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

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Dolmatova A. V., Goldobin D. S., Pikovsky A. S. Frequency entrainment and anti-entrainment of coupled active rotators synchronized by common noise. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 6, pp. 91-112. DOI: https://doi.org/10.18500/0869-6632-2019-27-6-91-112

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Language: 
Russian
Heading: 
Nonlinear Waves. Solitons. Autowaves. Self-Organization.
Article type: 
Article
UDC: 
537.86
DOI: 
https://doi.org/10.18500/0869-6632-2019-27-6-91-112

Frequency entrainment and anti-entrainment of coupled active rotators synchronized by common noise

Autors: 
Dolmatova Anastasija Vladimirovna, Institute of Continuum Mechanics of the Ural Branch of the Russian Academy of Sciences (IMSS UB RAS)
Goldobin Denis S., Institute of Continuum Mechanics of the Ural Branch of the Russian Academy of Sciences (IMSS UB RAS)
Pikovsky Arkady Samuilovich, Potsdam University
Abstract: 

Topic and Aim. We study the effect of common noise on the ensemble of coupled active rotators. Such a noise always has a synchronizing effect on the system, whereas the coupling may be attracting (synchronizing) or repulsing
(desynchronizing). For this reason, the case when both the common noise and the coupling simultaneously influence the dynamics of the system is of great interest. The paper aims are to construct a theory describing the behavior of the ensemble of coupled active rotators subject to common noise and to analyze all possible states in the system.

Methods. Phase description is adopted for active rotators. We develop an analytical approach based on a transformation to approximate angle–action variables and averaging over fast rotations that allows to analyze the dynamics of the system.

Results. For identical rotators, we describe a transition from full to partial synchrony at a critical value of repulsive coupling. For ensembles of nonidentical rotators, we show that although the common noise synchronizes the phases of the rotators, the frequency locking becomes imperfect due to the fact that the noise causes episodic phase slips. For moderate repulsive coupling, even more nontrivial effect occurs: strong common noise can lead to a high degree of synchrony, where a juxtaposition of phase locking with frequency repulsion (anti-entrainment) is observed. We show that the frequency repulsion obeys a nontrivial power law.

Discussion. Comparison of the results for the active rotators model with those for the Kuramoto–Sakaguchi system of coupled oscillators shows that the basic effects are similar in these setups. However, technically the analysis of the active rotators is more involved. The developed analytical theory is confirmed by the results of direct numerical simulation.
 

