Stochastic resonance in an autonomous system with a nonuniform limit cycle

Wouter-Jan Rappel; Steven H. Strogatz

doi:10.1103/PhysRevE.50.3249

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Stochastic resonance in an autonomous system with a nonuniform limit cycle

Wouter-Jan Rappel and Steven H. Strogatz

Laboratoire de Physique Statistique associé au CNRS et aux Universités Paris VI et Paris VII, 24 rue Lhomond, 75231 Paris Cedex 05, France
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

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Phys. Rev. E 50, 3249 – Published 1 October, 1994

DOI: https://doi.org/10.1103/PhysRevE.50.3249

Abstract

In a recent numerical study, Gang et al. [Phys. Rev. Lett. 71, 807 (1993)] presented the first example of stochastic resonance in an autonomous system. They considered a two-variable model in which a limit cycle is born as some parameter is varied. Their numerical experiments revealed various noise-induced effects, including a noise-induced shift in the frequency of the limit-cycle oscillations, and noise-induced oscillations in the absence of a deterministic limit cycle. We show that both of these effects are simple consequences of the nonuniformity of the motion along the limit cycle.

References (13)

P. Jung, Phys. Rep. 234, 175 (1993).
R. Benzi, A. Sutera and A. Vulpiani, J. Phys. A 14, 453 (1981).
For a review, see F. Moss, in An Introduction to Some Contemporary Problems in Statistical Physics, edited by G. H. Weiss (SIAM, Philadelphia, in press).
A. Simon and A. Libchaber, Phys. Rev. Lett. 68, 3375 (1992).
J. Douglass, L. Wilkens, E. Pantazelou and F. Moss, Nature 365, 337 (1993).
H. Braun, H. Wissing, K. Schafer and M. C. Hirsh, Nature 367, 270 (1994).
K. Wiesenfeld, D. Pierson, E. Pantazelou, C. Dames and F. Moss, Phys. Rev. Lett 72, 2125 (1994).
G. Perez and H. A. Cerdeira, Phys. Rev. A 46, 7492 (1992).
V. Hakim and W. J Rappel (unpublished).
H. Gang, T. Ditzinger, C.Z. Ning and H. Haken, Phys. Rev. Lett. 71, 807 (1993).
We have used Euler's forward integration method with a time step of dt=0.01. To obtain a reasonably smooth power spectrum, we have averaged over the spectra of 500 different time series.
For an experimental realization of this dynamics in a sensory neural system, along with a theory of stochastic resonance for excitable systems, see [7].
S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison Wesley, Reading, MA, 1994).

[c1] P. Jung, Phys. Rep. 234, 175 (1993).

[c2] R. Benzi, A. Sutera and A. Vulpiani, J. Phys. A 14, 453 (1981).

[c3] For a review, see F. Moss, in An Introduction to Some Contemporary Problems in Statistical Physics, edited by G. H. Weiss (SIAM, Philadelphia, in press).

[c4] A. Simon and A. Libchaber, Phys. Rev. Lett. 68, 3375 (1992).

[c5] J. Douglass, L. Wilkens, E. Pantazelou and F. Moss, Nature 365, 337 (1993).

[c6] H. Braun, H. Wissing, K. Schafer and M. C. Hirsh, Nature 367, 270 (1994).

[c7] K. Wiesenfeld, D. Pierson, E. Pantazelou, C. Dames and F. Moss, Phys. Rev. Lett 72, 2125 (1994).

[c8] G. Perez and H. A. Cerdeira, Phys. Rev. A 46, 7492 (1992).

[c9] V. Hakim and W. J Rappel (unpublished).

[c10] H. Gang, T. Ditzinger, C.Z. Ning and H. Haken, Phys. Rev. Lett. 71, 807 (1993).

[c11] We have used Euler's forward integration method with a time step of dt=0.01. To obtain a reasonably smooth power spectrum, we have averaged over the spectra of 500 different time series.

[c12] For an experimental realization of this dynamics in a sensory neural system, along with a theory of stochastic resonance for excitable systems, see [7].

[c13] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison Wesley, Reading, MA, 1994).

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Stochastic resonance in an autonomous system with a nonuniform limit cycle

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References (13)

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