Homogeneous-Like Generalized Cubic Systems

Abstract

We consider properties and center conditions for plane polynomial systems of the forms x over dot = -y -p(1)(x,y) -p(2)(x,y), y over dot = x +q(1) (x,y) + q(2) (x,y) where p(1),q(1) and p(2), q(2) are polynomials of degrees n and 2n - 1, respectively, for integers n >= 2. We restrict our attention to those systems for which yp(2) (x,y) + xq(2) (x,y) = 0. In this case the system can be transformed to a trigonometric Abel equation which is similar in form to the one obtained for homogeneous systems (p(2) = q(2) = 0). From this we show that any center condition of a homogeneous system for a given n can be transformed to a center condition of the corresponding generalized cubic system and we use a similar idea to obtain center conditions for several other related systems. As in the case of the homogeneous system, these systems can also be transformed to Abel equations having rational coefficients and we briefly discuss an application of this to a particular Abel equation.