The European Physical Journal Special Topics

, Volume 225, Issue 1, pp 171–186 | Cite as

A mathematical framework for amplitude and phase noise analysis of coupled oscillators

Regular Article Synchronization, Control and Dynamics of Chaotic Models
Part of the following topical collections:
  1. Synchronization and Control in Time-Delayed Complex Networks and Spatio-Temporal Patterns

Abstract

Synchronization of coupled oscillators is a paradigm for complexity in many areas of science and engineering. Any realistic network model should include noise effects. We present a description in terms of phase and amplitude deviation for nonlinear oscillators coupled together through noisy interactions. In particular, the coupling is assumed to be modulated by white Gaussian noise. The equations derived for the amplitude deviation and the phase are rigorous, and their validity is not limited to the weak noise limit. We show that using Floquet theory, a partial decoupling between the amplitude and the phase is obtained. The decoupling can be exploited to describe the oscillator’s dynamics solely by the phase variable. We discuss to what extent the reduced model is appropriate and some implications on the role of noise on the frequency of the oscillators.

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References

  1. 1.
    Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer–Verlag, Berlin, 2003)Google Scholar
  2. 2.
    A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization A universal concept in nonlinear sciences (Cambridge University Press, 2001)Google Scholar
  3. 3.
    E. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (MIT Press, 2006)Google Scholar
  4. 4.
    C.W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1985)Google Scholar
  5. 5.
    B. Øksendal, Stochastic Differential Equations (Springer, New York, 2003)Google Scholar
  6. 6.
    A. Demir, A. Mehrotra, J. Roychowdhury, IEEE T Circuits–I 47(5), 655 (2000)CrossRefGoogle Scholar
  7. 7.
    F.X. Kaertner, Int. J. Circ. Theor. App. 18, 485 (1990)CrossRefGoogle Scholar
  8. 8.
    K. Yoshimura, K. Arai, Phys. Rev. Lett. 101, 154101 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    M. Bonnin, F. Corinto, IEEE T Circuits–II 60(8), 2104 (2013)MathSciNetGoogle Scholar
  10. 10.
    M. Bonnin, F. Corinto, IEEE T Circuits–II 61(3), 158 (2014)Google Scholar
  11. 11.
    R. Benzi, G. Parisi, A. Sutera, A. Vulpiani, Tellus 34(1), 10 (1982)ADSCrossRefGoogle Scholar
  12. 12.
    K. Wiesenfeld, F. Moss, Nature 373, 33 (1995)ADSCrossRefGoogle Scholar
  13. 13.
    L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70(1), 223 (1998)ADSCrossRefGoogle Scholar
  14. 14.
    S.P. Beeby, M.J. Tudor, N.M. White, Meas. Sci. Technol. 17(2), 175 (2006)CrossRefGoogle Scholar
  15. 15.
    L. Gammaitoni, Contemp. Phys. 53(2), 119 (2012)ADSCrossRefGoogle Scholar
  16. 16.
    B. Hauschildt, N.B. Janson, A. Balanov, E. Schöll, Phys. Rev. E 74(5), 051906 (2006)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    H. Nakao, K. Arai, Y. Kawamura, Phys. Rev. Lett. 98(18), 184101 (2007)ADSCrossRefGoogle Scholar
  18. 18.
    M. Bonnin, F. Corinto, M. Gilli, IEEE T Circuits–II 59(10), 638 (2012)Google Scholar
  19. 19.
    G.S. Medvedev, Phys. Lett. A 374, 1712 (2010)ADSCrossRefGoogle Scholar
  20. 20.
    J. Hespanha, Moment closure for biochemical networks, 3rd International Symposium on Communications, Control and Signal Processing (2008), p. 152Google Scholar
  21. 21.
    C.S. Gillespie, IET Syst. Biol. 3(1), 52 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2016

Authors and Affiliations

  1. 1.Politecnico di Torino, Department of Electronics and TelecommunicationsTurinItaly
  2. 2.Normandie Univ. ULH, LMAH, CNRS 3335, ISCNLe HavreFrance

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