Geometrical and topological investigation of some families of quadratic differential systems possessing saddle-nodes or invariant ellipses

• Autor(a) principal: Mota, Marcos Coutinho
• Data de Publicação: 2021
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-18052021-121432/
• topic: Classificação geométrica e topológica; Configuração de elipses e retas invariantes; Configuration of invariant ellipses and lines; Geometrical and topological classification; Invariant polynomial; Invariante polinomial; Phase portrait; Quadratic system; Retrato de fase; Sistema quadrático

The study of quadratic polynomial differential systems on the plane have been shown a tough challenge, there exist hundreds of papers about them which are dated for over a century and until now there exist several topics to be studied and concluded. For instance, the complete characterization of phase portraits of quadratic systems remains unknown and the complete topological classification of such systems has been a complex work. It is well known that the greatest difficult of working with quadratic systems is the quantity of parameters. A (generic) quadratic system is defined by 12 parameters, however by using affine transformations and time rescaling one can reduce this number by five, but yet this is a very large number, once the corresponding bifurcation diagram is a fivedimensional euclidean space. So, it is convenient to use some tools (as the Invariant Theory) in order to study families of quadratic systems with specific properties (for instance, according to the structural stability or possessing classes of invariant algebraic curves) with the purpose of reducing even more (when it is possible) this quantity of parameters. The main goal of this thesis is to contribute to the classification of the quadratic systems on the plane. More precisely, we present the complete study (modulo islands) of the bifurcation diagram of two families of quadratic systems possessing specific properties on their singularities, we do the complete topological classification (modulo limit cycles) of all the phase portraits of two sets of quadratic systems of codimension two and we perform the classification of quadratic differential systems with invariant ellipses according to their configurations of invariant ellipses and invariant lines. It is worth mentioning that these three works represent three different approaches to the study of quadratic systems and each one of them uses different techniques, which all together are useful towards the final goal of classifying phase portraits.