Some Results Of The Alternating System

Topological dynamical system is an important branch of nonlinear science, the non-autonomous system is an important part of topological dynamical system, and the alternating system is the basis of the non-autonomous system. In this thesis, the definitions of the wandering point set, the chain recurrent point set and the asymptotic stability under the alternating system are given, and we obtained some conclusions.In chapter 1, we introduce the origin, the development of the dynamical system and the background of the alternating system.In chapter 2, to prepare for the research behind, we introduce the related notations and concepts of the classical discrete dynamical system and the alternating system.In chapter 3, we research the ω-limit set, the wandering point set and the chain recurrent point set under the alternating system, we proved that:(1)ω([f,g])=ω(g(?)f))∩ω(f(?)g); (2)ω([f,g])(?)Ω([(f,g])∪Ω([g,f]); (3) g(CR(f(?)g))=CR(g(?)f). We also give some counterexamples to tell the differences between the autonomous system and the alternating system, for example:there exist continuous maps f,g such that ω([f,g])(?) Ω([f,g]) and ω([f,g])(?)CR([f,g]).In chapter 4, we research the topological transitivity under the alternating system, we obtain some necessary conditions and counterexamples. For example:there exist continuous maps f, g satisfy: f, g are not topological transitive, but{Fn}n∈N is topological transitive.In chapter 5, we research the asymptotic stability under the alternating system, we proved that:If A is asymptotically stable under the alternating system, there exists an open set W(?)A such that (1) (g(?)f9(W)(?)W;(2)(?)Fn(W)(?)A.