Stability properties in dynamical systems

Abstract

The main goal of this research project is to expose the candidate to the modern techniques and problems in the discrete dynamical systems area, and, in particular, to the principles of the rotation vectors of surface homeomorphisms. By the end of the project, its expected from the student to, if he wishes to do so, continue his studies in this research line in a graduate stage efficiently. Minding that, studies will begin by the interval mappings dynamics and circle homeomorphisms, where its possible to explain and understand the fundamental concepts of the modern dynamical systems theory, and after move on to the study of circle endomorphism and questions relative to these mappings, such as degree and rotation set. The fundamental concepts of dynamical systems will be presented focusing on the stability of invariant sets and their hyperbolicity, both of an isolated system and of families in one and two parameters. This way, the student will be in touch with periodical orbits stability problems and equilibrium points, as well as will be presented to contemporary questions about hyperbolic sets stability and bifurcation theory.