Estimating Lyapunov Exponents from Time Series
Estimating Lyapunov Exponents from Time Series
复制标题
从时间序列估计李雅普诺夫指数
DOI：
10.1007/978-3-662-48410-4_1
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
U. Parlitz
中科院分区：
文献类型：
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作者：
U. Parlitz
李雅普诺夫指数是量化动力系统稳定性和确定性混沌的重要统计量。在这篇综述文章中，我们首先回顾使用模型方程计算李雅普诺夫谱。然后，利用状态空间重构（延迟坐标），提出了两种从时间序列估计李雅普诺夫指数的方法：基于重构流的雅可比矩阵近似的方法和评估邻近轨道距离演化的所谓直接方法。最直接的方法估计最大的李雅普诺夫指数，但作为一个优势，他们给用户的图形反馈，以确认指数发散。这个反馈提供了关于估计结果的有效性和准确性的有价值的信息。因此，我们重点研究了这种从时间序列中估计Lyapunov指数的算法，并通过（迭代）Henon映射、超混沌折叠毛巾映射、众所周知的混沌Lorenz-63系统和时间连续的6维Lorenz-96模型来说明其特征。这些例子表明，使用直接方法可以成功地估计低维混沌系统时间序列的最大Lyapunov指数。然而，随着吸引子维数的增加，需要更长的时间序列，并且只考虑延迟坐标空间中距离足够近的相邻轨迹段是至关重要的。
Lyapunov exponents are important statistics for quantifying stability and deterministic chaos in dynamical systems. In this review article, we first revisit the computation of the Lyapunov spectrum using model equations. Then, employing state space reconstruction (delay coordinates), two approaches for estimating Lyapunov exponents from time series are presented: methods based on approximations of Jacobian matrices of the reconstructed flow and so-called direct methods evaluating the evolution of the distances of neighbouring orbits. Most direct methods estimate the largest Lyapunov exponent, only, but as an advantage they give graphical feedback to the user to confirm exponential divergence. This feedback provides valuable information concerning the validity and accuracy of the estimation results. Therefore, we focus on this type of algorithms for estimating Lyapunov exponents from time series and illustrate its features by the (iterated) Henon map, the hyper chaotic folded-towel map, the well known chaotic Lorenz-63 system, and a time continuous 6-dimensional Lorenz-96 model. These examples show that the largest Lyapunov exponent from a time series of a low-dimensional chaotic system can be successfully estimated using direct methods. With increasing attractor dimension, however, much longer time series are required and it turns out to be crucial to take into account only those neighbouring trajectory segments in delay coordinates space which are located sufficiently close together.