Center Conditions for a Class of Rigid Quintic 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 Poincaré center-focus problem. So far, people has tried many ways to solve the problem of central focus. However, it is difficult to solve the center focus problem of higher order polynomial system. In this paper, we use the Poincaré and Alwash-Lloyd methods to study the center focus problem and derive the center conditions of the five periodic differential equation.