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Stochastically perturbed limit cycles

Published online by Cambridge University Press:  14 July 2016

Charles J. Holland
Charles J. Holland*
Affiliation:
Purdue University
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Abstract

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.

Keywords

STOCHASTIC DIFFERENTIAL EQUATIONSLIMIT CYCLESERGODIC MEASURES
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 2 , June 1978 , pp. 311 - 320
Copyright
Copyright © Applied Probability Trust 1978 

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References

[1] Fleming, W. (1974) Stochastically perturbed dynamical systems. Rocky Mountain J. Maths 4, 407–433.Google Scholar
[2] Friedman, A. (1964) Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
[3] Friedman, A. (1976) Stochastic Differential Equations and Applications, Vols 1, 2. Academic Press, New York.Google Scholar
[4] Gihman, I. and Skorokhod, A. (1972) Stochastic Differential Equations. Springer-Verlag, New York.Google Scholar
[5] Hale, J. (1969) Ordinary Differential Equations. Wiley-Interscience, New York.Google Scholar
[6] Holland, C. (1974) Ergodic expansions in small noise problems. J. Differential Eqns 16, 281–288.Google Scholar
[7] Holland, C. (1976) Stationary small noise problems. Internat. J. Nonlinear Mech. 11, 41–47.Google Scholar
[8] Khasminski, R. (1960) Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations. Theory Prob. Appl. 5, 179–196.Google Scholar
[9] Kushner, H. (1969) The Cauchy problem for a class of degenerate parabolic equations and asymptotic properties of the related diffusion processes. J. Differential Eqns 6, 209–231.Google Scholar
[10] Wonham, W. (1966) A Lyapunov method for the estimation of statistical averages. J. Differential Eqns 2, 365–377.Google Scholar
[11] Zakai, M. (1969) A Lyapunov criterion for the existence of stationary probability distributions for systems perturbed by noise. SIAM J. Control 7, 390–397.CrossRefGoogle Scholar