Bifurcation Of Limit Cycles And Global Phase Portraits Of Two Classes Of Planar Polynomial Systems

Isochronous systems and quasi-homogeneous systems are two important classes of planar polynomial differential systems,which are widely applied in both theoretical and practical problems.This paper mainly investigates the bifurcation of limit cycles for a class of isochronous systems and quasi-homogeneous systems,as well as the global phase portraits of planar cubic quasi-homogeneous systems.The paper is divided into five chapters.The first chapter mainly introduces the research status of planar polynomial differential systems at home and abroad in recent years,which including limit cycles,normal forms and global phase portrait and so on,and proposes the research framework of this paper.The second chapter introduces the basic concepts of planar quasi-homogeneous systems,the first-order averaging method,the quasihomogeneous blasting method,Poincaré-Lyapunov compaction and important lemmas used in this paper.The third chapter considers a class of 2n +3 order polynomial planar systems with isochronous centers,by using the averaging theory of first order,the maximum number of limit cycles that bifurcate from the periodic annulus of the isochronous center of the system is studied.When the system is perturbed inside the class of all polynomial differential systems of degree m,the maximum number of limit cycles is denoted by H(m).For any fixed natural number n and any positive integer k,the maximum number of limit cycles is proved to be satisfied that.For n =1,the upper bounds H(1)=H(2)=1,H(3)=H(4)=4,H(5)=H(6)=8 are given.The fourth chapter studies the bifurcation of limit cycles for a class of planar quasi-homogeneous systems with a global center.Under the perturbation of the(n,1)-quasi-homogeneous polynomial with weight degree m,we get the maximum numbers of limit cycles bifurcating from the period annulus of the system by using the averaging theory of first order,and the upper bound of these numbers which is proved to be reachable is given.What's more,the maximum number of the system that under the perturbation of polynomials with degree n is studied,and the obtained result proves that the upper bound of the maximum number of the system,which given in [Giné J,Grau M,Llibre J,JDE,2015.],is reachable.The fifth chapter studies the global dynamic structures of planar cubic quasi-homogeneous systems.Firstly,according to the canonical forms of cubic quasi-homogeneous systems,we use tools such as quasi-homogeneous blasting method and nilpotent singularity theorem to analyze the local dynamic property of the unique finite singularities of these systems;Secondly,the methods of the Poincaré-Lyapunov compaction is applied to study the types of singularities of the system at infinity,and the global phase portraits of all standard systems are determined by combining the properties of finite singularities.Finally,these global phase portraits are classified in the sense of topological equivalence.