Research On The Composition Conjecture Problem Of A Class Of Higher Degree Polynomial Differential Systems

The study of the Poincare-center focus problem is a very important research topic in the qualitative theory study of differential equations.So far,only the center focus problem of quadratic systems has been completely solved in planar polynomial differential systems.Only some special cases in cubic polynomial differential systems have better results.However,the center focus problem is more complicated for the general higher order system with more than three order.So far there is still no simple and effective way to solve it.In this paper,we have studied the Poincare-center focus problem of a class of polynomial differential systems of higher order(?)(here Pk(x,y)(k=3,m)are k order homogeneous polynomial with real coefficients).We first adopt a clever transformation and apply Langrange-Bürman formula specification to simplify the structure of the differential equation to be studied.Then the necessary conditions for it to be centered at the origin are derived by using Poincare post function method.In order to avoid a lot of complicated calculation and quickly get the expression of focus quantity in the process of deducing these conditions,we adopt the principle of searching for the rule of blocks and simplifying the calculation section by section.Finally,Alwash-Lloyd method is applied to prove that these conditions for the center,which further prove that the center is a composition center.This article consists of three chapters.In chapter 1,we introduce the development of differential equations and briefly describe the research background and present development situation of center-focus,composition center and composition conjecture of planar polynomial differential system.At the same time,we introduce the main research content of this paper and the thought method of researching this problem.In chapter 2,we introduce the concept of composition conditions and composition conjecture,and in order to prove the conditions of m+1 order differential system centered at the origin,we give five preliminary lemmas.In chapter 3,we mainly study the Poincare-center focus problem for m+1 order polynomial differential system above.We give the necessary and sufficient conditions for the system to be centered at the origin under several constraints.According to the theorem,we obtain the sufficient and necessary conditions for the eighth,ninth and tenth order polynomial differential systems centered on the origin and verify the correctness of the obtained conclusions with examples.