Centers And Limit Cycles For Several Classes Of Planar And Three-dimensional Differential Systems

This thesis is devoted to the problem of bifurcations of limit cycles, centerconditions and isochronous center conditions for several classes of planar andthree-dimensional differential systems. It is composed of six chapters.In Chapter1, we introduce the historical background and the present progress ofproblems that concern with bifurcations of limit cycles, centers and isochronous centersfor the differential systems. The main works of this paper are concluded as well.In Chapter2, we introduce some basic conception and analyzing methods ofqualitative theory of differential equation.In Chapter3, the qualitative for a class of quasi-seven-degree analytic system wasstudied. We obtained the first twenty-four singular point values and proved strictlysufficient conditions for the origin to be a center and isochronous center.In Chapter4, center conditions and bifurcation of limit cycles for a class of quinticpolynomial differential system in which the origin is a nilpotent singular point are studied.Under a small perturbation, we obtain a class of quintic polynomial differential system thatbifurcates12limit cycles enclosing the origin.In Chapter5, the research methods of Hopf bifurcation for a class ofthree-dimensional differential dynamic systems applied to calculate the implicit functionformal series on center manifold. By using the calculation methods, two examples arestudied.In Chapter6, Hopf bifurcation for a class of three-dimensional differential dynamicsystems is studied, the existence of5limit cycles bifurcated from the small neighborhoodof origin is proved.In the last chapter, the whole paper is summarized and the problems which are stillunsolved in the research are showed.