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Investigation of high order stochastic differential equations using averaging method

In the paper this method is applied to high order stochastic differential equations. The nonlinear oscillations in high order deterministic differential equations were investigated in the fundamental work of Prof. Nguyen Van Dao. As an application of high order stochastic differential equations the nonlinear oscillation of single degree of freedom systems subjected to the excitation of a class of colored noises is outline !. | Vietnam Journal of Mechanics, VAST, Vol. 29, No. 3 (2007), pp. 2!J9 - 255 Special Issue Dedicated to the l\1emory of Prof. Nguyen Van Dao INVESTIGATION OF HIGH ORDER STOCHASTIC DIFFERENTIAL EQUATIONS USING AVERAGING METHOD NGUYEN DONG ANIJ , NGO Till IIONC HUE Institute of Mechanics, VAST Abstract. The averaging method is an useful tool for investigating both deterministic and stochastic quasilinear system. In the stochastic problems, however, the method has often been developed only for mechanical systems subjected to white noise excitations. In the paper this method is applied to high order stochastic differential equations. The nonlinear oscillations in high order deterministic differential equations were invest igated in the fundamental work of Prof. Nguyen Van Dao. As an application of high order stochastic differential equations the nonlinear oscillation of single degree of freedom systems subjected to the excitation of a class of colored noises is outline~!. The results obtained show that the higher order averaging method can also be successfully extended to the cases of colored noise excitation. 1. INTRODUCTION Interest in the investigation of random phenomena is considerable over the recent years, due to various problems encountered in engineering applications. The well - kuown averaging method originally given by Krylov and I3ogolibov and then developed by l\'Iitropolskii is one of most popular methods for the approximate analysis of nonlinear systems [2]. The advantage of this method is that it reduces the dimension of the response coordinates. In the field of random vibration the averaging method was extended by Stratonovich [3] and has a mathematically rigorbus proof by Khasminskii [4]. It is well-known, however , that the effect of some nonlinear terms is lost during first - order averaging procedure . In order to overcome this insufficiency, the procedures for obtaining higher approxirnatc solutions in the stochastic averaging method were .