Efficient path integral solution of Fokker–Planck equation: response, bifurcation and periodicity of nonlinear systems

Response of nonlinear systems subjected to harmonic, parametric, and random excitation is of importance in the field of structural dynamics. The transitional probability density function (PDF) of the random response of nonlinear systems under white or coloured noise excitation (delta-correlated) is governed by both the forward Fokker–Planck (FP) and the backward Kolmogorov equations. This article presents a new approach for efficient numerical implementation of the path integral (PI) method in the solution of the FP equation for some nonlinear systems subjected to white noise, parametric and combined harmonic and white noise excitations. The modified PI method is based on a non-Gaussian transition PDF and the Gauss–Legendre integration scheme. The steady state PDF, jump phenomenon, noise induced state changes of PDF are studied by the modified PI method. Effects of white noise intensity, amplitude and frequency of harmonic excitation and the level of nonlinearity on stochastic jump and bifurcation behaviours of a hardening Duffing oscillator are also investigated.