Ergodic Theory of Chaos
Ergodic Theory of Chaos
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混沌遍历理论
DOI：
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发表时间：
1985
期刊：
影响因子：
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通讯作者：
D. Ruelle
中科院分区：
文献类型：
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作者：
D. Ruelle
确定性混沌存在于物理学中的各种非线性动力系统中，尤其是在光学中。人们现在已经从分叉和奇怪吸引子的几何角度对混沌的开始有了合理的理解。这种几何方法不适用于二维或三维以上的吸引子。然而，对于这些，遍历理论提供了新的概念：特征指数、熵、信息维度，这些概念可以从物理实验中重复估计出来。特征指数衡量动力系统附近轨迹的发散率，熵衡量系统创造信息的速度，信息维度是特别感兴趣的分形维。存在与特征指数相关的熵和信息维度之间的不等式(甚至恒等式)。这些遍历量的实验测量提供了混沌系统不稳定性的数值估计，以及它们所具有的“自由度”的数目。
Determinsistic chaos arises in a variety of nonlinear dynamical systems in physics, and in particular in optics. One has now gained a reasonable understanding of the onset of chaos in terms of the geometry of bifurcations and strange attractors. This geometric approach does not work for attractors of more than two or three dimensions. For these, however, ergodic theory provides new concepts: characteristic exponents, entropy, information dimension, which are reproducibly estimated from physical experiments. The Characteristic exponents measure the rate of divergence of nearby trajectories of a dynamical system, the entropy measures the rate of information creation by the system, and the information dimension is a fractal dimension of particular interest. There are inequalities (or even identities) relating the entropy and information dimension to the characteristic exponents. The experimental measure of these ergodic quantities provides a numerical estimate of the instability of chaotic systems, and of the number of "degrees of freedom" which they possess.