李雅普诺夫指数
雅可比矩阵与行列式
数学
李雅普诺夫函数
嵌入
李雅普诺夫方程
数学分析
维数(图论)
应用数学
纯数学
混乱的
非线性系统
计算机科学
物理
量子力学
人工智能
作者
Ramazan Gençay,W. Davis Dechert
出处
期刊:Studies in Nonlinear Dynamics and Econometrics
[De Gruyter]
日期:1996-01-01
卷期号:1 (3)
被引量:18
标识
DOI:10.2202/1558-3708.1018
摘要
The method of reconstructing an n-dimensional system from observations is to form vectors of m consecutive observations, which for m 2n, is generically an embedding. This is Takens's result. The Jacobian methods for Lyapunov exponents utilize a function of m variables to model the data, and the Jacobian matrix is constructed at each point in the orbit of the data. When embedding occurs at dimension m = n, the Lyapunov exponents of the reconstructed dynamics are the Lyapunov exponents of the original dynamics. However, if embedding only occurs for an m > n, then the Jacobian method yields m Lyapunov exponents, only n of which are the Lyapunov exponents of the original system. The problem is that as currently used, the Jacobian method is applied to the full m-dimensional space of the reconstruction, and not just to the n-dimensional manifold that is the image of the embedding map. Our examples show that it is possible to obtain spurious Lyapunov exponents that are even larger than the largest Lyapunov exponent of the original system.
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