Nonlinear Dynamics

, Volume 62, Issue 4, pp 743–750 | Cite as

Nonzero mean PDF solution of nonlinear oscillators under external Gaussian white noise

Original paper


This paper is concerned with the nonzero mean stationary probability density function (PDF) solution for nonlinear oscillators under external Gaussian white noise. The PDF solution is governed by the well-known Fokker–Planck–Kolmogorov (FPK) equation and this equation is numerically solved by the exponential-polynomial closure (EPC) method. Different types of oscillators are further investigated in the case of nonzero mean response. Either weak or strong nonlinearity is considered to show the effectiveness of the EPC method. When the polynomial order equals 2, the results of the EPC method are identical with those given by equivalent linearization (EQL) method. These results obtained with the EQL method differ significantly from exact solution or simulated results. When the polynomial order is 4 or 6, the PDFs obtained with the EPC method present a good agreement with the exact solution or simulated results, especially in the tail regions. The numerical analysis also shows that the nonzero mean PDF of the response is nonsymmetrically distributed about its mean unlike the case of the zero mean PDF reported in the references.


Nonlinear oscillator Nonzero mean Probability density function Gaussian white noise Stochastic processes 


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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of MacauMacao SARPR China
  2. 2.Department of Civil EngineeringTianjin UniversityTianjinPR China

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