Research On Chaoticity And Shadowing Properties Of Several Dynamical Systems

Chaos is a universally dynamical behavior of the nonlinear dynamical system,which is one of the main content in topological dynamical systems.Over the last few years,chaos has an increasing influence on the development of topological dynamical systems.Also,shadowing property is a hot topic and plays an essential role in dynamical systems.At present,elegant fruitful results which explore the chaoticity and shadowing properties of dynamical systems are achieved.The aim of this thesis is to study the chaoticity of several dynamical systems:nonau-tonomous discrete systems,g-fuzzification systems and iterated function systems.More-over,the relation between shadowing properties and topological transitivity on nonuni-formly expanding mapping is investigated.The main contributions in this thesis are as follows:1.In a nonautonomous discrete system which converges uniformly,the rigorous definition of weak?F₁,F₂?-sensitivity is proposed,and for a positive integer k,the prop-erties ???and ???of Furstenberg families are introduced.Based on the two proper-ties,consider the following chaotic behaviors:?F₁,F₂?-sensitivity,weakly?F₁,F₂?-sensitivity,?F₁,F₂?-chaos and?????s?,????t??-chaos,and it is proved that the four prop-erties are preserved under iterations.Finally,two examples are presented to illustrate some applications of the results.2.Compared to the hyperspace systems,some lemmas of e?U?and??U?in the g-fuzzification systems are obtained and a few chaotic properties of g-fuzzification sys-tem are discussed.It is proved that its g-fuzzification system is turbulent?resp.,erratic?if the crisp system is turbulent?resp.,erratic?.Also,it is shown that the crisp system is?-expansive if and only if the induced g-fuzzification system is?-expansive.Besides,we study the turbulence,Er-chaos and?-expansiveness of nonautonomous discrete dy-namical systems and its Zadeh's extension and the corresponding results compared with the g-fuzzification system are proved.3.Considering several sensitivities and transitivity of iterated function systems,it is shown that the sensitive dependence on initial conditions,Li-York sensitivity,syndetic sensitivity,cofinite sensitivity,ergodic sensitivity,multi-sensitivity and mixing property are preserved in product systems,but the topological transitivity,syndetic transitivity and weakly mixing property are not inherited.4.Many research works have been devoted to the relationship between shadowing property and?chain?transitivity.In this thesis,we focus on the nonuniformly expand-ing mapping.It is shown that the nonuniformly expanding system with d-shadowing property is topologically transitive.Furthermore,by using the connections of the various shadowing properties,a series of corollaries are obtained.In particular,the nonuniformly expanding mapping with d-shadowing property is not only topologically transitive,but also weakly mixing.