Spatial and Temporal Scaling Properties of Strange Attractors and Their Representations by Unstable Periodic Orbits
Spatial and Temporal Scaling Properties of Strange Attractors and Their Representations by Unstable Periodic Orbits
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奇异吸引子的时空尺度特性及其不稳定周期轨道的表示
DOI：
10.1143/ptp.79.296
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发表时间：
1988
影响因子：
--
通讯作者：
K. Tomita
中科院分区：
文献类型：
--
作者：
Terumitsu Morita;H. Hata;H. Mori;T. Horita;K. Tomita
从动力学的观点出发，以可逆二维映射为例，研究了混沌的标度性质。一个基本的作用是由一个小盒子沿着混沌轨道的概率的演化方程。利用附近轨道的局域膨胀率导出了奇异吸引子在膨胀和收缩方向上的偏维数之间的一个新关系。当映射的拉科比数为常数时，该关系可以用局部膨胀率在膨胀方向上的波动的势rJJ（q）来表示。对于双曲型吸引子，该势rJJ（q）与广义熵K（q）的关系为rJJ（q）=（q-l）K（q），并且上述关系简化为广义维数与广义熵之间的简单关系.此外，广义维数、势rJJ（q）和概率密度都用吸引子内不稳定周期轨道的局部膨胀率表示，从而为研究混沌提供了一种新的方法.
Scaling properties of chaos are studied from a dynamic viewpoint by taking invertible two· dimensional maps. A fundamental role is played by an evolution equation for the probability of a small box along a chaotic orbit. A new relation between the partial dimen~ions of strange attractors in the expanding and contracting direction is derived in terms of the local expansion rates of nearby orbits. When the lacobians of the maps are constant, this relation can be written in terms of a potential rJJ(q) for the fluctuations of the local expansion rate in the expanding direction. For hyperbolic attractors, this potential rJJ(q) is related to the generalized entropies K(q) by rJJ(q)=(q -l)K(q), and the above relation reduces to a simple relation between the generalized dimensions and the generalized entropies. Furthermore, the generalized dimensions, the potential rJJ(q) and the probability densities are expressed in terms of the local expansion rates of unstable periodic orbits within the attractors, leading to a new method of studying chaos.