Numerical solutions of stochastic Duffing-Van der Pol equations

Maha Hamed (), I. L. El-Kalla (), Mohamed A. El-Beltagy () and Beih S. El-desouky ()
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Maha Hamed: Mansoura University
I. L. El-Kalla: Mansoura University
Mohamed A. El-Beltagy: Cairo University
Beih S. El-desouky: Mansoura University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 1, 256-273

Abstract: Abstract In this paper, we introduce the numerical study of the stochastic Duffing-Van der Pol equation under both multiplicative and additive random forcing. The study involves the use of the Wiener-Chaos expansion (WCE) technique and the 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. Furthermore, we introduce a formula that facilitates handling the cubic nonlinear term of the polynomials Chaos. From the study, we do not only demonstrate the accuracy and ease of the WCE technique to solve the cubic nonlinear stochastic differential equations, but also, show the physical understanding of the dynamical behavior of the stochastic solutions through its statistical properties. Moreover, we investigate the WCE of the cubic term in the equation, the influence of the damping coefficients and the external force coefficient on both the used techniques and the behavior of the solution.

Keywords: Duffing-Van der Pol oscillator; Wiener-Chaos expansion; Wiener-Hermite expansion; White noise; Damping force; Multiplicative forcing; Additive forcing (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-022-00361-3

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