Journal of Statistical Physics, Vol. 15, No. 5, 1976 Studies in Nonlinear Stochastic Processes. II. The Dulling Oscillator Revisited A. B. Budgor, 1 K. Lindenberg, 1 and K. E. Shuler 1 Received April 27, 1976 We have applied the approximation method of statistical linearization and various higher order corrections thereto to the study of a nonlinear oscillator perturbed by Gaussian, delta-correlated noise. We compute the second- order statistics of the response, i.e., the variances, autocorrelation functions, and spectral densities for various forms of the nonlinearity and compare our results with the few more exact calculations which are available in the literature. We show that a very simple modification of statistical lineariza- tion, based upon the use of the variance as obtained from the appropriate Fokker-Planck equation, yields results which are in better agreement with the "exact" literature results than either statistical linearization or first-order corrections thereto. This modified method of statistical linearization has the significant advantage of great computational simplicity as compared to other attempts of accurate calculations of second-order statistics of nonlinear stochastic equations now in the literature. KEY WORDS: Stochastic processes; nonlinear stochastic equations; statistical linearization ; autocorrelation functions ; spectral densities. 1. INTRODUCTION In the first paper of this series/1l Budgor presented a general exposition of the method of "statistical linearization" and its higher order correction terms for the approximate solution of nonlinear stochastic differential equations. In this paper, we apply the method of statistical linearization to a study of the "stochastic" Duffing oscillator, i.e., an anharmonic damped oscillator This work was supported in part by the National Science Foundation under Grants MPS 72-04363 and CHE 75-20624. 1 Department of Chemistry, University of California--San Diego, La Jolla, California. 375 91976PlenumPublishing Corporation, 227 West 17th Street,New York, N.Y. I0011. No part of this publica- tion may be reproduced,stored in a retrieval system,or transmitted, in any form or by any means, electronic. mechanical,photocopying, microfilming, recording,or otherwise,without written permission of the publisher.