On the uniqueness of limit cycles surrounding one or more singularities for Lienard equations

In this paper we prove a theorem on the uniqueness of limit cycles surrounding one or more singularities for Lienard equations. By using this theorem we give a positive answer to the conjecture in Dumortier and Rousseau (1990 Nonlinearity 3 1015-39), completing the classification of the cubic Lienard equations with linear damping. It also finishes the study of the generic three-parameter unfoldings of the nilpotent focus in the plane.