Hyperspace Dynamical Systems And Dynamical Systems Of Varying Parameters Asymptotically Periodic Points And Topological Mixing

The study of the map f demenstrates how do points in Topological space X move. Nevertheless,in the field of the Biological species,Demography and Numerical simulation, this is not enough.Sometimes one needs to know how the subsets of X move.Then it is necessary to consider that the set-valued map(?) is associated with f.Therefore,the thesis focuses on the relationship of the strong topological mixing of(?) and f^N,(?)and f^×N,f and (?)|_Ω,as well as the sensitive dependence on initial conditions.Accordingly,some meaningful results are obtained.In Chapter 1,the notions and lemmas of the dynamical system are introduced,the research background and the importance of their research of hyperspace dynamical system and variable-parametric dynamical system are illustrated,then the definition and relationship among Li-Yorke chaos,Devaney chaos,strong chaos,topological transitivity and mixing are introduced systematically.In Chapter 2,a systematic definition and some related character of set-valued discrete dynamical systems are given at the beginning.Then the focus goes to the relationship of topological transitivity,mixing and chaos between the topological dynamical system(X,f) and its induced set-valued discrete dynamical systems(K(X),(?)).Finally,some significant conclusions are obtained:1.If(?) is topological mixing,f^N is topological mixing for any integer N＞0.2.If f^N has sensitive dependence on initial conditions for any integer N＞0,(?) has sensitive dependence on initial conditions.3.If(?) is topological mixing and only if f^×N is topological mixing for any integer N＞0.4.Suppose that(X,f) were a compact topological dynamical systems,d is the metric on X,Ωis a subspace of K(X) and(?)(Ω)(?)K(X),then we have:ⅰ) ifΩ₁(X)(?)Ω,(?)|_Ωis topological mixing,that implies f is topological mixing.ⅱ) ifΩ=Ω₁(X),(?)|_Ωhas sensitive dependence on initial conditions,which implies that f has sensitive dependence on initial conditions.In Chapter 3,notions of asymptotic cycle,mixing and weak mixing of the variableparametric discrete dynamical system are introduced in detail.we focus on the basic dynamical properties among the composite of Variable-parametric discrete dynamical system (X,F·G),the ordinary Variable-parametric discrete dynamical system(X,F) and (X,G).We proved that if F is transitivity,mixing,and strong transitivity,and G satisfies certain condition,it is true with F·G.In addition,we investigated the condition of the existence of its asymptotic cycle.Ultimately,we proved that if F is chaotic in the sense of Xiong which implies that F is chaotic in the sense of Li-Yorke in variable-parametric discrete dynamical system and that F is chaotic in the sense of Li-Yorke and only if G is also chaotic in the sense of Li-Yorke when(X,F) conjugate with(Y,G) and both of them are mixing.