Invariant Sets in differential Dynamical Systems: Periodic orbits, Invariant Tori and Algebraic surfaces.

Abstract

This research project has mainly three work lines. First, we shall develop results in the context of averaging theory for detecting periodic orbits. This method has one of the main tools for detecting periodic orbits in differential systems. Recently, methods like Brouwer degree and Lyapunov--Schimdt reduction were incorporated into this theory widely improving the set of differential systems for which averaging theory can be applied. Our aim is to provide a criterion for determining the stability of non-hyperbolic periodic orbits detected by the recently formulated version of the averaging theorem.Moreover, we shall improve the averaging theory in order to make possible the detection of invariant tori in planar non-autonomous systems and tridimensional autonomous systems. These bifurcations are known as Neimark-Sacker bifurcation and we shall prove that they can be precisely detected by means averaging theory.Finally, we will use the algebraic invariant surfaces of a given differential system to describe its behavior providing, even, the description of its orbits at infinity. This will be done using the concept of weight homogeneous polynomial, characteristic courves, and Poincaré compactification.All results developed here will be applied to study relevant physical systems. For instance, the Rossler system, Maxwell-Bloch system, and the generalized Van der system. This project has the collaboration of important researchers from Brazil, Portugal and Spain.