Study of the Duffing–Rayleigh oscillator subject to harmonic and stochastic excitations by path integration

• Published in: Applied mathematics and computation Vol. 172; no. 2; pp. 1212 - 1224
• Main Authors: Xie, W.X., Xu, W., Cai, L.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 15.01.2006; Elsevier
• Subjects: Distribution theory; Duffing–Rayleigh oscillator; Exact sciences and technology; Harmonic and stochastic excitations; Mathematical analysis; Mathematics; Measure and integration; Partial differential equations; Path integration; Probability and statistics; Probability density; Probability theory and stochastic processes; Sciences and techniques of general use; Harmonic and stochastic excitations; Probability density; Path integration; Duffing–Rayleigh oscillator; Harmonic and stochastic excitations: Probability density; Path integral; Periodic function; Probability distribution; Duffing-Rayleigh oscillator; Gauss formula; Stochastic excitation; Applied mathematics; Duffing Rayleigh oscillator; Fokker Planck equation; Probability density function; Sinusoidal excitation; Kolmogorov equation; Steady state solution; Harmonic oscillator
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2005.03.018

In this paper, the Duffing–Rayleigh oscillator subject to harmonic and stochastic excitations is investigated via path integration based on the Gauss–Legendre integration formula. The method can successfully capture the steady state periodic solution of probability density function. This path integration method, using the periodicity of the coefficient of associated Fokker–Planck–Kolmogorov equation, is extended to deal with the averaged stationary probability density, and is efficient to computation. Meanwhile, the changes of probability density caused by the intensities of harmonic and stochastic excitations, are discussed in three cases through the instantaneous probability density and the averaged stationary probability density.