On the Center Conditions of Certain Fifth 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.

So far, many methods have been tried to settle the problems of the center focus of the polynomial system. However, the center focus problem of high degree polynomial systems has not been completely solved. In this paper, we use the Poincaré method to study the center focus problem of the five periodic differential equation. Then we use the Alwash-Lloyd method [1,2,3] to derive the center conditions for this differential system.