高维动力学中的禁序模式
Forbidden ordinal patterns in higher dimensional dynamics
摘要：
禁止有序模式是指不能出现在线性有序空间上取值的映射所生成的轨道中的有序模式（或“秩块”），在这种情况下，我们说该映射具有禁止模式。一旦映射具有给定长度$L_{0}$的禁止模式，它就具有任何长度$L\ge L_{0}$的禁止模式，并且它们的数量随着$L$超指数地增长。本文利用拓扑置换熵的最新结果，研究了n维区间上自映射禁序模式的存在性及其基本性质。我们最适用的结论是，具有有限拓扑熵的可扩张区间映射必然具有禁止模式，尽管我们猜想在更一般的条件下也是如此。理论结果很好地说明了为n=2使用天真计数估计的禁止模式和超的估计的人口中的类的数量。禁止有序模式对观测白噪声的鲁棒性也说明了。
英文摘要：
Forbidden ordinal patterns are ordinal patterns (or `rank blocks') that cannot appear in the orbits generated by a map taking values on a linearly ordered space, in which case we say that the map has forbidden patterns. Once a map has a forbidden pattern of a given length $L_{0}$, it has forbidden patterns of any length $L\ge L_{0}$ and their number grows superexponentially with $L$. Using recent results on topological permutation entropy, we study in this paper the existence and some basic properties of forbidden ordinal patterns for self maps on n-dimensional intervals. Our most applicable conclusion is that expansive interval maps with finite topological entropy have necessarily forbidden patterns, although we conjecture that this is also the case under more general conditions. The theoretical results are nicely illustrated for n=2 both using the naive counting estimator for forbidden patterns and Chao's estimator for the number of classes in a population. The robustness of forbidden ordinal patterns against observational white noise is also illustrated.
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