This paper presents a PDE approach as an alternative to Monte Carlo simulations for computing the invariant measure of a white-noise-driven bilinear oscillator with hysteresis. This model is widely used in engineering to represent highly nonlinear dynamics, such as the Bauschinger effect. The study extends the stochastic elasto-plastic framework of Bensoussan et al. [SIAM J. Numer. Anal. 47 (2009), pp. 3374--3396] from the two-dimensional elasto-perfectly-plastic oscillator to the three-dimensional bilinear elasto-plastic oscillator. By constructing an appropriate Lyapunov function, the existence of an invariant measure is established. This extension thus enables the modelling of richer hysteretic behavior and broadens the scope of PDE alternatives to Monte Carlo methods. Two applications demonstrate the method's efficiency: calculating the oscillator's threshold crossing frequency (providing an alternative to Rice's formula) and probability of serviceability.

翻译：本文提出了一种偏微分方程方法，作为蒙特卡洛模拟的替代方案，用于计算具有滞回特性的白噪声驱动双线性振荡器的不变测度。该模型在工程领域被广泛用于表征高度非线性动力学行为，例如包辛格效应。本研究将Bensoussan等人[SIAM J. Numer. Anal. 47 (2009), pp. 3374--3396]的二维理想弹塑性振荡器随机框架，扩展至三维双线性弹塑性振荡器。通过构造适当的Lyapunov函数，确立了不变测度的存在性。这一扩展使得模型能够表征更丰富的滞回行为，并拓宽了偏微分方程方法作为蒙特卡洛方法替代方案的应用范围。两个应用案例展示了该方法的有效性：计算振荡器的阈值穿越频率（为Rice公式提供了替代方案）以及正常使用概率。