Recurrent Motions In Dynamical Systems

Recurrence is one of the most important motions in dynamical systems. Many experts and scholars, such as V.V.Nemytskii, V.V.Stepanov, N.P.Bhatia, G.P.Szego, investigated such motions, and got many interesting and valuable results. At the present time, recurrence is still paid an extensively attention. On the base of their investigations, I also make further study of recurrence motions. The main contents of this paper consist of three parts, which is discussed in chapter Ⅲ,Ⅳ,Ⅴ,respectively. In chapter Ⅲ, the main purpose of this chapter is to discuss the relationship between Poisson stability and Lyapunov stability in dynamical systems, and also the relationship between Poisson stability and periodicity, and get some results of Poisson stability and periodicity. One of the main results is that let X be a locally compact Haudorff space, and let x∈X be positively Poisson stable but not periodic, then the set ω(x)-γ(x) is dense in ω(x), i.e. ω(x)-γ(x)=ω(x)=γ(x). The other is that let X be Lyapunov stable, and let every point in P X be positively Poisson stable or negatively Poisson stable, then every in P is Poisson stable. In chapter Ⅳ, We make further discussion about the relation between periodicity and near periodicity. And we give a necessary and sufficient condition that the periodicity is equivalent to near periodicity, which generalizes the results of [6]. In addition, we give a result of near periodicity. The main result is that let ( R n,π) be asymptotically Lyapunov stable in x ∈Rn (n >2), then x is periodic if and only if x is nearly periodic. The other main result is that let ( X ,π) be positively nearly periodic in γ+( x), then every point in γ+( x) is recurrent. In chapter V, The main purpose of this chapter is to discuss the relationship between attraction and Lyapunov stability in dynamical systems. Some conditions are given for the pairwise equivalences among the three notions of weak attraction, attraction and uniform attraction. Under certain conditions, it is shown that the notions of (weak, uniform) attractor and global (weak, uniform) attractor coincide, which generalizes the results of [19]. The main result is that let X be a locally compact and connected metric space, and let ( X ,π) be positively Lyapunov stable, if M ? X is a positively invariant closed set, then M is a weak attractor ? attractor ? uniform attractor ? globally weak attractor ? globally attractor ? globally uniform attractor.