Research On Stability Of A Class Of Planar Differential Systems

The study of stability problem is one of the important research directions for the qualit.ative theory of differential systems.There are many methods for the study of stability problems,and many basic conclusions are obtained.This paper is a basic study on the stability of a class of planar differential systems.The work of this mainly includes two aspects.For the first aspect,The prob-lem of singular point index calculation for planar vector field on Cl is studied,and A special homogeneous polynomials is constructed by using polynomial prime theorem.A new and convenient method of calculated index is obtained applying the geometric significance for index,which is different from algebraic of Cauchy-index.This method is more concise and effective in calculation.For the second aspect,we introduce the Dulac function,construct a positive in-variant set,a negative invariant set,and study the stability problem of a class of planar different,ial systems by using the properties of the Dulac function and the positive invariant set and t.he negative invariant set.And effectively extend the study area from the entire plane to the bounded or unbounded connected area,improved Bendixson discriminant method for discriminating stability given by C.C.Mccluske and J.S.Muldowney,If Bendixson's criterion is satisfied on R2,then the non-constant periodic solution of the equation is nonexist.Thus,every positive semi-orbit of a equation satisfying Bendixson's criterion is either unbounded or its ? limit set contains an equilibrium.The full text is divided into five chapt,ers.The first chapt,er gives the re-search background and research work plan.The second chapter gives the basic concepts and lemma of this paper;The third chapter draws a new method for calculating the singularity index;the fourth chapter studies the new conclusion of the Dulac function in the study of the stability of the differential system,and improves the results of C.C.Mccluske and J.S.Muldowney;the fifth chap-ter is looking forward to the future work,We propose t.he question that we shall study next.