Introdução a sistemas suaves por partes com duas ou três zonas no plano

Abstract

This work had as its objective the introduction to the Qualitative Theory of Differential Equations and Discontinuous Systems. Firstly, contact with the basic definitions and principal theorems of the theory is necessary. Subsequently, the study of piecewise smooth systems begins, in particular, in the search for limit cycles in piecewise linear systems. Traditional techniques of mathematical research were used, consisting of the focused study of the proposed bibliographical references and follow-up meetings with the development of the problems suggested through analyses and discussions of the main definitions and proofs of theorems that encompass the theme, through the study of the article [12], we investigate the existence of limit cycles in piecewise differential systems formed by linear centers and Hamiltonian saddles. Initially, continuous or discontinuous systems divided by a single line are considered, where the non-existence of limit cycles is proven. Next, the study proceeds to continuous and discontinuous systems with three zones, separated by two parallel lines. In this context, it is observed that in the continuous cases there are no limit cycles, contrary to the discontinuous cases, which present at most one limit cycle. In this way, this research plan provided contact and theoretical deepening regarding the content of dynamic systems, in addition to the introduction to the scientific research environment, working on critical study and the capacity for investigation.