查看“︁李亚普诺夫指数”︁的源代码

在数学领域中，'''李亚普诺夫指数'''（{{lang|en|Lyapunov exponent}}）或'''李亚普诺夫特征指数'''（{{lang|en|Lyapunov characteristic exponent}}）用于量化[[动力系统]]中无限接近的[[轨迹]]之间的分离率。具体而言，[[相空间]]中初始间隔<math>\delta \mathbf{Z}_0</math>的两条轨迹的分离率为（假定分离可按线性近似来处理）
:<math> | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 | \, </math>
其中<math>\lambda</math>即为李亚普诺夫指数。

当初始分离向量的方向不同时，其分离率也不同。因而存在'''李亚普诺夫指数谱'''（{{lang|en|spectrum of Lyapunov exponents}}），其数量与相空间的维度相同。通常将其中最大的称为'''最大李亚普诺夫指数'''（{{lang|en|Maximal Lyapunov exponent}}，简称{{lang|en|MLE}}），因为它决定了动力系统的可预测性。正的{{lang|en|MLE}}通常表明系统是[[混沌]]的（假定其他条件满足，如相空间的紧致性）。需要注意的是，任意初始分离向量一般包括了{{lang|en|MLE}}所在方向的部分分量，由于其随指数增长的特征，其他分量的效果随着时间最终会被掩盖。

李亚普诺夫指数是以俄罗斯数学家[[亚历山大·李亚普诺夫]]的名字命名的。

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== 最大李亚普诺夫指数 ==
'''最大李亚普诺夫指数'''定義為

: <math> \lambda = \lim_{t \to \infty} \lim_{\delta \mathbf{Z}_0 \to 0}
\frac{1}{t} \ln\frac{| \delta\mathbf{Z}(t)|}{|\delta \mathbf{Z}_0|}.</math>

極限<math>\delta \mathbf{Z}_0 \to 0</math>確保任何時間線性近似的可行性<ref name="cencini">{{Cite book|first=M.|last=Cencini|title=Chaos From Simple models to complex systems|editor=World Scientific|year=2010|isbn=978-981-4277-65-5|id=|display-authors=etal}}</ref>。

對離散時間系統（映射或迭代）<math> x_{n+1} = f(x_n) </math>和以<math>x_0</math>為起始的軌跡，上式可以轉換成

: <math>
\lambda (x_0) = \lim_{n \to \infty}  \frac{1}{n} \sum_{i=0}^{n-1}  \ln | f'(x_i)|. 
</math>

== 参考文献 ==

* Cvitanović P., Artuso  R., Mainieri R., Tanner G.  and  Vattay G.[https://web.archive.org/web/20110725154552/http://www.chaosbook.org/ Chaos: Classical and Quantum] Niels Bohr Institute, Copenhagen 2005 – ''textbook about chaos available under [[Free Documentation License]]''

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[[Category:动力系统]]