Study On The Composition Problem Of A Class Of Sixth Degree Ploynomial Differential Systems

In planar polynomial differential systems,Poincare center-focus question is an important part of the study of plane qualitative theory,also is the premise to solve the problem of Hilbert 16th.At present,the center-focus question of quadratic polynomial differential system has been completely solved.However,for cubic and more cubic differential systems,relevant conclusions are rare,due to the huge amount of computation and the lack of effective methods.In this paper,we have studied the center-focus problem of a class of sextic planar polynomial differential systems,and give the necessary and sufficient conditions that the center is centered at the origin.At the same time,we show that the center is a composition center,and prove that the corresponding composition conjecture for the system is correct.This paper consists of three chapters.In chapter 1,we give a comprehensive review of the definition,historical background and research status of the center-focus,composition center and composition conjecture problem of planar differential systems.At the same time,the main work of this paper is briefly introduced.In chapter 2,we have mainly described the concepts of composition center and composition conjecture,and we proved several lemmas.This paper lays a foundation for the composition conjecture of a class of sextic planar polynomial differential systems.In chapter 3,we mainly study the center-focus and composition conjecture for a class of sextic planar polynomial differential systems(rigid system)(?)(1)Where P_k(x,y)=(?).We give the necessary and sufficient condition that the differential system is centered at(0,0).(?)In this paper,we apply the Poincaré method to derive the necessary condition that differential system(1)is centered on(0,0).By applying some unique ideas,it is proved that P2,P5 satisfies the composition conditions under these conditions.Thus,the composition conjecture for this differential system is proved to be true.