Invariant curves on differential systems defined in Rn, n ≥ 2

• Autor(a) principal: Lima, Camila Aparecida Benedito Rodrigues de
• Data de Publicação: 2019
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: http://www.teses.usp.br/teses/disponiveis/55/55135/tde-27032019-105434/
• topic: Algebraic invariant curve; Averaging method; Curva algébrica invariante; Darboux invariant; Filippovs systems; Invariante de Darboux; Lyapunov-Schmidt reduction method; Método de redução de Lyapunov-Schmidt; Método do averaging; Sistemas de Filippov

Differential systems appear modelling many natural phenomena in different branches of sciences, in biological and physical applications among other areas. Differential systems usually have invariant curves and we can obtain a better description of the qualitative behaviour of their solutions using them. Such invariant curves may be algebraic or not and in the case where they are closed, isolated in the set of periodic orbits and without singular points, they are called limit cycles. There is a very famous problem, proposed by David Hilbert in 1900 what ask about the maximum number of limit cycle that all polynomial differential systems of a given degree could present. In this work we investigate the existence of some invariant curves in quadratic polynomial differential systems and in discontinuous piecewise differential systems (they are known as Filippovs systems). Even after hundreds of studies on the phase portraits of real planar quadratic vector fields the complete characterization of their phase portraits is a quite complex task, they depend on twelve parameters, after affine transformations and time rescaling, we have families with five parameters, which is still a large number. So many subclasses have been considered instead of the complete system. In this work we investigate conditions under the parameters of the system for a planar quadratic differential system present invariant algebraic curve of degree 3 (a cubic curve) and a Darboux invariant and obtain all the topological non-equivalent phase portraits of these systems. The increasing interest in the theory of nonsmooth vector fields has been mainly motivated by their strong relation with physics, engineering, biology, economy, and other branches of sciences. In the study of the Filippovs systems, we investigate the number of periodic orbits that they can present. In this study we apply the averaging theory. Such theory is used to study some classical models and we also present generalization of such technique for a class of nonsmooth systems. In addition, we also show how the LyapunovSchmidt reduction method can be used to consider cases where the averaging theory is not sufficient to study periodic solutions.