Parametric Analysis of Stochastic Oscillators by the Statistical Modeling Method
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metric Analysis of Stochastic Oscillators by the Statistical Modeling Method M. A. Yakunin* Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences, pr. Akad. Lavrent’eva 6, Novosibirsk, 630090 Russia Received November 19, 2018; in final form, May 19, 2019; accepted April 16, 2020
Abstract—We use the statistical modeling method to investigate the influence of the Wiener and Poisson random noise on the behavior of linear and Van der Pol oscillators. In the case of linear oscillator, an analytical expression of the autocovariance function of the solution to a stochastic differential equation (SDE) is obtained. This expression along with the formulas for the mathematical expectation and variance of the SDE solution allows us to carry out parametric analysis and investigate the accuracy of estimates for the moments of SDE numerical solution obtained based on the generalized explicit Euler method. In the case of the Van der Pol oscillator, the Poisson component influence on the nature of oscillations of the first and the second moments of the SDE solution at large jumps is investigated numerically. DOI: 10.1134/S1995423920030088 Keywords: stochastic differential equations, Wiener and Poisson components, generalized Euler method, stochastic oscillators.
INTRODUCTION Analysis of the influence of random noise on the behavior of oscillating solutions to stochastic differential equations (SDEs) was carried out in [1–7]. The complexity of analysis of nonlinear system of SDEs consists in the impossibility of obtaining a finite system of ordinary differential equations (ODEs) even for the first moment of the solution, whereas in a linear case, such system is readily obtained from the associated ODE system, i.e., an SDE system with zero noise intensities. In [1] there are given many examples of mathematical models of physical processes in the form of SDE systems describing linear and nonlinear stochastic oscillations; numerical analysis of solutions to these systems is carried out; new statistical parameters, such as frequency analogs of the integral curve and phase trajectory, are introduced. In [2, 3], the first two moments of harmonic oscillator with additive and multiplicative noise are studied. The variants of random damping and random frequency of oscillations are considered, and the influence of external periodic force on the moments is analyzed. In [4–6], for some nonlinear oscillators with external stochastic and multiplicative noise, including Van der Pol type oscillators, approximations of distributions of their solutions are constructed; accurate results are obtained for a class of oscillators with limit cycles. A universal method of numerical analysis of solutions to nonlinear SDEs is the method of statistical modeling. This paper continues work [7], which used the generalized explicit Euler method to investigate the influence of Wiener and Poisson random noise on the behavior of linear oscillator and Van der Pol oscillator, as well as estimates for functionals of CDE numer
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