SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 577-618, January 2020.
Differential equations with random parameters have gained significant prominence in recent years due to their importance in mathematical modeling and data assimilation. In many cases, random ordinary differential equations (RODEs) are studied by using Monte Carlo methods or by direct numerical simulation techniques using polynomial chaos (PC), i.e., by a series expansion of the random parameters in combination with forward integration. Here we take a dynamical systems viewpoint and focus on the invariant sets of differential equations such as steady states, stable/unstable manifolds, periodic orbits, and heteroclinic orbits. We employ PC to compute representations of all these different types of invariant sets for RODEs. This allows us to obtain fast sampling, geometric visualization of distributional properties of invariants sets, and uncertainty quantification of dynamical output such as periods or locations of orbits. We apply our techniques to a predator-prey model, where we compute steady states and stable/unstable manifolds. We also include several benchmarks to illustrate the numerical efficiency of adaptively chosen PC depending upon the random input. Then we employ the methods for the Lorenz system, obtaining computational PC representations of periodic orbits, stable/unstable manifolds, and heteroclinic orbits.

中文翻译：

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使用多项式混沌计算随机微分方程的不变集

SIAM应用动力系统杂志，第19卷，第1期，第577-618页，2020年1月。
近年来，具有随机参数的微分方程由于在数学建模和数据同化中的重要性而备受关注。在许多情况下，通过使用蒙特卡洛方法或通过使用多项式混沌（PC）的直接数值模拟技术，即通过随机参数的级数展开并结合前向积分，研究随机常微分方程（RODE）。在这里，我们从动力学系统的角度出发，关注微分方程的不变集，例如稳态，稳定/不稳定流形，周期轨道和异斜轨道。我们使用PC来计算RODE的所有这些不同类型的不变集的表示形式。这样一来，我们就可以快速获取样本，对不变量集的分布特性进行几何可视化，动态输出的不确定性量化，例如轨道的周期或位置。我们将技术应用于捕食者－被捕食者模型，在其中我们计算稳态和稳定/不稳定流形。我们还包括几个基准，以说明根据随机输入自适应选择的PC的数值效率。然后，我们将这些方法用于Lorenz系统，获得周期轨道，稳定/不稳定流形和异斜轨道的计算PC表示。