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首页> 外文期刊>International journal of bifurcation and chaos in applied sciences and engineering >A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND OF SCIENCE. PART IX: QUASI-ERGODICITY
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A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND OF SCIENCE. PART IX: QUASI-ERGODICITY

机译：伍尔弗姆新型科学的非线性动力学视角。第九部分：准适度

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Our scientific odyssey through the theory of 1-D cellular automata is enriched by the definition of quasi-ergodicity, a new empirical property discovered by analyzing the time-1 return maps of local rules. Quasi-ergodicity plays a key role in the classification of rules into six groups: in fact, it is an exclusive characteristic of complex and hyper Bernoulli-shift rules. Besides introducing quasi-ergodicity, this paper answers several questions posed in the previous chapters of our quest. To start with, we offer a rigorous explanation of the fractal behavior of the time-1 characteristic functions, finding the equations that describe this phenomenon. Then, we propose a classification of rules according to the presence of Isles of Eden, and prove that only 28 local rules out of 256 do not have any of them; this result sheds light on the importance of Isles of Eden. A section of this paper is devoted to the characterization of Bernoulli basin-tree diagrams through modular arithmetic; the formulas obtained allow us to shorten drastically the number of cases to take into consideration during numerical simulations. Last but not least, we present some theorems about additive rules, including an analytical explanation of their scale-free property.
机译：通过一维元胞自动机的理论，我们的科学旅程更加丰富了准遍历性的定义，准遍历性是通过分析局部规则的时间-1回报图而发现的一种新的经验属性。拟遍历在将规则分为六类中起着关键作用：实际上，它是复杂和超伯努利移位规则的独有特征。除了介绍准遍历性之外，本文还回答了我们探索的前几章提出的几个问题。首先，我们对时间1特征函数的分形行为进行了严格的解释，找到了描述这种现象的方程式。然后，我们根据伊甸岛的存在对规则进行分类，并证明256条规则中只有28条没有规则。这一结果阐明了伊甸岛的重要性。本文的一部分专门通过模块化算法来描述伯努利盆地图。所获得的公式使我们能够大大缩短在数值模拟中要考虑的案例数。最后但并非最不重要的一点是，我们提出了一些关于加性规则的定理，包括对它们的无标度特性的分析性解释。

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