Research On System Analytical Problems Based On Symbolic Dynamical System Theory

The studying of dynamical system experienced a long process. Dynamical system is the mathematical subject that has something to do with system evolution laws. In recent decades, dynamical system research has achieved remarkable progress, with new research direction gradually generating. Nowadays, dynamical system includes roughly differential dynamical system, Hamilton dynamical system, topological dynamical system, complex dynamic system, ergodic theory, stochastic dynamical system, symbolic dynamical system and so on. Among these branches, symbolic dynamical system is considered as an important tool in mathematical analysis. It plays an important role in the analysis of all kinds of systems. However, it has essential difference from other systematic analysis methods. The main idea of this method is to divide the system state with symbolic rules, and to indirectly obtain some certain properties of the system itself according to the theory of symbolic dynamical systems.Chaos theory has received extensive concern since it was established, and the symbolic dynamical system theory has also gradually become a common tool when studying the theory of chaos. This paper is based on the former research, putting forward some methods that study chaotic systems with symbolic dynamical system theory, and making some improvement to the original control methods of chaotic systems. The basic content is as follows:The first chapter basically introduces the development history and research status of symbolic dynamical system theory and chaotic control theory, the application of chaotic control theory and the ongoing work of this paper. The second chapter basically introduces some basic concepts and prepared knowledge related to symbolic dynamical system and chaotic dynamical system so as to form a complete theoretical system. The third chapter basically makes some certain symbolic division for one-dimensional Logistic mapping making use of symbolic dynamical system theory, and makes some improvement to the original control methods. Then we study two-dimensional Logistic mapping with the same method. At last, the system simulation analysis for one-dimensional Logistic mapping control method is carried out. The fourth chapter mainly studies Rossler system using symbolic dynamical system theory. A kind of symbolic division is shown for Rossler system. Meanwhile, symbolic dynamical analysis of Rossler system is also given. The fifth chapter summarizes the main conclusions obtained in this paper, and prospects further research.