Theory and Computation of Covariant Lyapunov Vectors
Theory and Computation of Covariant Lyapunov Vectors
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DOI：
10.1007/s00332-012-9126-5
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发表时间：
2011-05
影响因子：
3
通讯作者：
P. V. Kuptsov;U. Parlitz
中科院分区：
文献类型：
--
作者：
P. V. Kuptsov;U. Parlitz
李雅普诺夫指数是描述在不同状态空间方向上施加到动力系统的轨迹的扰动的增长率的众所周知的特征数。协变（或特征）李雅普诺夫向量指示这些方向。虽然这些向量的概念已经知道了很长一段时间，但由于Ginelli等人[Phys.Rev.Lett. 99，2007，130601]和Wolfe和Samelson [Tellus 59A，2007，355]。鉴于协变李雅普诺夫向量及其广泛的潜在应用的极大兴趣，在这篇文章中，我们总结了可用的信息相关的李雅普诺夫向量，并提供了详细的解释的理论基础和数值算法。我们引入伴随协变李雅普诺夫向量的概念。这些向量与原始协变向量之间的角度是范数无关的，并且可以被认为是特征数。此外，我们提出并详细研究了一种改进的方法计算协变李雅普诺夫向量。此外，我们描述了如何可以测试混沌动力学的双曲性，而无需显式计算协变向量。
Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.