Sistemas dinâmicos e difeomorfismos no círculo: hiperbolicidade e estabilidade

Abstract

This work discusses some initial concepts of the theory of dynamic systems, with the circle as its main focus. Several important concepts are studied, such as fixed points, orbit, hyperbolicity and stability. The text is exploratory in nature, seeking an introduction to the subjects of dynamic systems. We start by introducing what dynamic systems are and explaining iterations. The circle is presented as a metric space and with points 0 and 1 identified. The first properties of dynamic systems are then presented, such as fixed and periodic points, in addition to the orbits of a point. We address what hyperbolic points are and classify them as attractor or repulse hyperbolic points. We define what Morse-Smale diffeomorphisms are and find all their dynamics, through the properties seen previously. Finally, we work with the conjugation of diffeomorphisms, observing their stability and demonstrating that Morse-Smale diffeomorphisms are structurally stable.