Pseudo-orbit Tracing Study In The Power System

If a pseudo orbit under a self-mapping of a compact metric space can be traced by a true orbit, then we say this map has the pseudo-orbit tracing property (POTP). POTP is closely related with the stability and plays an important role in the general qualitative theory of dynamical systems. Theory of shadowing was developed intensively in recent years and has a lot of deep results.This thesis consists of three chapters.In chapter â… , we introduce the background and some known results about the POTP.In chapter â…¡, we give some properties of the POTP and discuss relations between the POTP and other notions. In section 2.2, it is proved that if an expansive homeomorphism of a compact metric space have the POTP, then it has the POTP in its basic sets. We show that if f is positively expansive and has the POTP, then there is no proper nonempty invariant subset of basic sets in which f has the POTP. Moreover, if f has only one basic set, then this basic set is the whole space. In the end, we give a sufficient condition for f having the POTP in the closure of periodic points set. In section 2.3, it is proved that a equicontinuous self-mapping of a compact and connected metric space can not have the POTP. In section 2.4, a sufficient condition is given for a distal homeomorphism not having the POTP. In the end of this section, we describe the distal homeomorphisms of intervals and circles.In chapter â…¢, we study some other shadowing properties. In section 3.2, it is provedthat if a homeomorphism f on a compact metric space with the POTP is distal, thenf does not have the average shadowing property. A sufficient condition is given for a homeomorphism with the the average shadowing property to be minimal. In section 3.3, we prove that if/has the POTP and is equicontinuous, then/has the limit shadowingproperty. We also give some propertys of the limit shadowing property. In section 3.4, a necessary and sufficient condition is given for continuous self-mappings of compact metric spaces having the POTP. In the end, we give some properties of the strong shadowing.