We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable.

中文翻译：

---

通过微扰技术逼近具有弱非线性的随机微分方程的密度

我们将随机扰动方法与最大熵原理相结合，构造一类非线性振荡器稳态解的第一概率密度函数的近似值，该非线性解在非线性项下受到较小的扰动并由随机激励驱动。非线性取决于位置和速度，并且激发是由具有某些附加属性的平稳高斯随机过程给出的。此外，我们近似求解高阶矩，方差和解的相关函数。通过一些数值实验说明了理论发现，证实了我们的近似值是可靠的。