New Properties Of The Omega Limit Of A Planar Dynamical System

In this paper, we consider the properties of the omega limit set of a planar system, using the stability of an equilibrium and the general nature of the plane dynamic system, the introduction of a Dulac function and positive(negative) invariant set constructed,given the state of theωlimit set and a limit set. we study the Poincare-Bendixson theorem and obtain the following some results,these results are news.Theorem 2.3.1 If the equilibrium point of the system (1.1.1) keeps (?)P/(?)x+(?)Q/(?)y having the same sign,then the limit set of the semiorbit within the bound region could only be one of the following two types (1) a single equilibrium set (2) a closed orbit.Theorem3.2.1 For the plane system(1.1.1), Let simple connected region V+∈Ω+-(V-∈Ω-+) in V+(V-),,div[V-1P,V-1Q) V+(V-) equals to zero in any sub-domain, and the equilibrium of (1.1.1) is not equal to zero, then for any non-equilibrium point the limit set of P, there(a) Ifω(p)(?)V+(V-), thenω(p) is a single-point set or empty set;(b) ifω(p)∩(?)V+≠φ(ω)(p)∩(?)V-≠φ, thenφ(p)∩(?)V+ consist of singular point.Theorem3.2.2 For the plane system(1.1.1), Let simple connected region V+∈Ω++(V-∈Ω--) in V+(V-),, div(V-lP,V-1Q)V+(V-) equals to zero in any sub-domain, and the equilibrium of (1.1.1) is not equal to zero, then for any non-equilibrium point the limit set of P, there(a) Ifω(p)(?)V+(V-), thenω(p) is a single-point set or empty set;(b) ifω(p)∩(?)V+≠φ(ω)(p)∩(?)V-≠φ), thenω(p)∩(?)V+ consist of singular point.The paper is divided into three chapters. The first chapter,we introduce the background and the value of studying this paper and give the basic definitions and lemma. The second chapter is the part of the major content,we introduce on a note of the Poincar6-Bendixson theorem in the plane,discussing and proving theorem 2.3.1 and theorem 2.3.2, giving the type of the omega limit set.The third chapter introduces a new property of the omega limit set in the plane.