Identifying the exactly or approximately explicit expression of the stationary response probability density for general nonlinear stochastic dynamical systems is of great significance in the fields of stochastic dynamics and control. Almost all the existing methods are devoted to determine the exact or approximate solution for specific values of system and excitation parameters. Herein, aimed at stochastic systems with polynomial nonlinearity and excited by Gaussian white noises, a novel method is proposed to identify the stationary response probability density which explicitly includes system and excitation parameters. The stationary probability density is first written as an exponential function according to the maximum entropy principle, the power of the exponential function is then expressed as a linear combination of prescribed nondimensional parameter clusters constituted by system and excitation parameters, and state variables, with the coefficients to be determined. The undetermined coefficients are derived by minimizing the residual of the associated Fokker-Planck- Kolmogorov equation. The application and efficacy of the proposed method are illustrated by a typical numerical example.

确定一般非线性随机动力系统的平稳响应概率密度的精确或近似显式在随机动力学和控制领域中具有重要意义。几乎所有现有方法都致力于确定系统和励磁参数特定值的精确或近似解。在此，针对具有多项式非线性并且被高斯白噪声激励的随机系统，提出了一种新的方法来识别平稳响应概率密度，该方法明确包括系统参数和激励参数。首先根据最大熵原理将平稳概率密度写为指数函数，然后，将指数函数的幂表示为由系统参数和励磁参数以及状态变量构成的规定的无量纲参数簇与待确定系数的线性组合。通过最小化关联的Fokker-Planck-Kolmogorov方程的残差来得出不确定的系数。通过一个典型的数值例子说明了该方法的应用和有效性。