Journal of Statistical Physics

, Volume 174, Issue 6, pp 1263–1287 | Cite as

Practical Synchronization of Winfree Oscillators in a Random Environment

  • Dongnam KoEmail author


We present a practical synchronization estimate of Winfree oscillators in a random environment. In a large coupling regime, the deterministic Winfree model exhibits the oscillator death, emerging with a convergence of the phase ensemble. The additive noise, however, is expected to destroy the stability of an equilibrium. In this paper, we estimate the running maximum of the phase processes, and conclude that the escaping probability from a small interval is in the order of \(TN^{-1}\exp (-\kappa /\Vert \varSigma \Vert ^2)\) over a time interval [0, T], where \(\kappa \), N and \(\Vert \varSigma \Vert \) denote the coupling strength, number of oscillators and noise strength, respectively. This result explains the robustness of the practical synchronization, which indicates that the finite-time emergent behavior from finite oscillators is close to the synchronization phenomena when \(\kappa \) is large enough. Our approach produces explicit bounds on probabilities, relying on comparisons with the Ornstein–Uhlenbeck processes. It is hence optimal in the sense that the linearized model gives the same order.


Additive noise Ornstein–Uhlenbeck process Phase model Synchronization Winfree model 

Mathematics Subject Classification

70F99 92B25 



The work of D. Ko is supported by National Research Foundation of Korea grant (NRF-2017R1A2B2001864) funded by the Korean government. I would like to thank Prof. Seung-Yeal Ha for suggesting this problem and I also would like to thank Mr. Doheon Kim for careful reading of the manuscripts.


  1. 1.
    Aeyels, D., Rogge, J.A.: Existence of partial entrainment and stability of phase locking behavior of coupled oscillators. Prog. Theor. Phys. 112, 921–942 (2004)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Antonsen, T.M., Faghih, R.T., Girvan, M., Ott, E., Platig, J.: External periodic driving of large systems of globally coupled phase oscillators. Chaos 18, 037112 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ariaratnam, J.T., Strogatz, S.H.: Phase diagram for the Winfree model of coupled nonlinear oscillators. Phys. Rev. Lett. 86, 4278–4281 (2001)ADSCrossRefGoogle Scholar
  4. 4.
    Balmforth, N.J., Sassi, R.: A shocking display of synchrony. Physica D 143, 21–55 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benedetto, D., Caglioti, E., Montemagno, U.: On the complete phase synchronization for the Kuramoto model in the mean-field limit. Commun. Math. Sci. 13, 1775–1786 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berglund, N., Gentz, B.: A sample-paths approach to noise-induced synchronization: stochastic resonance in a double-well potential. Ann. Appl. Probab. 12, 1419–1470 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Berglund, N., Gentz, B.: Noise-Induced Phenomena in Slow–Fast Dynamical Systems. A Sample-Paths Approach. Springer, New York (2006)zbMATHGoogle Scholar
  8. 8.
    Bowong, S., Tewa, J.: Practical adaptive synchronization of a class of uncertain chaotic systems. Nonlinear Dynam. 56, 57–68 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buck, J., Buck, E.: Biology of synchronous flashing of fireflies. Nature 211, 562 (1966)ADSCrossRefGoogle Scholar
  10. 10.
    Carrillo, J.A., Klar, A., Martin, S., Tiwari, S.: Self-propelled interacting particle systems with roosting force. Math. Mod. Meth. Appl. Sci. 20, 1533–1552 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chiba, H.: A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model. Ergod. Theory Dynam. Syst. 35, 762–834 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Choi, Y.-P., Ha, S.-Y., Yun, S.-B.: Complete synchronization of Kuramoto oscillators with finite inertia. Physica D 240, 32–44 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Choi, S.-H., Cho, J., Ha, S.-Y.: Practical quantum synchronization for the Schrdinger–Lohe system. J. Phys. A: Math. Theor. 49, 205203 (2016)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    DeVille, L.: Transitions amongst synchronous solutions in the stochastic Kuramoto model. Nonlinearity 25, 1473–1494 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ding, X., Wu, R.: A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales. Stoch. Process. Appl. 78, 155–171 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dong, J.-G., Xue, X.: Synchronization analysis of Kuramoto oscillators. Commun. Math. Sci. 11, 465–480 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dörfler, F., Bullo, F.: Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators. SIAM J. Control Optim. 50, 1616–1642 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Erban, R., Hakovec, J., Sun, Y.: A Cucker–Smale model with noise and delay. SIAM J. Appl. Math. 76, 1535–1557 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fax, J.A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 49, 1465–1476 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gentz, B., Ha, S.-Y., Ko, D., Wiesel, C.: Emergent dynamics of Kuramoto oscillators under the effect of additive white noises. PreprintGoogle Scholar
  21. 21.
    Goldstein, R.E., Polin, M., Tuval, I.: Noise and synchronization in pairs of beating eukaryotic flagella. Phys. Rev. Lett. 103, 168103 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    Ha, S.-Y., Kim. D.: Robustness and asymptotic stability for the Winfree model on a general network under the effect of time-delay. PreprintGoogle Scholar
  23. 23.
    Ha, S.-Y., Noh, S.E., Park, J.: Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Netw. Heterog. Media 10, 787–807 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ha, S.-Y., Park, J., Ryoo, S.W.: Emergence of phase-locked states for the Winfree model in a large coupling regime. Discret. Contin. Dynam. Syst. A 35, 3417–3436 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ha, S.-Y., Ko, D., Park, J., Ryoo, S.W.: Emergent dynamics of Winfree oscillators on locally coupled networks. J. Differ. Equ. 260, 4203–4236 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ha, S.-Y., Ko, D., Park, J., Ryoo, S.W.: Emergence of partial locking states from the ensemble of Winfree oscillators. Q. Appl. Math. 75, 39–68 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, London (2009)CrossRefzbMATHGoogle Scholar
  28. 28.
    Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, Sec edn. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  29. 29.
    Kuramoto, Y.: International symposium on mathematical problems in mathematical physics. Lect. Notes Theor. Phys. 30, 420 (1975)ADSCrossRefGoogle Scholar
  30. 30.
    Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  31. 31.
    Oukil, W., Thieullen, P., Kessi, A.: Invariant cone and synchronization state stability of the mean field models. arXiv:1806.10916v1 [math.DS] (2018)
  32. 32.
    Saber, R.O., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49, 1520–1533 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Strogatz, S.H.: Human sleep and circadian rhythms: a simple model based on two coupled oscillators. J. Math. Biol. 25, 327–347 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Winfree, A.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42 (1967)CrossRefGoogle Scholar
  35. 35.
    Winfree, A.: 24 hard problems about the mathematics of 24 hour rhythms. Nonlinear oscillations in biology. In: Proceedings of the Tenth Summer Sem. Appl. Math., Univ. Utah, Salt Lake City, Utah, 1978, pp. 93–126. Lectures in Appl. Math., 17, Amer. Math. Soc., Providence, RI (1979)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Research Institute of MathematicsSeoul National UniversitySeoulRepublic of Korea

Personalised recommendations