Composition Conjecture For The High-Order Polynomial Differential Systems

This paper is devoted to composition conjecture for the high-order polynomial differential systems.It consists of two parts.In the first part,we summarize the historical background and the present conditions for the problems about composition conjecture in planar polynomial differential systems.At the same time,the main work of this thesis is introduced.In the second part,by studying the following high-order polynomial differential system#12 where(?)are real number.We give some specific restrictions to make sure that the center of this differential system is a composition center.Next,we use some positive and effective calculation techniques to imply these necessary and sufficient conditions for the differential system(1)to be centered on the origin.Simultaneously,we can expound that composition conjecture for this system is correct.We mainly study the high-order differential systems with m=4 and m=6.During the progress of the work,we first use Poincare return map method to imply necessary conditions centered around r=0 for two periodic differential equations equivalent to(1)respectively dr/d?=r(P2(cos?,sin?)r+P4(cos?,sin?)r2+P10(cos?,sin?)r5)(2)and dr/d?=r(P2(cos?,sin?)r+P6(cos?,sin?)r3+P14(cos?,sin?)r7)(3)(where(?)).Then we use some analytical techniques to prove that these necessary conditions are also sufficient.We can further clarify that the center is a composition center.At the same time,the specific restrictions on the composition conjecture of differential equations(2)and(3)are givenMoreover,the research on the composition conjecture of this high-order differential system has been extended the center focus problems about low-order differential systems of Alwash.