Information Dimension and the Probabilistic Structure of Chaos
Information Dimension and the Probabilistic Structure of Chaos
复制标题
信息维度与混沌的概率结构
DOI：
10.1515/zna-1982-1117
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发表时间：
1982
期刊：
影响因子：
--
通讯作者：
J. Farmer
中科院分区：
文献类型：
--
作者：
J. Farmer
从物理的角度回顾了熵和维的概念在动力系统中的应用。信息维度是衡量概率密度中包含的信息随分辨率缩放的速率，它填补了吸引子在度量熵、分形维数和拓扑熵方面分类的逻辑空白。给出了几个在概率分布上具有自相似几何尺度结构的混沌吸引子的例子；这些吸引子的信息维数和分形维数不同。正如度量（Kolmogorov-Sinai）熵对在一系列测量中获得的信息设置了上界，信息维可以用来估计在孤立测量中获得的信息。度量熵可以用由一系列测量结果构成的概率分布的信息维来表示。提出了一种实验确定信息维数和度量熵的算法。
The concepts of entropy and dimension as applied to dynamical systems are reviewed from a physical point of view. The information dimension, which measures the rate at which the information contained in a probability density scales with resolution, fills a logical gap in the classification of attractors in terms of metric entropy, fractal dimension, and topological entropy. Several examples are presented of chaotic attractors that have a self similar, geometrically scaling structure in their probability distribution; for these attractors the information dimension and fractal dimension are different. Just as the metric (Kolmogorov-Sinai) entropy places an upper bound on the information gained in a sequence of measurements, the information dimension can be used to estimate the information obtained in an isolated measurement. The metric entropy can be expressed in terms of the information dimension of a probability distribution constructed from a sequence of measurements. An algorithm is presented that allows the experimental determination of the information dimension and metric entropy.