Stochastic dynamics of hysteresis systems under harmonic and Poisson excitations

Hysteresis, a common nonlinear phenomenon in engineering structures, has been extensively studied. However, the nonlinear behavior of hysteretic systems under combined deterministic and random excitations remains insufficiently explored. This paper investigates the stochastic response and P-bifurcation of hysteretic systems under harmonic and Poisson white noise excitations. The generalized Fokker–Planck–Kolmogorov (GFPK) equation governing the probability density function (PDF) of the system is solved using a radial basis function neural network (RBFNN) method. Specifically, the trial solution of the GFPK equation is represented by a set of standard Gaussian functions. The loss function incorporates both the residual of the GFPK equation and a normalization constraint. Optimization of the weighting coefficients is transformed into solving a system of algebraic equations, which significantly accelerates the training process. The resulting PDF solutions are used to reveal stochastic P-bifurcation phenomena in both Bouc–Wen and integrable Duhem hysteretic systems. Bifurcation shifts are observed as the random excitation transitions from Poisson to Gaussian noise. The proposed approach is validated by close agreement with Monte Carlo simulation (MCS) results, demonstrating its effectiveness for analyzing complex stochastic dynamics under combined harmonic and non-Gaussian excitations.