Coeficientes de Lyapunov e bifurcação de Hopf para sistemas diferenciais analíticos no plano

Abstract

The Qualitative Theory of Ordinary Differential Equations addresses fundamental problems in the study of the dynamic behavior of systems, among which the "Center-Focus Problem" and ``Hilbert's 16th Problem'' stand out. These two problems are deeply interconnected through the Lyapunov Coefficients, an essential tool for analyzing the local behavior of differential systems around singular points. In this work, we present basic concepts of the Qualitative Theory of ODEs as a way to establish notation and study objects for the main chapters on the Lyapunov Constant and Hopf Bifurcation. Furthermore, we demonstrate and implement the Lyapunov Coefficients, exploring their relationship with another relevant problem in the theory: the bifurcation of limit cycles from a Hopf point. This connection highlights the importance of the coefficients in investigating central issues of nonlinear dynamics.