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Goldobin D. S., Dolmatova A. V. Reduced cumulant models for macroscopic dynamics of Kuramoto ensemble with multiplicative intrinsic noise. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 2, pp. 288-301. DOI: 10.18500/0869-6632-2021-29-2-288-301

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621.37; 537.862; 517.925.42

Reduced cumulant models for macroscopic dynamics of Kuramoto ensemble with multiplicative intrinsic noise

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)

The purpose of this work is developing reduced models describing the macroscopic dynamics of the Kuramoto ensemble with multiplicative intrinsic noise on the basis of the method of circular cumulants. Methods. The dynamics of the system is considered within the framework of the phase reduction. The dynamics equations are obtained by the method of circular cumulants. Stability of the asynchronous state is considered on the basis of linear analysis. Results are verified by the numerical simulation. Results. The infinite cumulant equation chain is derived for the Kuramoto ensemble with intrinsic multiplicative noise. Two closures of the cumulant series are proposed to construct reduced models of the ensemble dynamics. Conclusion. For a phase oscillator population with Kuramoto global coupling, the case of a multiplicative noise converges to the case of an additive one only in the high-frequency limit. Moreover, for low frequencies, the instability of the asynchronous state to formation of a macroscopic collective mode becomes monotonous. Two-cumulant model reductions provide a reasonable accuracy for the macroscopic description of the population dynamics. Meanwhile, the Ott–Antonsen ansatz and the Gaussian approximation fail to represent the system dynamics accurately for non-high frequencies.

