STATIONARY PROBABILITY DENSITY OF A HIGH-ORDER RAYLEIGH OSCILLATOR UNDER WEAK-TO-STRONG NOISE: ANALYSIS VIA STOCHASTIC AVERAGING AND MIXTURE DENSITY NETWORK METHOD

For wind-induced vibration of slender flexible structures such as antenna masts in turbulent wind fields, the dynamic behavior can be described by a Rayleigh-type nonlinear oscillator with higher-order velocity terms. The stochastic averaging method is first employed to analyze the stationary probability density and the bifurcation characteristics induced by higher-order damping coefficients under weak noise conditions. To address the significant prediction deviation and the difficulty in balancing accuracy and efficiency of the stochastic averaging method under strong stochastic excitation, a parameterized model based on Gaussian mixture representation and mixture density networks is constructed. A nonlinear mapping between system parameters and probability density distribution parameters is established, enabling direct reconstruction of the stationary probability density within the parameter space. Structural output constraints are imposed to guarantee non-negativity and normalization, forming a unified representation framework for both one-dimensional amplitude distributions and two-dimensional joint distributions. Taking a tenth-power Rayleigh oscillator subjected to combined white noise and correlated colored noise as a numerical example, comparative results demonstrate that the proposed approach accurately captures peak structures, tail behavior, and joint correlation characteristics in the strong-noise regime where the stochastic averaging method becomes ineffective. Quantitative evaluations based on the Kullback-Leibler divergence, Kolmogorov-Smirnov distance, and mean error further verify that the proposed model provides an efficient approach for analyzing stationary statistical characteristics of nonlinear stochastic vibration systems under strong stochastic excitation.