Statistical linearization and cumulant method in the analysis of non-linear stochastic systems

The article examines the methods of analysis of non-linear stochastic systems based on statistical linearization and the apparatus of cumulants. Two approaches to determining statistical linearization coefficients based on criteria of coincidence of statistical characteristics and minimization of mean square error of approximation are considered. Analytical expressions were obtained for the parameters of statistically linearized models, as well as the features of their application in the analysis of non-Gaussian random processes were investigated. Various statistical approximation methods were compared using a non-linear dynamic system under the influence of white Gaussian noise. The analysis of the cumulants method was carried out; the features of using the Edgewart decomposition to construct a closed system of differential equations relative to the first cumulants of a random process were considered. It is shown that the statistical linearization method is a special case of the cumulant method with the Gaussian approximation of the probability distribution. A comparative assessment of the accuracy of the methods considered was carried out, which makes it possible to determine the areas of their effective application in the study of nonlinear stochastic systems.