ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Goldobin D. S., Dolmatova A. V., Rosenblum M. G., Pikovsky A. S. Synchronization in kuramoto–sakaguchi ensembles with competing influence of common noise and global coupling. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, iss. 6, pp. 5-37. DOI: 10.18500/0869-6632-2017-25-6-5-37

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537.86, 001.891.57, 621.37

Synchronization in kuramoto–sakaguchi ensembles with competing influence of common noise and global coupling

Goldobin Denis S., Institute of Continuum Mechanics of the Ural Branch of the Russian Academy of Sciences (IMSS UB RAS)
Dolmatova Anastasija Vladimirovna, Institute of Continuum Mechanics of the Ural Branch of the Russian Academy of Sciences (IMSS UB RAS)
Rosenblum Mihail Grigorevich, Lobachevsky State University of Nizhny Novgorod
Pikovsky Arkady Samuilovich, Potsdam University

We study the effects of synchronization and desynchronization in ensembles of phase oscillators with the global Kuramoto–Sakaguchi coupling under common noise driving. Since the mechanisms of synchronization by coupling and by common noise are essentially different, their interplay is of interest. In the thermodynamic limit of large number of oscillators, employing the Ott–Antonsen approach, we derive stochastic equations for the order parameters and consider their dynamics for two cases: (i) identical oscillators and (ii) small natural frequency mismatch. For identical oscillators, the stability of the perfect synchrony state is studied; a strong enough common noise is revealed to prevail over a moderate negative (repelling) coupling and to synchronize the ensemble. An inequality between the states of maximal asynchrony (zero-value of the order parameter) and perfect synchrony is found; the former can be only weakly stable, while the latter can become adsorbing (the transition to the synchrony becomes unidirectional). The dependence of the temporal dynamics of the transition on the system parameters is investigated. For nonidentical oscillators the perfect synchrony state becomes impossible and an absorbing state disappears; on its place, only a weakly stable state of imperfect synchrony remains. A nontrivial effect of the divergence of individual frequencies of oscillators with different natural frequencies is revealed and studied for moderate repelling coupling; meanwhile, the order parameter remains non-small for this case. In Appendix we provide an introduction to the theories of Ott–Antonsen and Watanabe–Strogatz.

