Research On The Center Of A Class Of Stiff Differential Systems

The problem of determining necessary and sufficient conditions on P and Q for system x=-y+P(x,y),y=x+Q(x,y).to have a center at the origin is known as the Poincare center-focus problem.It is closely related to discussing the qualitative behavior of the solution of the differential system,and it is also a hot topic in the research field of differential theory.Many experts and scholars all over the world have made unremitting efforts to carry out research on it.Unfortunately,the known results are still very limited.So far,only the center-focus problem of quadratic polynomial differential systems has been solved,but when it is a cubic polynomial,this problem is still not completely solved.Even with the help of computers,people's research progress on the central-focus problem of higher-order polynomial differential systems is very slow.In this paper,we focus on two types of rigid systems x=-y+x(P1(x,y)+P3(x,y)+P7(x,y)),y=x+y(P1(x,y)+P3(x,y)+P7(x,y)),and x=-y+x(P1(x,y)+P4(x,y)+P9(x,y)),y=x+y(P1(x,y)+P4(x,y)+P9(x,y)),when to take the origin as the center.We have obtained the periodic differential equation equivalent to the two differential systems under several restrictions by using the Poincare method and the Alwash Lloyd method r=r(P1(cos ?,sin ?)r+P3(cos ?,sin ?)r3+P7(cos ?,sin ?)r7),and r=r(P1(cos ?,sin ?)r+P4(cos ?,sin ?)r4+P9(cos ?,sin ?)r9),taking r=0 as the combination center.The necessary and sufficient conditions for the corresponding rigid system to take(0,0)as the center are obtained.In this paper,although the number of systems we have studied is relatively high,we do not need computer assistance here,and we only use some mathematical analysis techniques to obtain these central conditions.Moreover,the expressions of these conditions are concise and beautiful,the methods are novel,and the ideas are unique.Thus,it is easier to calculate the focal points of these two systems,and the expressions are concise.