Dynamics Analysis Of Two Discrete Dynamical Systems

Dynamical system is a discipline on studying the limit behaviors of the nature phenomena with evolution over time.Due to the studying of Poincar?e,Birkhoff and Lyapunov,dynamical system has become important part of modern mathematics.Cellular automata are a class of spatially,temporally and state discrete mathematical model,which were introduced by John von Neumann in 1951.Each cellular automaton is a discrete dynamical system in the form of performance.Through designing different local rules,cellular automata exhibit numerous complex dynamical behaviors.Furthermore,various types of cellular automata,such as hybrid cellular automata and totalistic cellular automata,have attracted the attention of many scholars.The mathematical theory has been developed rapidly,and the achievements are abundant.In this paper,dynamical properties of hybrid totalistic cellular automata rule 2 and39(HTCA(2,39))and hybrid cellular automata rule 37 and 156(HCA(37,156))are discussed under the perspective of symbolic dynamics,we focus on the symbolic dynamical traits of the system,such as topological entropy,topological transitivity and topological mixing,which enrich and develop the theory of symbolic dynamics and discrete dynamical system theory.The main contents of this paper are as follows: In chapter 1,the development of dynamical system,chaos,cellular automata,and some fundamental concepts of cellular automata are introduced.In chapter 2,the definitions and theorems of symbolic dynamics used in this paper are listed.In chapter 3,under the hybrid mechanism and totalistic mechanism,the dynamical properties of HTCA(2,39)with three gliders in its three subsystems are analyzed by means of symbolic dynamical system,and the symbolic dynamical properties of the subsystems are strictly described,for example,topologically transitive,topologically mixing,positive topological entropy.Thus,we demonstrate that the subsystems of HTCA(2,39)are chaotic in the sense of both Li-Yorke and Devaney.In chapter 4,HCA(37,156)is qualitatively analyzed by means of computer simulation and a series of tools,such as characteristic function,power spectrum and time-tau mapping,we prove that HCA(37,156)is chaotic in the sense of both Li-Yorke and Devaney on its specific subset by means of symbolic dynamics.Finally,chapter 5 makes a brief summary on this paper,and points out some prospects for future studies.