计算协变向量、李雅普诺夫向量、Oseledets向量和二分法投影：比较数值研究
Computing covariant vectors, Lyapunov vectors, Oseledets vectors, and dichotomy projectors: a comparative numerical study
摘要：
协变向量，李雅普诺夫向量，或Oseledets向量越来越多地被用于各种模型分析领域，如偏微分方程，非自治可微动力系统，和随机动力系统。这些向量确定特定的渐近增长率的空间变化方向，并遵守等方差原理。近年来，由于模型复杂性的增加和对精度的更高要求，已经开发了用于近似Oseledets向量的新计算方法。本文介绍了基于奇异值分解和指数二分的两种新方法，并对Ginelli等人（2007）和Wolfe and Samelson（2007）的两种新方法进行了比较和改进。我们通过三个案例研究非常不同的动力学对称性，光谱分离，和尺寸的四种方法的性能进行比较。我们还研究了哪些方法在有限的数据下表现良好。
英文摘要：
Covariant vectors, Lyapunov vectors, or Oseledets vectors are increasingly being used for a variety of model analyses in areas such as partial differential equations, nonautonomous differentiable dynamical systems, and random dynamical systems. These vectors identify spatially varying directions of specific asymptotic growth rates and obey equivariance principles. In recent years new computational methods for approximating Oseledets vectors have been developed, motivated by increasing model complexity and greater demands for accuracy. In this numerical study we introduce two new approaches based on singular value decomposition and exponential dichotomies and comparatively review and improve two recent popular approaches of Ginelli et al. (2007) and Wolfe and Samelson (2007). We compare the performance of the four approaches via three case studies with very different dynamics in terms of symmetry, spectral separation, and dimension. We also investigate which methods perform well with limited data.
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