Key words: 
synchronization
stochastic processes
common noise
active rotators
Reference: 
  1. Benz S.P., Burroughs C.J. Coherent emission from two-dimensional Josephson junction arrays // Appl. Phys. Lett. 1991. Vol. 58. P. 2162–2164. doi: 10.1063/1.104993
  2. Nixon M., Ronen E., Friesem A.A., Davidson N. Observing geometric frustration with thousands of coupled lasers // Phys. Rev. Lett. 2013. Vol. 110. 184102. doi: 10.1103/PhysRevLett.110.184102
  3. Kiss I., Zhai Yu., Hudson J.L. Emerging coherence in a population of chemical oscillators // Science. 2002. Vol. 296. P. 1676–1678. doi: 10.1126/science.1070757 
  4. 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. doi: 10.1103/PhysRevE.85.015204
  5. 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. doi: 10.1103/PhysRevE.87.062917
  6. Acebron J.A., Bonilla L.L., Vicente C.J.P., Ritort F., Spigler R. The Kuramoto model: A simple paradigm for synchronization phenomena // Rev. Mod. Phys. 2005. Vol. 77. P. 137–185. doi: 10.1103/RevModPhys.77.137
  7. Pikovsky A., Rosenblum M. Dynamics of globally coupled oscillators: Progress and perspectives // Chaos. 2015. Vol. 25. 097616. doi: 10.1063/1.4922971
  8. Kuramoto Y. Self-entrainment of a population of coupled non-linear oscillators // International Symposium on Mathematical Problems in Theoretical Physics. January 23–29, 1975, Kyoto University, Kyoto, Japan/ Ed. H. Araki. Springer Lecture Notes in Physics. No. 39. New York: Springer, 1975. P. 420–422.
  9. Pikovskii A.S. Synchronization and stochastization of nonlinear oscillations by external noise // Nonlinear and Turbulent Processes in Physics. Vol. 3 / Ed. R.Z. Sagdeev. Chur: Harwood Academic, 1984. P. 1601–1604.
  10. Pikovskii A.S. Synchronization and stochastization of array of self-excited oscillators by external noise. Radiophys. Quantum Electron., 1984, vol. 27, pp. 390–395.
  11. Garc´ia-Alvarez D., Bahraminasab A., Stefanovska A., McClintock P.V.E. Competition between noise and coupling in the induction of synchronisation // Europhys. Lett. 2009. Vol. 88. 30005. doi: 10.1209/0295-5075/88/30005
  12. Nagai K.H., Kori H. Noise-induced synchronization of a large population of globally coupled nonidentical oscillators // Phys. Rev. E. 2010. Vol. 81. 065202. doi: 10.1103/PhysRevE.81.065202
  13. Braun W., Pikovsky A., Matias M.A., Colet P. Global dynamics of oscillator populations under common noise // Europhys. Lett. 2012. Vol. 99. 20006. doi: 10.1209/0295-5075/99/20006
  14. 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. doi: 10.1038/srep38518
  15. Goldobin D.S., Dolmatova A.V., Rosenblum M., Pikovsky A. Synchronization in KuramotoSakaguchi ensembles with competing influence of common noise and global coupling. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, no. 6, pp. 5–37 (in Russian). doi: 10.18500/0869-6632-2017-25-6-5-37
  16. Goldobin D.S., Dolmatova A.V. Frequency repulsion in ensembles of general limit-cycle oscillators synchronized by common noise in the presence of global desynchronizing coupling. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, no. 3, pp. 33–60 (in Russian).
  17. Shinomoto S., Kuramoto Y. Phase transitions in active rotator systems // Prog. Theor. Phys. 1986. Vol. 75. P. 1105–1110. doi: 10.1143/PTP.75.1105
  18. Sakaguchi H., Shinomoto S., Kuramoto Y. Phase transitions and their bifurcation analysis in a large population of active rotators with mean-field coupling // Prog. Theor. Phys. 1988. Vol. 79. P. 600–607. doi: 10.1143/PTP.79.600 
  19. Park S.H., Kim S. Noise-induced phase transitions in globally coupled active rotators // Phys. Rev. E. 1996. Vol. 53. P. 3425–3430. doi: 10.1103/PhysRevE.53.3425
  20. Tessone C.J., Scire A., Toral R., Colet P. Theory of collective firing induced by noise or diversity in excitable media // Phys. Rev. E. 2007. Vol. 75. 016203. doi: 10.1103/PhysRevE.75.016203
  21. Zaks M.A., Neiman A.B., Feistel S., Schimansky-Geier L. Noise-controlled oscillations and their bifurcations in coupled phase oscillators // Phys. Rev. E. 2003. Vol. 68. 066206. doi: 10.1103/PhysRevE.68.066206
