Qualitative theory of polynomial vector fields

Abstract

The theory of dynamical systems is one of the most important tools in quantitative and qualitative study of mathematical models of applied sciences. Most of these models are formulated using differential equations (continuous dynamical systems). Since the first published work by Poincaré in the late nineteenth century, the qualitative theory of differential equations have experienced a significant expansion. While old problems still resisting, new techniques were developed to solve some old problems, and gave rise to new questions as intriguing and fascinating as the old ones. One of the most studied objects are the vector fields defined by ordinary differential equations in the plane or in surfaces. However these issues are far from fully understood and many problems remain unresolved, as the 16th Hilbert problem, the problemthe Center -Focus, the problem of integrity etc. This project aims to study the qualitative theory of ordinary differential equations in various lines of research. These involve: a) Polynomial vector fields in two or more variables as well as on surfaces or hypersurfaces; b) Study of the Center-Focus problem in the plane or on surfaces; c) discontinuous vector fields. Each of the research areas and its objectives are described in detail in the research project. (AU)