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

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  • Synchronization, Control and Dynamics of Chaotic Models
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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. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer–Verlag, Berlin, 2003)

  2. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization A universal concept in nonlinear sciences (Cambridge University Press, 2001)

  3. E. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (MIT Press, 2006)

  4. C.W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1985)

  5. B. Øksendal, Stochastic Differential Equations (Springer, New York, 2003)

  6. A. Demir, A. Mehrotra, J. Roychowdhury, IEEE T Circuits–I 47(5), 655 (2000)

    Article  Google Scholar 

  7. F.X. Kaertner, Int. J. Circ. Theor. App. 18, 485 (1990)

    Article  Google Scholar 

  8. K. Yoshimura, K. Arai, Phys. Rev. Lett. 101, 154101 (2008)

    Article  ADS  Google Scholar 

  9. M. Bonnin, F. Corinto, IEEE T Circuits–II 60(8), 2104 (2013)

    MathSciNet  Google Scholar 

  10. M. Bonnin, F. Corinto, IEEE T Circuits–II 61(3), 158 (2014)

    Google Scholar 

  11. R. Benzi, G. Parisi, A. Sutera, A. Vulpiani, Tellus 34(1), 10 (1982)

    Article  ADS  Google Scholar 

  12. K. Wiesenfeld, F. Moss, Nature 373, 33 (1995)

    Article  ADS  Google Scholar 

  13. L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70(1), 223 (1998)

    Article  ADS  Google Scholar 

  14. S.P. Beeby, M.J. Tudor, N.M. White, Meas. Sci. Technol. 17(2), 175 (2006)

    Article  Google Scholar 

  15. L. Gammaitoni, Contemp. Phys. 53(2), 119 (2012)

    Article  ADS  Google Scholar 

  16. B. Hauschildt, N.B. Janson, A. Balanov, E. Schöll, Phys. Rev. E 74(5), 051906 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  17. H. Nakao, K. Arai, Y. Kawamura, Phys. Rev. Lett. 98(18), 184101 (2007)

    Article  ADS  Google Scholar 

  18. M. Bonnin, F. Corinto, M. Gilli, IEEE T Circuits–II 59(10), 638 (2012)

    Google Scholar 

  19. G.S. Medvedev, Phys. Lett. A 374, 1712 (2010)

    Article  ADS  Google Scholar 

  20. J. Hespanha, Moment closure for biochemical networks, 3rd International Symposium on Communications, Control and Signal Processing (2008), p. 152

  21. C.S. Gillespie, IET Syst. Biol. 3(1), 52 (2009)

    Article  MathSciNet  Google Scholar 

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Bonnin, M., Corinto, F. & Lanza, V. A mathematical framework for amplitude and phase noise analysis of coupled oscillators. Eur. Phys. J. Spec. Top. 225, 171–186 (2016). https://doi.org/10.1140/epjst/e2016-02617-8

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  • DOI: https://doi.org/10.1140/epjst/e2016-02617-8

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