  22. Sonnenschein B., Zaks M., Neiman A., Schimansky-Geier L. Excitable elements controlled by noise and network structure // Eur. Phys. J.: Spec. Top. 2013. Vol. 222. P. 2517–2529. doi: 10.1140/epjst/e2013-02034-7
  23. Ionita F., Meyer-Ortmanns H. Physical aging of classical oscillators // Phys. Rev. Lett. 2014. Vol. 112. 094101. doi: 10.1103/PhysRevLett.112.094101
  24. Goldobin D.S., Pikovsky A.S. Synchronization of periodic self-oscillations by common noise. Radiophys. Quantum Electron., 2004, vol. 47, no. 10–11, pp. 910–915. doi: 10.1007/s11141-005-0031-8
  25. Goldobin D.S., Pikovsky A.S. Synchronization of self-sustained oscillators by common white noise // Phys. A. 2005. Vol. 351, № 1. P. 126–132. doi: 10.1016/j.physa.2004.12.014
  26. 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. doi: 10.1103/PhysRevLett.93.204103
  27. Sakaguchi H. Synchronization in coupled phase oscillators // J. Korean Phys. Soc. 2008. Vol. 53. P. 1257–1264. doi: 10.3938/jkps.53.1257
  28. Bacic I., Yanchuk S., Wolfrum M., Franovic I. Noise-induced switching in two adaptively coupled excitable systems // Eur. Phys. J.: Spec. Top. 2018. Vol. 227. P. 1077–1090. doi: 10.1140/epjst/e2018-800084-6
  29. Marvel S.A., Strogatz S.H. Invariant submanifold for series arrays of Josephson junctions // Chaos. 2009. Vol. 19. 013132. doi: 10.1063/1.3087132
  30. Laing C.R. Derivation of a neural field model from a network of theta neurons // Phys. Rev. E. 2014. Vol. 90. 010901. doi: 10.1103/PhysRevE.90.010901
  31. O’Keeffe K.P., Strogatz S.H. Dynamics of a population of oscillatory and excitable elements // Phys. Rev. E. 2016. Vol. 93. 062203. doi: 10.1103/PhysRevE.93.062203
  32. Luke T.B., Barreto E., So P. Macroscopic complexity from an autonomous network of networks of theta neurons // Front. Comput. Neurosci. 2014. Vol. 8. 145. doi: 10.3389/fncom.2014.00145
  33. Montbrio E., Pazo D., Roxin A. Macroscopic description for networks of spiking neurons // Phys. Rev. X. 2015. Vol. 5. 021028. doi: 10.1103/PhysRevX.5.021028
  34. Ott E., Antonsen T.M. Low dimensional behavior of large systems of globally coupled oscillators // Chaos. 2008. Vol. 18. 037113. doi: 10.1063/1.2930766
  35. Tyulkina I.V., Goldobin D.S., Klimenko L.S., Pikovsky A. Dynamics of noisy oscillator populations beyond the Ott-Antonsen ansatz // Phys. Rev. Lett. 2006. Vol. 120. 264101. doi: 10.1103/PhysRevLett.120.264101
  36. Goldobin D.S., Tyulkina I.V., Klimenko L.S., Pikovsky A. Collective mode reductions for populations of coupled noisy oscillators // Chaos. 2018. Vol. 28. 101101. doi: 10.1063/1.5053576 
  37. Goldobin D.S., Tyulkina I.V., Klimenko L.S., Pikovskii A. Towards the description of collective dynamics in ensembles of real oscillators. Bulletin of Perm University. Physics, 2018, no. 3 (41), pp. 5–7 (in Russian). doi: 10.17072/1994-3598-2018-3-05-07
  38. Stratonovich R.L. Topics in the Theory of Random Noise. New York: Gordon and Breach, 1967.
  39. Nayfeh A.H. Introduction to Perturbation Techniques. New York, John Wiley & Sons, 1981, 519 p. doi: 10.1002/zamm.19810611224
  40. Benzi R., Sutera A., Vulpiani A. The mechanism of stochastic resonance // J. Phys. A. 1981. Vol. 14. P. L453–L457. doi: 10.1088/0305-4470/14/11/006
  41. Gang H., Ditzinger T., Ning C.Z., Haken H. Stochastic resonance without external periodic force // Phys. Rev. Lett. 1993. Vol. 71. P. 807–810. doi: 10.1103/PhysRevLett.71.807
  42. Gammaitoni L., Hanggi P., Jung P., Marchesoni F. Stochastic resonance // Rev. Mod. Phys. 1998. Vol. 70. P. 223–287. doi: 10.1103/RevModPhys.70.223
  43. Pikovsky A.S., Kurths J. Coherence resonance in a noise-driven excitable system // Phys. Rev. Lett. 1997. Vol. 78. P. 775–778. doi: 10.1103/PhysRevLett.78.775
  44. Goldobin D.S., Pikovsky A. Synchronization and desynchronization of self-sustained oscillators by common noise // Phys. Rev. E. 2005. Vol. 71. 045201. doi: 10.1103/PhysRevE.71.045201
  45. Goldobin D.S., Pikovsky A. Antireliability of noise-driven neurons // Phys. Rev. E. 2006. Vol. 73. 061906. doi: 10.1103/PhysRevE.73.061906
Received: 
18.09.2019
Accepted: 
24.10.2019
Published: 
02.12.2019
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