  1. Ott E, Antonsen TM. Low dimensional behavior of large systems of globally coupled oscillators. Chaos. 2008;18(3):037113. DOI: 10.1063/1.2930766.
  2. Ott E, Antonsen TM. Long time evolution of phase oscillator systems. Chaos. 2009;19(2):023117. DOI: 10.1063/1.3136851.
  3. Kuramoto Y. Self-entrainment of a population of coupled non-linear oscillators. In: Araki H. (eds) International Symposium on Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol. 39. Springer, Berlin, Heidelberg; 1975. P. 420–422. DOI: 10.1007/BFb0013365.
  4. Kuramoto Y. Chemical Oscillations, Waves, and Turbulence. Springer, Berlin, Heidelberg; 1984. 158 p. DOI: 10.1007/978-3-642-69689-3.
  5. Acebron JA, Bonilla LL, Vicente CJP, Ritort F, Spigler R. The Kuramoto model: A simple ´ paradigm for synchronization phenomena. Rev. Mod. Phys. 2005;77(1):137–185. DOI: 10.1103/RevModPhys.77.137.
  6. Pikovsky A, Rosenblum M. Dynamics of globally coupled oscillators: Progress and perspectives. Chaos. 2015;25(9):097616. DOI: 10.1063/1.4922971.
  7. Watanabe S, Strogatz SH. Integrability of a globally coupled oscillator array. Phys. Rev. Lett. 1993;70(16):2391–2394. DOI: 10.1103/PhysRevLett.70.2391.
  8. Watanabe S, Strogatz SH. Constants of motion for superconducting Josephson arrays. Physica D. 1994;74(3–4):197–253. DOI: 10.1016/0167-2789(94)90196-1.
  9. Pikovsky A, Rosenblum M. Partially integrable dynamics of hierarchical populations of coupled oscillators. Phys. Rev. Lett. 2008;101(26):264103. DOI: 10.1103/PhysRevLett.101.264103.
  10. Marvel SA, Mirollo RE, Strogatz SH. Identical phase oscillators with global sinusoidal coupling evolve by Mobius group action. Chaos. 2009;19(4):043104. ¨ DOI: 10.1063/1.3247089.
  11. Martens EA, Thutupalli S, Fourriere A, Hallatschek O. Chimera states in mechanical oscillator ´ networks. PNAS. 2013;110(26):10563–10567. DOI: 10.1073/pnas.1302880110.
  12. Totz JF, Rode J, Tinsley MR, Showalter K, Engel H. Spiral wave chimera states in large populations of coupled chemical oscillators. Nature Phys. 2018;14(3):282–285. DOI: 10.1038/s41567-017-0005-8.
  13. Pietras B, Daffertshofer A. Network dynamics of coupled oscillators and phase reduction techniques. Phys. Rep. 2019;819:1–105. DOI: 10.1016/j.physrep.2019.06.001.
  14. Tyulkina IV, Goldobin DS, Klimenko LS, Pikovsky A. Dynamics of noisy oscillator populations beyond the Ott-Antonsen ansatz. Phys. Rev. Lett. 2018;120(26):264101. DOI: 10.1103/PhysRevLett.120.264101.
  15. Goldobin DS, Tyulkina IV, Klimenko LS, Pikovsky A. Collective mode reductions for populations of coupled noisy oscillators. Chaos. 2018;28(10):101101. DOI: 10.1063/1.5053576.
  16. Tyulkina IV, Goldobin DS, Klimenko LS, Pikovsky AS. Two-bunch solutions for the dynamics of Ott-Antonsen phase ensembles. Radiophys. Quantum El. 2019;61(8–9):640–649. DOI: 10.1007/s11141-019-09924-7.
  17. Pazo D, Montbri ´ o E. Low-dimensional dynamics of populations of pulse-coupled oscillators. Phys. ´ Rev. X. 2014;4(1):011009. DOI: 10.1103/PhysRevX.4.011009.
  18. Montbrio E, Paz ´ o D, Roxin A. Macroscopic description for networks of spiking neurons. Phys. ´ Rev. X. 2015;5(2):021028. DOI: 10.1103/PhysRevX.5.021028.
  19. Ullner E, Politi A, Torcini A. Ubiquity of collective irregular dynamics in balanced networks of spiking neurons. Chaos. 2018;28(8):081106. DOI: 10.1063/1.5049902.
  20. di Volo M, Torcini A. Transition from asynchronous to oscillatory dynamics in balanced spiking networks with instantaneous synapses. Phys. Rev. Lett. 2018;121(12):128301. DOI: 10.1103/PhysRevLett.121.128301.
  21. Goldobin DS. Anharmonic resonances with recursive delay feedback. Phys. Lett. A. 2011; 375(39):3410–3414. DOI: 10.1016/j.physleta.2011.07.059.
  22. Goldobin DS. Uncertainty principle for control of ensembles of oscillators driven by common noise. Eur. Phys. J. Spec. Top. 2014;223(4):677–685. DOI: 10.1140/epjst/e2014-02133-y.
  23. Goldobin DS, Dolmatova AV. 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;27(3):33–60 (in Russian). DOI: 10.18500/0869-6632-2019-27-3-33-60.
  24. Gardiner CW. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo; 1983. 442 p.
  25. Daido H. Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: bifurcation of the order function. Physica D. 1996;91(1–2):24–66. DOI: 10.1016/0167-2789(95)00260-X.
  26. Crawford JD. Amplitude expansions for instabilities in populations of globally-coupled oscillators. J. Stat. Phys. 1994;74(5–6):1047–1084. DOI: 10.1007/BF02188217.
  27. Zaks MA, Neiman AB, Feistel S, Schimansky-Geier L. Noise-controlled oscillations and their bifurcations in coupled phase oscillators. Phys. Rev. E. 2003;68(6):066206. DOI: 10.1103/PhysRevE.68.066206.
  28. Sonnenschein B, Schimansky-Geier L. Approximate solution to the stochastic Kuramoto model. Phys. Rev. E. 2013;88(5):052111. DOI: 10.1103/PhysRevE.88.052111.
  29. Sonnenschein B, Peron TKD, Rodrigues FA, Kurth J, Schimansky-Geier L. Collective dynamics in two populations of noisy oscillators with asymmetric interactions. Phys. Rev. E. 2015;91(6): 062910. DOI: 10.1103/PhysRevE.91.062910.
  30. Hannay KM, Forger DB, Booth V. Macroscopic models for networks of coupled biological oscillators. Sci. Adv. 2018;4(8):e1701047. DOI: 10.1126/sciadv.1701047.
  31. Goldobin DS, Dolmatova AV. Ott-Antonsen ansatz truncation of a circular cumulant series. Phys. Rev. Research. 2019;1(3):033139. DOI: 10.1103/PhysRevResearch.1.033139.
  32. Lukacs E. Characteristic Functions. 2nd edition. Griffin, London; 1970. 350 p.
  33. Goldobin DS, Klimenko LS. On relationships between the distribution of Watanabe–Strogatz phases and circular cumulants. Bulletin of Perm University. Physics. 2019(2):24–34 (in Russian). DOI: 10.17072/1994-3598-2019-2-24-34.
  34. Ratas I, Pyragas K. Noise-induced macroscopic oscillations in a network of synaptically coupled quadratic integrate-and-fire neurons. Phys. Rev. E. 2019;100(5):052211. DOI: 10.1103/PhysRevE.100.052211.