  1. Пиковский А., Розенблюм М., Куртс Ю. Синхронизация. Фундаментальное нелинейное явление. М: Техносфера, 2003. 496 с.
  2. Crawford J.D. Amplitude expansions for instabilities in populations of globally-coupled oscillators // J. Stat. Phys. 1994. Vol. 74. Pp. 1047–1084.
  3. Strogatz S.H., Abrams D.M., McRobie A., Eckhardt B., Ott E. Theoretical mechanics: Crowd synchrony on the Millennium Bridge // Nature. 2005. Vol. 438. Pp. 43–44.
  4. Golomb D., Hansel D., Mato G. Mechanisms of synchrony of neural activity in large networks // Handbook of Biological Physics. Volume 4: Neuroinformatics and Neural Modelling. Ed. by F. Moss and S. Gielen. Amsterdam: Elsevier, 2001. Pp. 887–968.
  5. Пиковский А.С. Синхронизация и стохастизация ансамбля автогенераторов внешним шумом // Изв. вузов. Радиофизика. 1984. Т. 27. С. 390–395.
  6. Mainen Z.F., Sejnowski T.J. Reliability of spike timing in neocortical neurons // Science. 1995. Vol. 268. Pp. 1503–1506.
  7. Uchida A., McAllister R., Roy R. Consistency of nonlinear system response to complex drive signals // Phys. Rev. Lett. 2004. Vol. 93. 244102.
  8. Grenfell B.T., Wilson K., Finkenstadt B.F., Coulson T.N., Murray S., Albon S.D., Pemberton J.M., Clutton-Brock T.H., Crawley M.J. Noise and determinism in synch-ronized sheep dynamics // Nature. 1998. Vol. 394. Pp. 674–677.
  9. Ritt J. Evaluation of entrainment of a nonlinear neural oscillator to white noise // Phys. Rev. E. 2003. Vol. 68. 041915.
  10. Голдобин Д.С., Пиковский А.С. О синхронизации периодических автоколебаний общим шумом // Изв. вузов. Радиофизика. 1984. Т. 47. С. 1013–1019.
  11. Teramae J.N., Tanaka D. Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators // Phys. Rev. Lett. 2004. Vol. 93. 204103.
  12. Goldobin D.S., Pikovsky A.S. Synchronization of self-sustained oscillators by common white noise // Physica A. 2005. Vol. 351, No 1. Pp. 126–132.
  13. Goldobin D.S., Pikovsky A. Synchronization and desinchronization of self-sustained oscillators by common noise // Phys. Rev. E. 2005. Vol. 71. 045201(R).
  14. Goldobin D.S., Pikovsky A. Antireliability of noise-driven neurons // Phys. Rev. E. 2006. Vol. 73. 061906.
  15. Маляев В.С., Вадивасова Т.Е., Анищенко В.С. Стохастический резонанс, стохастическая синхронизация и индуцированный шумом хаос в осцилляторе  Дуффинга // Изв. вузов. Прикладная нелинейная динамика. 2007. Т. 15, No 5. С. 74–83.
  16. Wieczorek S. Stochastic bifurcation in noise-driven lasers and Hopf oscillators // Phys. Rev. E. 2009. Vol. 79. 036209.
  17. Goldobin D.S., Teramae J.-N., Nakao H., Ermentrout G.-B. Dynamics of limit-cycle oscillators subject to general noise // Phys. Rev. Lett. 2010. Vol. 105. 154101.
  18. Goldobin D.S. Uncertainty principle for control of ensembles of oscillators driven by common noise // Eur. Phys. J. ST. 2014. Vol. 223, No 4. Pp. 677–685.
  19. Голдобин Д.С. Принцип неопределенности для ансамблей осцилляторов с общим шумом // Вестник Пермского университета. Физика. 2014. Т. 27–28, вып. 2–3. С. 33–41.
  20. Braun W., Pikovsky A., Matias M.A., Colet P. Global dynamics of oscillator populations under common noise // EPL. 2012. Vol. 99. 20006.
  21. Pimenova A.V., Goldobin D.S., Rosenblum M., Pikovsky A. Interplay of coupling and common noise at the transition to synchrony in oscillator populations // Sci. Rep. 2016. Vol. 6. 38518.
  22. Garcia-Alvarez D., Bahraminasab A., Stefanovska A., McClintock P.V.E. Competition between noise and coupling in the induction of synchronisation // EPL. 2009. Vol. 88. 30005.
  23. Nagai K.H., Kori H. Noise-induced synchronization of a large population of globally coupled nonidentical oscillators // Phys. Rev. E. 2010. Vol. 81. 065202.
  24. Wiener N. Cybernetics: Or Control and Communication in the Animal and the Machine. 2nd Ed. Cambridge (MA): MIT Press, 1965. 212 p.
  25. Martens E.A., Thutupalli S., Fourriere A., Hallatschek O. Chimera states in mechanical oscillator networks // Proc. Natl. Acad. Sci. 2013. Vol. 110. Pp. 10563–10567.
  26. Temirbayev A.A., Zhanabaev Z.Z., Tarasov S.B., Ponomarenko V.I., Rosenblum M. Experiments on oscillator ensembles with global nonlinear coupling // Phys. Rev. E. 2012. Vol. 85. 015204(R).
  27. Temirbayev A.A., Nalibayev Y.D., Zhanabaev Z.Z., Ponomarenko V.I., Rosenblum M. Autonomous and forced dynamics of oscillator ensembles with global nonlinear coupling: An experimental study // Phys. Rev. E. 2013. Vol. 87. 062917.
  28. Watanabe S., Strogatz S.H. Constant of Motion for Superconducting Josephson Arrays // Physica D. 1994. Vol. 74. Pp. 197–253.
  29. Pikovsky A., Rosenblum M. Partially integrable dynamics of hierarchical populations of coupled oscillators // Phys. Rev. Lett. 2008. Vol. 101. 2264103.
  30. Marvel S.A., Mirollo R.E., Strogatz S.H. Identical phase oscillators with global sinusoidal coupling evolve by Mobius group action // Chaos. 2009. Vol. 19. 043104. 
  31. Ott E., Antonsen T.M. Low dimensional behavior of large systems of globally coupled oscillators // Chaos. 2008. Vol. 18. 037113.
  32. Флетчер К. Численные методы на основе метода Галёркина. М: Мир, 1988. 352 с.
  33. Marvel S.A., Strogatz S.H. Invariant submanifold for series arrays of Josephson junctions // Chaos. 2009. Vol. 19. 013132.
  34. Shinomoto Sh., Kuramoto Y. Phase transitions in active rotator systems // Prog. Theor. Phys. 1986. Vol. 75, No 5. Pp. 1105–1110.
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