Dynamical systems in engineering and physics are often subject to irregular excitations that are best modeled as random. Monte Carlo simulations are routinely performed on such random models to obtain statistics on their long-term response. Such simulations, however, are prohibitively expensive and time consuming for high-dimensional nonlinear systems. Here we propose to decrease this numerical burden significantly by reducing the full system to very low-dimensional, attracting, random invariant manifolds in its phase space and performing the Monte Carlo simulations on that reduced dynamical system. The random spectral submanifolds (SSMs) we construct for this purpose generalize the concept of SSMs from deterministic systems under uniformly bounded random forcing. We illustrate the accuracy and speed of random SSM reduction by computing the SSM-reduced power spectral density of the randomly forced mechanical systems that range from simple oscillator chains to finite-element models of beams and plates.

翻译：工程与物理中的动力系统常受不规则激励作用，此类激励最适合建模为随机过程。对此类随机模型通常采用蒙特卡洛模拟来获取系统长期响应的统计特性。然而，对于高维非线性系统而言，此类模拟的计算成本极高且耗时巨大。本文提出通过将完整系统降阶至其相空间中极低维、具有吸引性的随机不变流形，并在降阶后的动力系统上进行蒙特卡洛模拟，从而显著降低计算负担。为此构建的随机谱子流形（SSM）将确定性系统中谱子流形的概念推广至一致有界随机激励作用下的系统。通过计算从简单振子链到梁与板有限元模型等一系列随机激励机械系统的SSM降阶功率谱密度，验证了随机SSM降阶方法的精度与计算效率。