Prolongations And Asymptotic Behaviour In Dynamical Systems

The content of this paper consists of two parts,one discusses prolongations in dynamical systems,the other discusses asymptotic behaviour in dynamical systems.Prolongation is one of the most important notions in dynamical systems, It is closely linked with stability of dynamical systems. Many experts and scholars (such as T.Ura, J.Auslander, N.P.Bhatia, K.S.Sibirsky, etc.) investigated it,and got many famous and valuable results.At the present time, prolongation is still paid an extensively attention. The first part of this paper focuses on the discussions about prolongations in normal topology space.In the articles of J.Auslander, N.P.Bhatia, K.S.Sibirsky (such as [1], [3], [4], [6], [9] etc.), dynamical systems was limited in compact metric spaces or locally compact metric spaces. Our phase spaces are relatively compact sets of a normal topology space X , thus the results are more general. The main contents of this part are related to basic facts of dynamical systems.In the course of the discussion,a key role is the concept of filter.In Section 1 of the first part, We mainly introduce some basic concepts of dynamical systems, discuss the properties of limit sets of a point or a set, get the relations of thet limit sets, trajectories of points and trajectory clousures of points. We also discusse the compactness and connectedness of limit sets. At the same time properties of invariant sets and clousure invariant sets are also investigated,and we get necessary and sufficient conditions for their equivalence. In section 2,we discuss the first positive prolongation of a set M in a relatively compact set A of a normal topology space X . It is one of the most important contents.We make discussion about the relations of M , the first positive prolongation of M , trajectories of M and trajectory clousures of M and the relations of the first positive prolongation of a set M and the first positive prolongation of a P in M .The connectedness of the...