Center conditions for a lopsided quintic polynomial vector field

Abstract

We consider in this article, a lopsided quintic polynomial vector field\(X = - x\frac{\partial }{{\partial y}} + y\frac{\partial }{{\partial x}} + \sum\nolimits_{i + j = 5} {(a_{ij} x^i y^j \frac{\partial }{{\partial x}})} \). We first compute the first nonzero derivative of the return mapr→L(r,e). We study then the necessary and sufficient conditions for the existence of a center.