本文選題：混合集權(quán)細胞自動機　切入點：混合細胞自動機　出處：《杭州電子科技大學(xué)》2017年碩士論文

【摘要】：動力系統(tǒng)是一門研究自然現(xiàn)象隨時間演變的極限行為的學(xué)科。經(jīng)過Poincar′e、Birkhoff、Lyapunov等人的研究,動力系統(tǒng)已成為現(xiàn)代數(shù)學(xué)的重要組成部分。細胞自動機是由John von Neumann于1951年正式提出的一種時間、空間和狀態(tài)都是離散的數(shù)學(xué)模型。在型態(tài)表現(xiàn)上,每個細胞自動機都是一個離散的動力系統(tǒng)。通過設(shè)計一些特定的局部規(guī)則,細胞自動機可以展示出豐富、復(fù)雜的動力學(xué)行為。因此,各種類型的細胞自動機如混合細胞自動機、集權(quán)細胞自動機等吸引了許多學(xué)者的關(guān)注,其數(shù)學(xué)理論研究得到了迅速發(fā)展,成果豐富。本文從符號動力學(xué)的觀點討論混合集權(quán)細胞自動機規(guī)則2和39(HTCA(2,39)),以及混合細胞自動機規(guī)則37和156(HCA(37,156))的動力學(xué)性質(zhì),著眼于刻畫系統(tǒng)的符號動力性狀,如拓撲熵、拓撲傳遞性與拓撲混合性等,由此豐富與發(fā)展了符號動力理論與離散動力系統(tǒng)理論。本文主要內(nèi)容如下:第一章介紹了動力系統(tǒng)、混沌、細胞自動機的發(fā)展及其基本概念。第二章簡單羅列了本文用到的符號動力學(xué)中的定義、定理等。第三章以符號動力系統(tǒng)為工具,在混合機制和集權(quán)機制下,討論了混合集權(quán)細胞自動機規(guī)則2和39所具有的三種滑翔機,并嚴(yán)格刻畫了其子系統(tǒng)的符號動力性狀,如拓撲傳遞性、拓撲混合性、有正拓撲熵等。由此,本章證明了它們在Li-Yorke和Devaney意義下都是混沌的。第四章通過計算機模擬并借助特征函數(shù)、功率譜、time-τ映射等一系列工具對HCA(37,156)進行定性分析,借助符號動力學(xué)相關(guān)理論證明了HCA(37,156)在其特定子集上是在Li-Yorke和Devaney意義下混沌的。第五章總結(jié)全文,并對未來的研究做了展望。
[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.
【學(xué)位授予單位】：杭州電子科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O19

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