This thesis is based on the following paper "Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields " , "I.E . Colak , J . Llibre , C . Valls" . Determining limit cycles and distinguishing when a singular point is either a focus or a center are two of the main problems in the qualitative theory of real planar polynomial differential systems . Poincare , in [21] , defines a center for a vector field on the real plane as a singular point having a neighborhood filled with periodic orbits with the exception of the singular point . If an analytic system has a center , then after an affine change of variables and a rescaling of the time variable , it can be written in one of the following three forms : called a linear type center; called a nilpotent center; called a degenerate center , where and are real analytic functions without constant and linear terms , defined in a neighborhood of the origin . An algorithm for the characterization of linear type centers is provided by Poincare [22] and Lyapunov [16] , see also Chazy [5] and Moussu [18] . There is also an algorithm for the characterization of the nilpotent and some dir=ltr The dir=ltr differential systems of the form linear with homogeneous nonlinearities of degree 3 were characterized by Malkin [17] , and by Vulpe and Sibiirski [25]. For polynomial differential systems of the form linear with homogeneous nonlinearities of degree greater than 3 the linear type centers are not dir=ltr In this work , we dir=ltr Theorem 1.4 A Hamiltonian planar polynomial vector field with linear plus cubic homogeneous terms has a nilpotent center at the origin if and only if , after a linear change of variables and a rescaling of its independent variable , it can be written as one of six dir=ltr The contents of this thesis are presented in four chapters. The first chapter was devoted to an introduction of differential systems centers and theorem on the Hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields . In the second chapter we give basic concepts of qualitative theory of differential systems . Then in the third chapter of this thesis we give a prove of the main theorem described in the first chapter . Finally , the fourth chapter contains a code related to calculations of Gr?bner basis .