Dulac's Problem and of the Focus Center on Two-Dimensional Manifolds

Abstract

In this project our goal is the study of some current problems of the Qualitative Theory of the Ordinary Differentials Equations. We will study the Center-Focus Problem in central manifolds of vector fields in R3 and the Dulac's Problem for monodromic singular points of piecewise analytic vector fields in the plane.We will develop a method that allows us to computate the Lyapunov coefficients of a given system in R3 using the Lyapunov coefficients of simpler subsystems. We will use this method to study the cyclicity of centers and foci in central manifolds. We will then focus our efforts on the characterization of centers of some families of rigidi systems in R3. We hope to develop an algorithm to characterize them based on the existence of a comutator for these system.We will characterize the singular monodromic points of piecewise analytic vector fields and hope to solve the Dulac's Problem for these points.The problems will be developed in partnership with the group of dynamic systems of the Autonomous University of Barcelona, which has researchers of recognized scientific trajectory. We can mention the professors Armengol Gasull ("Host Researcher"), Jaume Llibre, and Joan Torregrosa.