两个离散动力系统的动力学性质分析

发布时间：2018-03-30 01:27

本文选题：混合集权细胞自动机　切入点：混合细胞自动机　出处：《杭州电子科技大学》2017年硕士论文

【摘要】：动力系统是一门研究自然现象随时间演变的极限行为的学科。经过Poincar′e、Birkhoff、Lyapunov等人的研究,动力系统已成为现代数学的重要组成部分。细胞自动机是由John von Neumann于1951年正式提出的一种时间、空间和状态都是离散的数学模型。在型态表现上,每个细胞自动机都是一个离散的动力系统。通过设计一些特定的局部规则,细胞自动机可以展示出丰富、复杂的动力学行为。因此,各种类型的细胞自动机如混合细胞自动机、集权细胞自动机等吸引了许多学者的关注,其数学理论研究得到了迅速发展,成果丰富。本文从符号动力学的观点讨论混合集权细胞自动机规则2和39(HTCA(2,39)),以及混合细胞自动机规则37和156(HCA(37,156))的动力学性质,着眼于刻画系统的符号动力性状,如拓扑熵、拓扑传递性与拓扑混合性等,由此丰富与发展了符号动力理论与离散动力系统理论。本文主要内容如下:第一章介绍了动力系统、混沌、细胞自动机的发展及其基本概念。第二章简单罗列了本文用到的符号动力学中的定义、定理等。第三章以符号动力系统为工具,在混合机制和集权机制下,讨论了混合集权细胞自动机规则2和39所具有的三种滑翔机,并严格刻画了其子系统的符号动力性状,如拓扑传递性、拓扑混合性、有正拓扑熵等。由此,本章证明了它们在Li-Yorke和Devaney意义下都是混沌的。第四章通过计算机模拟并借助特征函数、功率谱、time-τ映射等一系列工具对HCA(37,156)进行定性分析,借助符号动力学相关理论证明了HCA(37,156)在其特定子集上是在Li-Yorke和Devaney意义下混沌的。第五章总结全文,并对未来的研究做了展望。
[Abstract]:The dynamical system is a subject that studies the ultimate behavior of natural phenomena over time, which has been studied by Poincare Birkhoffon Lyapunov et al. Dynamic system has become an important part of modern mathematics. Cellular automata is a discrete mathematical model proposed by John von Neumann in 1951. Each cellular automaton is a discrete dynamic system. By designing specific local rules, cellular automata can exhibit rich and complex dynamic behaviors. Therefore, various types of cellular automata such as hybrid cellular automata, The centralized cellular automata has attracted the attention of many scholars, and its mathematical theory has developed rapidly. From the point of view of symbolic dynamics, this paper discusses the dynamical properties of hybrid centralized cellular automata (rule 2 and 39) and hybrid cellular automata (rule 37 and 156), focusing on the characterization of symbolic dynamic properties, such as topological entropy, of the system. The symbolic dynamical theory and discrete dynamic system theory are enriched and developed. The main contents of this paper are as follows: in Chapter 1, the dynamical system, chaos, and chaos are introduced. The development of cellular automata and its basic concepts. Chapter two briefly lists the definitions and theorems of symbolic dynamics used in this paper. In this paper, three kinds of gliders with mixed centralization weight cellular automata rule 2 and 39 are discussed, and the symbolic dynamic properties of their subsystems, such as topological transitivity, topological mixing, positive topological entropy and so on, are strictly described. It is proved in this chapter that they are chaotic in the sense of Li-Yorke and Devaney. In Chapter 4, a series of tools, such as computer simulation, power spectrum time- 蟿 mapping and eigenfunction, are used to qualitatively analyze HCA3 7156). With the help of symbolic dynamics correlation theory, it is proved that HCAN 37156) is chaotic in the sense of Li-Yorke and Devaney on its particular subset. Chapter 5 summarizes the full text and makes a prospect for future research.
【学位授予单位】：杭州电子科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O19

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