Abstract
In this paper, we introduce the study of the general form of stochastic Van der Pol equation (SVDP) under an external excitation described by Gaussian white noise. The study involves the use of Wiener-Chaos expansion technique (WCE) and Wiener-Hermite expansion (WHE) technique. The application of these techniques results in a system of deterministic differential equations (DDEs). The resulting DDEs are solved by the numerical techniques and compared with the results of Monte Carlo (MC) simulations. Also, we introduce a new formula that facilitates handling the cubic nonlinear term of van der Pol equations. The main results of this study are: 1) WCE technique is more accurate, programmable compared with WHE and for the same order, WCE consumes less time. 2) The number of Gaussian random variables (GRVs) is more effective than the order of expansion. 3) The agreement of the results with the MC simulations reflects the validity of the forms obtained through theorem 3.1.
Highlights
In recent years, vibration systems represented by oscillations have received considerable attention
We introduce the study of the general form of stochastic Van der Pol equation (SVDP) under an external excitation described by Gaussian white noise
We investigated the SVDP equation under an external excitation described by Gaussian white noise
Summary
Vibration systems represented by oscillations have received considerable attention. In 1920, the Dutch scientist named Van der Pol found stable oscillation in the electric circuit employing a vacuum tube [1] This oscillation was modeled mathematically by a second-order differential equation. The stochastic system can be modeled by a mathematical model and called the SVDP system This system has been studied by many authors such as using the stochastic averaging method with the Fokker-Planck equation to find the probability density function of the stationary solution [14] [15] [16]. The spectral decomposition techniques have been used for solving nonlinear stochastic differential equations (NSDEs) One of those techniques is the WHE technique which was suggested by Norbert Wiener [17].
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