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Asymptotic analysis for interactive oscillators of the van der Pol type

Published online by Cambridge University Press:  01 July 2016

Kiyomasa Narita
Kiyomasa Narita*
Affiliation:
Kanagawa University
*
∗Postal address: Department of Mathematics, Faculty of Technology, Kanagawa University, Rokkakubashi Kanagawa-ku, Yokohama 221, Japan.
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Abstract

We consider the N-oscillator system of the van der Pol type, which contains a small positive parameter ε multiplying the non-linear damping and the random disturbance. For a formulation of the output we take the solution X(t) = (Xi(t))i=1···,N of the system of 2N-dimensional stochastic differential equations. Rotating each component Xi(t) about the origin of the plane by an angle t, we find that on time scales of order 1/ε together with sufficiently large N each Xi(t) behaves as the equi-ultimately bounded solution of an equation of the McKean type admitting a stationary probability distribution.

Keywords

VAN DER POL EQUATIONSTOCHASTIC DIFFERENTIAL EQUATIONEXPLOSION TIMECOMPACTNESS OF MEASURESMARTINGALE PROBLEMMCKEAN EQUATIONPROPAGATION OF CHAOSSTATIONARY PROBABILITY DISTRIBUTION
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 1 , March 1987 , pp. 44 - 80
Copyright
Copyright © Applied Probability Trust 1987 

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