Abstract
Astochastic model for some class ofnonlinear oscillators, which includes a van der Pol-type oscillator with random parameters, is analyzed in thediffusion limit. That is, small random fluctuations and long time are considered, while the nonlinearity is also assumed to be small. We show that there existstationary distributions, independent of the phase of the oscillator, a result proved earlier by R. L. Stratonovich assuming the random perturbations of the frequency to be delta correlated. The time behavior of the moments of the displacement of the oscillator from its rest position is also investigated and the results are compared with the corresponding ones for the linear random oscillator. A numerical study is also performed for the first two moments and plots are given.
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References
G. Blankenship and G. C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbances. I,SIAM J. Appl. Math. 34:437–476 (1978).
W. Feller, The parabolic differential equations and the associated semi-groups of transformations,Ann. Math. 55:468–519 (1952).
E. Hille, Le probabilités continues en chaîne,C. R. Acad. Sci. Paris 230:34–35 (1950).
M. Kac, On distributions of certain Wiener functionals,Trans. Amer. Math. Soc. 65:1–13 (1949).
R. Z. Khas'minskii, A limit theorem for the solutions of differential equations with a random right hand side,Theor. Probability Appl. 11:390–406 (1966).
G. Papanicolaou and J. B. Keller, Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media,SIAM J. Appl. Math. 21:287–305 (1971).
G. Papanicolaou, Wave propagation in a one dimensional random medium,SIAM J. Appl. Math. 21:13–18 (1971).
G. C. Papanicolaou, D. Stroock, and S. R. S. Varadhan, Martingale approach to some limit theorems, inStatistical Mechanics, Dynamical Systems and the Duke Turbulence Conference, D. Ruelle ed., Duke Univ. Math. Series, Vol. 3 (Duke University Press, Durham, North Carolina, 1977).
J. J. Stoker,Nonlinear Vibrations in Mechanical and Electrical Systems (Wiley-Interscience, New York, 1950).
R. L. Stratonovich,Topics in the Theory of Random Noise, Vol. II (Gordon and Breach, New York, 1967).
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on leave from the University of Padua, Italy.
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Spigler, R. Nonlinear parametric oscillations in certain stochastic systems: A random van der Pol oscillator. J Stat Phys 41, 175–200 (1985). https://doi.org/10.1007/BF01020608
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DOI: https://doi.org/10.1007/BF01020608