#### For citation:

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, iss. 3, pp. 33-60. DOI: 10.18500/0869-6632-2019-27-3-33-60

# Frequency repulsion in ensembles of general limit-cycle oscillators synchronized by common noise in the presence of global desynchronizing coupling

Topic. We study the interaction of two fundamentally different synchronization mechanisms: by means of coupling and by means of the driving by a random signal, which is identical for all oscillators – common noise. Special attention is focused on the effect of frequency divergence arising from the competition between these mechanisms. Aim. The aim of the paper is to construct a universal theory describing such an interaction for a general class of smooth limit-cycle oscillators with a global coupling. The effect of intrinsic noise, which is individual for each oscillator, is also to be taken into account. We also plan to assess how the results obtained earlier for the systems of the Ott–Antonsen type reflect the situation in general case. Method. For a general class of oscillators, the phase description is introduced. For the Fokker–Planck equation, which corresponds to the stochastic equations of phase dynamics, a rigorous averaging procedure is performed in the limit of high oscillation frequency (the conventional multiple scale method is used). With the derived equations one can obtain the conditions for synchronization of ensembles of identical oscillators, and for weakly nonidentical oscillators, one can find the average oscillation frequencies in quadratures. Analytical results are verified by direct numerical simulations for large but finite ensembles of van der Pol, Rayleigh and van der Pol–Duffing oscillators, as well as for FitzHugh–Nagumo neuronlike oscillators. Results. For the case of identical oscillators without intrinsic noise, a sufficiently strong common noise synchronizes an ensemble with a repulsive global coupling, and the dynamics of localization of the oscillator distribution is investigated. The latter clearly indicates that during the transition to the state of perfect synchrony, the distribution of oscillators possesses «heavy» power-law tails, even with an arbitrary strong attracting global coupling – without common noise, such tails do not apper. For the case of oscillators with intrinsic noise, the equilibrium distribution of phase differences always possesses «heavy» power-law tails, and the parameters of these tails are determined. The asymptotic behavior for the average frequency of an oscillator as a function of the natural frequency is derived analytically; in particular, the effect of divergence of the average frequencies is reported to accompany the synchronization by common noise in the presence of a repulsive global coupling. Examples of the application of the constructed theory for van der Pol, Rayleigh, van der Pol–Duffing and FitzHugh–Nagumo oscillators are presented. The results of direct numerical simulation for large finite ensembles of these oscillators are consistent with the theory. Discussion. Arbitrary weak general noise, on the one hand, increases the stability of the synchronous state, and on the other hand, it always creates «heavy» power-law tails for the distribution of phase differences. This indicates a significantly intermittent character of synchronization by common noise –epoches of synchronous behavior are interrupted by the phase difference slips – and is consistent with the fact that in the presence of common noise, a perfect frequency locking becomes impossible. For a repulsive coupling, a nontrivial effect occurs: sufficiently strong common noise synchronizes the states of oscillators, but their average frequencies are mutually repelled. The effect of individual intrinsic noise on the average frequencies is effectively equivalent to the effect of synchrony imperfection.

