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Radiophysics and Quantum Electronics

, Volume 61, Issue 8–9, pp 672–680 | Cite as

Correlations of the States of Non-Entrained Oscillators in the Kuramoto Ensemble with Noise in the Mean Field

  • A. S. Pikovsky
  • A. V. Dolmatova
  • D. S. GoldobinEmail author
Article
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We consider the dynamics of the Kuramoto ensemble oscillators not included in a common synchronized cluster, where the mean field is subject to fluctuations. The fluctuations can be either related to the finite size of the ensemble or superimposed on the mean field in the form of common noise due to the constructive features of the system. It is shown that the states of such oscillators with close natural frequencies appear correlated with each other, since the mean-field fluctuations act as common noise. We quantify the effect with the synchronization index of two oscillators, which is calculated numerically and analytically as a function of the frequency difference and noise intensity. The results are rigorous for large ensembles with additional noise superimposed on the mean field and are qualitatively true for the systems where the mean-field fluctuations are due to the finite size of the ensemble. In the latter case, the effect is found to be independent of the number of oscillators in the ensemble.

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References

  1. 1.
    A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Fundamental Nonlinear Phenomenon [in Russian], Tekhnosfera, Moscow (2003).CrossRefzbMATHGoogle Scholar
  2. 2.
    H. Daido, J. Phys. A, 20, No. 10, 1629 (1987).CrossRefGoogle Scholar
  3. 3.
    H. Hong, H. Chate, L. H. Tang, and H. Park, Phys. Rev. E, 92, No. 3, 022122 (2015).ADSCrossRefGoogle Scholar
  4. 4.
    F. Peter and A. Pikovsky, Phys. Rev. E, 97, 032310 (2018).ADSCrossRefGoogle Scholar
  5. 5.
    A. S. Pikovsky, Radiophys. Quantum Electron., 27, No. 5, 390 (1984).ADSCrossRefGoogle Scholar
  6. 6.
    D. S. Goldobin and A. S. Pikovsky, Radiophys. Quantum Electron., 47, Nos. 10–11, 910 (2004).ADSCrossRefGoogle Scholar
  7. 7.
    J. N. Teramae and D. Tanaka, Phys. Rev. Lett., 93, No. 20, 204103 (2004).ADSCrossRefGoogle Scholar
  8. 8.
    D. S. Goldobin and A. S. Pikovsky, Phys. Rev. E, 71, No. 4, 045201(R) (2005).ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Y. Kuramoto, in: H. Araki, ed., Lecture Notes in Phys. 39, Springer, New York (1975), p. 420.Google Scholar
  10. 10.
    J.D. Crawford, J. Stat. Phys., 74, Nos. 5–6, 1047 (1994).ADSCrossRefGoogle Scholar
  11. 11.
    A. H. Nayfeh, Perturbation Methods, John Wiley, New York (1973).zbMATHGoogle Scholar
  12. 12.
    A. V. Dolmatova, D. S. Goldobin, and A. Pikovsky, Phys. Rev. E, 96, No. 6, 062204 (2017).ADSCrossRefGoogle Scholar
  13. 13.
    G. F. Zharkov and Yu. K. Al’tudov, Sov. Phys. JETP, 47, No. 5, 901 (1978).ADSGoogle Scholar
  14. 14.
    S. Butz, P. Jung, L. V. Filippenko, et al., Opt. Express, 21, No. 19, 22540 (2013).ADSCrossRefGoogle Scholar
  15. 15.
    P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, Appl. Phys. Lett ., 102, No. 6, 062601 (2013).ADSCrossRefGoogle Scholar
  16. 16.
    V. Pierro and G. Filatrella, Physica C, 517, 37 (2015).ADSCrossRefGoogle Scholar
  17. 17.
    A.V. Pimenova, D. S. Goldobin, M. Rosenblum, and A. Pikovsky, Sci. Rep., 6, 38518 (2016).ADSCrossRefGoogle Scholar
  18. 18.
    H. Nakao, J.-N. Teramae, D. S. Goldobin, and Y. Kuramoto, Chaos, 20, No. 3, 033126 (2010).ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    D. S. Goldobin, J.-N. Teramae, H. Nakao, and G.-B. Ermentrout, Phys. Rev. Lett ., 105, No. 15, 154101 (2010).ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • A. S. Pikovsky
    • 1
    • 2
  • A. V. Dolmatova
    • 3
  • D. S. Goldobin
    • 3
    • 4
    Email author
  1. 1.Potsdam UniversityPotsdamGermany
  2. 2.N. I. Lobachevsky State University of Nizhny NovgorodNizhny NovgorodRussia
  3. 3.Institute for Mechanics of Continuous MediaUral Branch of the Russian Academy of SciencesPermRussia
  4. 4.State University of PermPermRussia

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