On the Omega Limit Sets for Analytic Flows

Abstract. In this paper, we describe the characterizations of omega limit sets (= ! -limitset) on R 2 in detail. For a local real analytic °ow ' by z 0 = f ( z ) on R 2 , we prove the ! -limit set from the basin of a given attractor is in the boundary of the attractor. Usingthe result of Jim¶enez-L¶opez and Llibre [9], we can completely understand how both theattractors and the ! -limit sets from the basin. 1. IntroductionAttractors and ! -limit sets, arising from their ubiquitous applications in Dy-namical Systems, have played an important role in the fleld with the useful proper-ties. Especially, these are used to describe the time behavior for dynamical systems,and to provide the dynamicists with certain notions for localizing the complexity.For a manifold M with a vector fleld on it, and a point q 2 M , we denote the inte-gral curve (with the initial point q ) by Z q ( t ). Let an open interval ( a q ;b q ), possiblywith a q = i1 or b q = 1 , be the maximal domain on which Z q (