- Pikovsky A., Rosenblum M., Kurths J. Synchronization. A Universal Concept in Nonlinear Sciences. Cambridge: Cambridge University Press, 2003. 432 p
- Winfree A.T. Biological Rhythms and the Behavior of Populations of Coupled Oscillators // J. Theoret. Biol. 1967. Vol. 16. P. 15–42.
- 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. Araki H. Springer Lecture Notes in Physics No. 39. New York: Springer, 1975. P. 420–422.
- Kuramoto Y. Chemical Oscillations, Waves and Turbulence. New York: Dover, 2003.
- Kawamura Y., Shirasaka Sh., Yanagita T., Nakao H. Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems // Phys. Rev. E. 2017. Vol. 96.012224.
- Taira K., Nakao H. Phase-response analysis of synchronization for periodic flows // J. Fluid Mech. 2018. Vol. 846. R2. https://doi.org/10.1017/jfm.2018.327
- Nakao H., Yasui Sh., Ota M., Arai K., Kawamura Y. Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations // Chaos. 2018. Vol. 28. 045103.
- Pikovskii A.S. Synchronization and stochastization of nonlinear oscillations by external noise //Nonlinear and Turbulent Processes in Physics. Vol. 3. Ed. Sagdeev R.Z. Chur: Harwood Academic, 1984. P. 1601–1604.
- Pikovskii A.S. Synchronization and stochastization of array of self-excited oscillators by external noise. Radiophys. Quantum Electron., 1984, vol. 27, pp. 390–395.
- Ritt J. Evaluation of entrainment of a nonlinear neural oscillator to white noise // Phys. Rev. E. 2003. Vol. 68. 041915.
- 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.
- Goldobin D.S., Pikovsky A.S. Synchronization of periodic self-oscillations by common noise. Radiophys. Quantum Electron., 2004, vol. 47, pp. 910–915.
- Pakdaman K., Mestivier D. Noise induced synchronization in a neuronal oscillator // Phys. D. 2004. Vol. 192. P. 123–137.
- Snyder J., Zlotnik A., Hagberg A. Stability of entrainment of a continuum of coupled oscillators// Chaos. 2017. Vol. 27. 103108.
- Goldobin D.S., Pikovsky A. Synchronization and desynchronization of self-sustained oscillators by common noise // Phys. Rev. E. 2005. Vol. 71. 045201.
- 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.
- Nagai K.H., Kori H. Noise-induced synchronization of a large population of globally coupled nonidentical oscillators // Phys. Rev. E. 2010. Vol. 81. 065202.
- Ott E., Antonsen T.M. Low dimensional behavior of large systems of globally coupled oscillators //Chaos. 2008. Vol. 18. 037113.
- 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.
- Goldobin D.S., Dolmatova A.V., Rosenblum M., Pikovsky A. Synchronization in Kuramoto– Sakaguchi 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).
- Dolmatova A.V., Goldobin D.S., Pikovsky A. Synchronization of coupled active rotators by common noise // Phys. Rev. E. 2017. Vol. 96. 062204.
- Wiener N. Cybernetics: Or Control and Communication in the Animal and the Machine. 2nd ed. Cambridge: MIT Press, 1965.
- Watanabe S., Strogat S.H. Constant of motion for superconducting josephson arrays // Phys. D. 1994. Vol. 74. P. 197–253.
- Pikovsky A., Rosenblum M. Partially integrable dynamics of hierarchical populations of coupled oscillators // Phys. Rev. Lett. 2008. Vol. 101. 264103.
- 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. ̈
- 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.
- 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.
- Totz J.F., Rode J., Tinsley M.R., Showalter K., Engel H. Spiral wave chimera states in large populations of coupled chemical oscillators // Nature Physics. 2018. Vol. 14. P. 282–285.
- Goldobin D.S., Pikovsky A. Antireliability of noise-driven neurons // Phys. Rev. E. 2006. Vol. 73.061906.
- Wieczorek S. Stochastic bifurcation in noise-driven lasers and Hopf oscillators // Phys. Rev. E. 2009. Vol. 79. 036209.
- Yoshimura K., Arai K. Phase reduction of stochastic limit cycle oscillators // Phys. Rev. Lett. 2008. Vol. 101. 154101.
- Goldobin D.S., Teramae J.N., Nakao H., Ermentrout G.B. Dynamics of limit-cycle oscillator subject to general noise // Phys. Rev. Lett. 2010. Vol. 105. 154101.
- Bensoussan A., Lions J.L., Papanicolaou G. Asymptotic Analysis for Periodic Structures. Amsterdam: North-Holland, 1978.
- FitzHugh R.A. Impulses and physiological states in theoretical models of nerve membrane // Biophys. J. 1961. Vol. 1. P. 445–466.
- Nagumo J., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon // Proc. IRE. 1962. Vol. 50. P. 2061–2070.
- Goldobin D.S. Anharmonic resonances with recursive delay feedback // Phys. Lett. A. 2011. Vol. 375. P. 3410–3214.
- Peter F., Pikovsky A. Transition to collective oscillations in finite Kuramoto ensembles // Phys. Rev. E. 2018. Vol. 97. 032310.
- Pikovsky A., Dolmatova A.V., Goldobin D.S. Correlations of states of non-entrained oscillators in Kuramoto ensemble with noise in the mean field. Radiophys. Quantum Electron., 2019, vol. 61, no. 8–9, pp. 672–680.

- 2196 reads