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

Abstract

We study the number of limit cycles that bifurcate from the periodic orbits of the center x ̇ = − y R ( x , y ) , 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.