I Tight Mixed In The Power System With N Initial Value Sensitive Dependence

The main issue of dynamical system(X,f) is to study the asymptotic prop-erty of the points under the iteration of f in the compact space X, while topologi-cal transitivity, topological mixing and sensitive dependence on initial conditions all reflect the complexity of the system from different aspects. However, the space of Rn which we are familiar with is not compact, thus it becomes really difficult for us to study its dynamical properties. Fortunately, Wang Yangeng and Wei Guo[1] have introduced the definitions of the co-compact mixing, co-compact weakly mixing and co-compact transitivity in the Hausdorff space. Based on the definitions, the co-compact mixing, co-compact weakly mixing and co-compact transitivity are further studied and some novel conclusions are obtained in this paper. In addition, the sensitive dependence on initial conditions of n is also explored in the compact dynamical systems. The result is that it is invariants for topological conjugacy and the sensitive dependence on initial conditions of n is extended from compact spaces to the Hausdorff locally compact second count-able spaces(HLCSC), which helps to extend the scope of the study of dynamical systems and lays a theoretical basis for the future applications. The specific arrangement of this paper is as follows:In chapter 1, we briefly introduce the developing process of the dynamical systems. Then we introduce several concepts systematically, which are used to describe the dynamical systems, and their mutual relations. Finally, this paper gives an account of research background and main contents.In chapter 2, on the basis of the definitions and properties of the co-compact mixing, co-compact weakly mixing and co-compact transitivity, we study their relations to mixing, weakly mixing and transitivity. Eventually, some theorems are proved to be true:1.If f is co-compact mixing, for any positive integer p, fp is also co-compact mixing.2.If f is co-compact weakly mixing, and for any co-compact subset U of X satisfying f(U) (?) U, then U is dense.3.If f is co-compact mixing, and g is absolutely expansive, then f o g is also co-compact mixing.4. If f is co-compact weakly mixing, and g is absolutely expansive, then f o g is also co-compact weakly mixing.5.If f is co-compact transitive, and g is absolutely expansive, then f o g is also co-compact transitive.In chapter 3, we firstly describe how to expand dynamical systems of Haus-dorff locally compact second countable spaces(HLCSC) to the compact dynami-cal systems, and then study the sensitive dependence on initial conditions of n Finally, we come to the conclusion that the sensitive dependence on initial condi-tions of n is invariants for topological conjugacy and it is extended from compact spaces to the Hausdorff locally compact second countable spaces(HLCSC).At last, we summarize our results and raise a question for further study.