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Master's Dissertation
DOI
https://doi.org/10.11606/D.55.2020.tde-09112020-185555
Document
Author
Full name
Maíra Duran Baldissera
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2020
Supervisor
Committee
Oliveira, Regilene Delazari dos Santos (President)
Angles, Claudia Valls
Diaz, José Claudio Vidal
Mello, Luis Fernando de Osório
Title in Portuguese
Integrabilidade em sistemas planares e existência de ciclos limites para o sistema de Rayleigh generalizado
Keywords in Portuguese
Ciclos limites
Compactificação de Poincaré
Curvas algébricas invariantes
Integrabilidade Darbouxiana
Sistemas diferenciais polinomiais planares
Abstract in Portuguese
Esta dissertação esta dividida em duas partes, ambas relacionados ao problema do foco centro e a ciclicidade em sistemas diferenciais planares, temas que pretendemos investigar em futuro próximo. Na primeira parte, a existência de uma integral primeira Darbouxiana em sistemas diferenciais polinomiais planares é investigada. O estudo realizado foi aplicado em um exemplo retirado da classificação dos sistema quadráticos com elipse invariantes. Na segunda parte da dissertação, investigamos a existência de ciclos limites no sistema diferencial de Rayleigh generalizado. Em (LINS; MELO; PUGH, 1977) os autores afirmam que este sistema possui um único ciclo limite. No entanto a prova apresentada por eles traz algumas falhas. Neste texto apresentamos uma prova completa e correta da existência e unicidade de ciclo limite para esta classe de sistemas.
Title in English
Integrability in planar systems and existence of limit cycle in the generalized Rayleigh systems
Keywords in English
Darboux integrability
Invariant algebraic curve
Limit cycles
Planar polynomial differential systems
Poincaré compactification
Abstract in English
This dissertation is divided in two parts, both of them are related with the center problem and the ciclicity in planar differential systems, subjects that we intent to investigate in short term. In the first part of this dissertation, the existence of a Darboux first integral in planar polynomial differential systems is investigated. Such study is applied in an example took from the classification of quadratic systems with invariant ellipses. In the second part of this dissertation, we investigate the existence of limit cycles in the generalized Rayleigh systems. In (LINS; MELO; PUGH, 1977) the authors stated that such system has a unique limit cycle. But the proof has some gaps. In this work we present a complete and correct proof of the existence and uniqueness of limit cycle for such class of systems.
 
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Publishing Date
2020-11-11
 
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