Asymptotic methods in the theory of nonlinear stochastic oscillations

Abstract

We study applications of asymptotic methods of nonlinear mechanics and the method of Fokker-PlanckKolmogorov equations to stochastic oscillations in quasilinear oscillating systems with random perturbations.In practice, we often encounter the action of random forces on oscillating systems and stochastic oscillations in these systems. Electric fluctuations, noises in radio-engineering systems, random vibrations of aircrafts, oscillations of various elastic structures under earthquakes, etc., can serve as examples of random oscillations. For this reason, the study of the influence of random forces on oscillating systems and stochastic oscillations in them is very important for practical problems related to an increase in the sensitivity and interference immunity of radio and measuring devices, stability of structural units under random influence, etc. As a rule, oscillating systems encountered in practice are nonlinear.In numerous problems, one encounters simpler systems close to linear ones and subjected to the action of weak random forces. These problems were thoroughly investigated from both theoretical and applied points of view.The analytic investigation of nonlinear stochastic oscillations meets serious difficulties. In this connection, approximate methods of investigation are of great importance. Methods of nonlinear mechanics, in particular the asymptotic method, averaging method, method of harmonic balance, and method of equivalent linearization, proposed and developed by Krylov, Bogolyubov, and Mitropol'skii proved to be especially efficient [1,2].Investigations in the theory of nonlinear stochastic oscillations are carried out mainly in the direction of creation of methods for the asymptotic integration of nonlinear stochastic systems and their mathematical justification.The present paper is devoted mainly to the development of asymptotic methods associated with the averaging method that are used in the theory of nonlinear stochastic differential equations and nonlinear stochastic oscillations.For the first time, stochastic differential equations were considered by Bernstein in [3]. Problems of investigation of mechanical systems subjected to the action of random forces led to the necessity of the study of stochastic equations. In [4], Krylov and Bogolyubov considered the behavior of such systems under the assumption that, in the limit, random forces turn into processes with independent increments. In contemporary terminology, these processes are called "white noises." In this case, the limit process is a Markov process; for the transition probability density of this process, the Fokker-Planck-Kolmogorov partial differential equation was deduced.Investigating the limit behavior of a linear oscillating system subjected to the action of a random force that turns into a white noise in the limit, Bogolyubov showed that the motion of this system is described by a Markov process whose transition probability density satisfies the Fokker-Planck-Kolmogorov partial differential equation [5].The Fokker-Planck-Kolmo...

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