205 (2010)

    MathSciNet  Google Scholar 

  61. Shena, J., Hizanidis, J., Kovanis, V., Tsironis, G.P.: Turbulent chimeras in large semiconductor laser arrays. Sci. Rep. 7, 42116 (2017)

    Google Scholar 

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Acknowledgements

The reported study was funded by the Russian Science Foundation (project no. 20-12-00119).

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  1. Saratov National Research State University, Astrakhanskaya Street 83, Saratov, Russia, 410012

    Igor A. Shepelev & Tatiana E. Vadivasova

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  1. Igor A. Shepelev
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  2. Tatiana E. Vadivasova
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Correspondence to Igor A. Shepelev.

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Shepelev, I.A., Vadivasova, T.E. Effects of external global harmonic influence on chimera states. Nonlinear Dyn 102, 417–430 (2020). https://doi.org/10.1007/s11071-020-05874-2

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