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Practical Synchronization of Winfree Oscillators in a Random Environment

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Abstract

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.

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References

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Ariaratnam, J.T., Strogatz, S.H.: Phase diagram for the Winfree model of coupled nonlinear oscillators. Phys. Rev. Lett. 86, 4278–4281 (2001)

    Article  ADS  Google Scholar 

  4. Balmforth, N.J., Sassi, R.: A shocking display of synchrony. Physica D 143, 21–55 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berglund, N., Gentz, B.: Noise-Induced Phenomena in Slow–Fast Dynamical Systems. A Sample-Paths Approach. Springer, New York (2006)

    MATH  Google Scholar 

  8. Bowong, S., Tewa, J.: Practical adaptive synchronization of a class of uncertain chaotic systems. Nonlinear Dynam. 56, 57–68 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Buck, J., Buck, E.: Biology of synchronous flashing of fireflies. Nature 211, 562 (1966)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Choi, Y.-P., Ha, S.-Y., Yun, S.-B.: Complete synchronization of Kuramoto oscillators with finite inertia. Physica D 240, 32–44 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  14. DeVille, L.: Transitions amongst synchronous solutions in the stochastic Kuramoto model. Nonlinearity 25, 1473–1494 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dong, J.-G., Xue, X.: Synchronization analysis of Kuramoto oscillators. Commun. Math. Sci. 11, 465–480 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Erban, R., Hakovec, J., Sun, Y.: A Cucker–Smale model with noise and delay. SIAM J. Appl. Math. 76, 1535–1557 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fax, J.A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 49, 1465–1476 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gentz, B., Ha, S.-Y., Ko, D., Wiesel, C.: Emergent dynamics of Kuramoto oscillators under the effect of additive white noises. Preprint

  21. Goldstein, R.E., Polin, M., Tuval, I.: Noise and synchronization in pairs of beating eukaryotic flagella. Phys. Rev. Lett. 103, 168103 (2009)

    Article  ADS  Google Scholar 

  22. Ha, S.-Y., Kim. D.: Robustness and asymptotic stability for the Winfree model on a general network under the effect of time-delay. Preprint

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, London (2009)

    Book  MATH  Google Scholar 

  28. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, Sec edn. Springer, New York (1998)

    Book  MATH  Google Scholar 

  29. Kuramoto, Y.: International symposium on mathematical problems in mathematical physics. Lect. Notes Theor. Phys. 30, 420 (1975)

    Article  ADS  Google Scholar 

  30. Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  33. Strogatz, S.H.: Human sleep and circadian rhythms: a simple model based on two coupled oscillators. J. Math. Biol. 25, 327–347 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  34. Winfree, A.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42 (1967)

    Article  Google Scholar 

  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)

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Acknowledgements

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.

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  1. Research Institute of Mathematics, Seoul National University, Gwanakro 1, Gwanak-gu, Seoul, 08826, Republic of Korea

    Dongnam Ko

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Ko, D. Practical Synchronization of Winfree Oscillators in a Random Environment. J Stat Phys 174, 1263–1287 (2019). https://doi.org/10.1007/s10955-019-02234-2

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