Several Problems On Planar Quasi-Homogeneous And Semi-Quasi-Homogeneous Polynomial Systems

Planar quasi-homogeneous and semi-quasi-homogeneous systems have important applications in both theoretical and practical problems.This paper mainly investigates the bifurcation of limit cycles for a class of planar quasi-homogeneous polynomial differential systems,and,the limit cycles,as well as the global phase portraits of planar quadratic and cubic semi-quasi-homogeneous systems.The paper is divided into five chapters.The first chapter mainly introduces some problems on planar polynomial differential systems,which have been studied in recent years,especially the quasi-homogeneous systems and semi-quasi-homogeneous systems on integrability,canonical form,limit cycle,and global phase portrait.The second chapter mainly proposes the basic concepts of planar quasi-homogeneous and semi-quasi-homogeneous systems,Abelian integral,blow-up technique,Poincaré compactation,and important lemmas used in this paper.The third chapter studies a class of(m,1)planar quasi-homogeneous systems with global center.Under respectively the perturbation of polynomial with degree n,and the(n,1)-quasi-homogeneous polynomial,we get the number of limit cycles bifurcating from the period annulus of the center of the system by discussing the number of zeros of Abelian integral.Moreover,the upper bound of these numbers which is proved to be reachable is given.The forth chapter study the limit cycle and global phase portraits of planar quadratic semi-quasi-homogeneous systems.We firstly get the canonical forms of these systems from the existing reference.And,by using some means such as the blow-up technique and the nilpotent singularity,we analyze the structure of the trajectory near the unique finite singularity of these canonical systems and thus obtain the local phase portrait.Next,we use the Poincaré compactation to discuss the types of infinite singularities.Then we consider the existence of limit cycle for these systems.Combined with the above discussion,we obtain all the global phase portraits of them.Finally,we classify these global phase portraits.Under the meaning of topological equivalence,the quadratic semi-quasi-homogeneous systems have 6 different classes global phase portraits.The fifth chapter firstly discuss the limit cycle for several kinds of semi-quasi-homogen-eous systems,including proving the nonexistence of limit cycle for the cubic homogeneous and quasi-homogeneous systems,and the existence of limit cycle for the cubic semi-homoge-neous and semi-quasi-homogeneous systems.On this basis,we also obtain the canonical form of cubic semi-quasi-homogeneous system with a unique stable limit cycle.Furthermore,we extend the expression of this system to the more general semi-quasi-homogeneous polynomial system with odd degree which has a unique stable limit cycle.Finally,by using the method of Chapter 4,it is proved that,under the meaning of topological equivalence,the cubic semi-quasi-homogeneous systems have 43 different classes global phase portraits.