Research On The Central Conditions Of Several Types Of Higher Order Differential Systems

There are two basic problems in the qualitative theory of planar nonlinear differential equations,namely the center focus and the limit cycle.The decision problem of the center focus is also the premise and foundation of the study on the limit cycle.Therefore,the study of the center focus and the limit cycle constitutes an independent branch of mathematics.In the research and development of the qualitative behavior of the solution to the autonomous system,there is a problem that is closely related to solving the sixteenth problem of Hilbert,that is,the research on the center focus of the polynomial system.So far,many methods have been tried,but only the center focus problem of the quadratic polynomial systems and some special cubic polynomial systems has been settled.However,the center focus problem of general cubic polynomial systems and higher degree polynomial systems has not been completely solved.In this paper,we use the Poincare method to study the five periodic differential equation dr/d?=r3(P2(cos?,sin?)+ P4(cos?,sin?)r2),and the four periodic differential equation dr/d?=r2(P1(cos?,sin?)+ rP2(cos?,sin?)+r2P3(cos?,sin?)),(here Pn(cos?,sin?)=(?)Pij cosi ? sinj?,Pij are real number,n = 1,2,3,4;i,j=0,1,2,3,4)whose condition centered around r = 0,that is,when these polynomial differential equations have periodic solution bounds around r = 0.More importantly,we obtain the equivalent plane polynomial differential systems:and whose necessary conditions centered around r = 0.At the same time,we use the Alwash-Lloyd method to prove that the necessary conditions for the differential systems(1),(2)and(3)are also sufficient,that is,when these conditions are set up the systems(1),(2)and(3)with(0,0)as combination center.Furthermore,our research on the center of differential system(4)has been extended the corresponding results of Alwash.