On the number of limit cycles surrounding a unique singular point for polynomial differential systems of arbitrary degree

We study the number of limit cycles that bifurcate from the periodic orbits of the center over(x, ̇) = - y R (x, y), over(y, ̇) = x R (x, y) where R is a convenient polynomial of degree 2, when we perturb it inside the class of all polynomial differential systems of degree n. We use averaging theory for computing this number. As a consequence of our study we provide the biggest number of limit cycles surrounding a unique singular point in terms of the degree of the system, known up to now. © 2007 Elsevier Ltd. All rights reserved.