Nilpotent centers on the center manifolds

Abstract

In this project our goal is the study of monodrome nilpotent singular points in center manifolds of vector fields in R3. We will look for a result that characterizes these points according to the parameters of the system and without the necessity of restricting it to a center manifold. Once characterized, we will focus on finding techniques to solve the center-focus problem for these singularities and also avoid the need to restrict the system to a center manifold. The main idea is to make a change of coordinates using the so called generalized trigonometric functions introduced by Lyapunov to study the nilpotent centers in the plane. We also intend to extend to this case a result in the plan which shows that nilpotent centers are limites of non-degenerate centers. Finally we will use the techniques obtained to classify nilpotent centers on the central manifold of certain families of polynomial systems and we will seek lower bounds for the number of limit cycles that bifurcate from these centers. (AU)