Multiple-peak probability density function of non-linear oscillators under Gaussian white noise

Abstract

An exponential-polynomial closure (EPC) method is introduced to solve the Fokker–Planck equation for the stationary multiple-peak probability density function (PDF) of the response of non-linear oscillators under Gaussian white noise. The EPC method uses Gaussian PDFs as a part of the weighting functions in the solution procedure. The Gaussian PDFs can be obtained by a standard equivalent linearization (EQL) method. This study considers the case that the EQL method gives multiple Gaussian PDFs. An improvement is proposed to make the weighting functions be independent of the choice of the Gaussian PDFs. To assess the effectiveness of the proposed method, a non-linear oscillator with a three-peak PDF of displacement is examined. Both low-level excitation and high-level excitation are considered. Comparison with the exact PDF solution shows that the improved EPC method can present a unique exact stationary PDF solution to the Fokker–Planck equation no matter which Gaussian PDF is adopted from the EQL method.