Chaos Theory And Its Application In Economic Systems

Chaos theory is not only one of the most active branches for dynamical system, but also one of important topics of nonlinear scientific field. It is applied widely in physics, biology and economics etc, and has become a hot-spot theory in many fields.In studying a dynamical system, we know that there frequently exist some kinds of disturbance or false phenomenon, namely that some chaotic sets are included in the sets with absolute measure zero. From the viewpoint of ergodic theory, a set with absolute measure zero is negligible. In order to remove these disturbance and false phenomenon, Z. L. Zhou introduced the concept of measure center, and pointed out that the measure center of a minimal system is itself. The structure of the measure center is totally determined by the set of weakly almost periodic points. Consequently, it is meaningful to discuss problems on measure center and the set of weakly almost periodic points.With the development of chaos theory, it is applied widely in the field of economics. Ever since Day introduced nonlinear dynamics into economics, tremendous efforts have been devoted in the economic models. The theory of chaos in economics has brought valuable perspective into the study of economic systems and helped people to deeply un-derstand some economic phenomena. As we all know, researchers always analyze chaotic behavior by numerical simulations and few has given the theoretical proof for the existence of chaos in economic models.The mains purpose of this thesis is to analyze the chaoticity, ergodicity and other important properties of dynamical systems on their measure centers. At the same time, in order to have a better analysis about the meaning of economic models, we make the theoretical analysis about the existence of chaos in the models. The research of this thesis mainly is divided into two parts. First, we study the chaoticity, ergodicity and other properties(for example, mixing, uniform rigidity, etc.) of different dynamical systems on their measure centers. Second, we establish two new duopoly systems, and analyze their dynamical properties, including stability and chaoticity. The results obtained in this thesis are significant theoretically to reveal the essence of chaos and extend its application in economics.A detailed introduction is as follow:In chapter 1, we mainly introduce the background and development status of the research at home and abroad. And the main work completed in this thesis is shown.In chapter 2, we introduce some preliminary knowledge in topologically dynamical system which will be used in this paper. The different definitions of chaos are introduced, and the relationships among the mentioned chaos are summarized in this section.In chapter 3, we mainly research the dynamical properties of different systems on their measure centers and complete the following three tasks:1. We draw the conclusions that the one-sided shift a has an uncountable distribu-tionally scrambled set S(?)W(σ)-A(σ). Then we give a sufficient condition for the compact dynamic system (X,f) to exhibit distributional chaos on measure center, and a sufficient condition is given such that the map f:X �'X in Banach space (X,||·||) has an uncountable distributionally scrambled set W(?)W(f)-A(f).2. In this part, we study the chaoticity and ergodicity of the minimal subshifts and the set-valued shifts induced by them on symbolic space.Firstly, we prove that there exists a minimal subset Y(?)Σ2 such that σ|Y is M-system, uniformly rigid, topologically mixing, topologically ergodic, topologically double ergodic, chaotic in the sense of Wiggins and chaotic in the sense of Martelli in the one-sided symbolic system (Σ2,σ). We also find that there are two kinds of subshift:one is distributionally chaotic, the other is strong distributionally chaotic in a sequence but not distributionally chaotic. But they are both uniformly rigid, topologically mixing, topologically double ergodic and not almost equicontinuous.Secondly, we research a special shift in symbolic space——the delayed shift, and draw the following conclusions. Let(Σ2,ρ) be an one-sided symbolic space, and T Σ2�'Σ2 is the delayed shift. T is topologically double ergodic and topologically ergodic; and we prove that there exists a minimal set M such that T|M is topologically mixing, chaotic in the strong sense of Kato, chaotic in the strong sense of Li-Yorke, chaotic in the sense of Ruelle-Takens, chaotic in the sense of Martelli, and stictly ergodic. Then we get that the set-valued map induced by T is topologically mixing, topologically transitive, topologically double ergodic and topologically ergodic. At last, we find that the set-valued system (κ(M),T) induced by T|M is topologically mixing, distributionally chaotic, totally maximum sensitive, and Li-Yorke sensitive.3. Let (Σ2×S1, d) be a metric space and f:Σ2×S1�'Σ2×S1, we prove that there exists a minimum set M such that f|M is chaotic in the sense of Wiggins and chaotic in the sense of Martelli.In chapter 4, we construct two new duopoly models, and analyze the dynamic prop-erties of the systems.1. We construct a duopoly model of technological innovation based on constant conjectural variation under the assumption of bounded rationality. The stability of the system is investigated. We discuss the optimal technical contents in the models under the two different situations, non-cooperative technological innovation and cooperative technological innovation. Then we prove the system displays distributional chaos and Li-Yorke chaos by employing the snap-back repeller theory. At last, numerical simulations are used to illustrate the dynamical properties of the model. Numerical simulations with respect to the output adjusted coefficient and technical content are done in four different economical situations, namely four different conjectural variations.2. We construct a duopoly model of technological innovation under the assumption of delayed bounded rationality. We investigate the stability of the system and prove that the system displays distributional chaos and Li-Yorke chaos. At last, in the part of numerical simulation, we numerically simulate with respect to the output adjusted coefficient and technical content respectively; at the same time, the delayed bounded rationality model and the non-delayed bounded rationality model are compared in the numerical simulations.