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Doctoral Thesis
DOI
10.11606/T.45.2016.tde-23082016-103753
Document
Author
Full name
Alexander Fernandes da Fonseca
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2016
Supervisor
Committee
Mello, Luis Fernando de Osório (President)
Lima, Mauricio Firmino da Silva
Martins, Ricardo Miranda
Mereu, Ana Cristina de Oliveira
Salomão, Pedro Antonio Santoro
Title in Portuguese
Estabilidade assintótica global e continuação de soluções periódicas em sistemas suaves por partes com duas zonas no plano
Keywords in Portuguese
Ciclo limite
Função de Melnikov
Método de regularização
Sistema suave por partes
Abstract in Portuguese
Nesta tese estudamos um dos principais problemas na teoria qualitativa das equações diferenciais planares: o problema de determinar a bacia de atração de um ponto de equilíbrio. Damos uma prova rigorosa de que para sistemas lineares por partes de costura com duas zonas no plano, definidas por matrizes Hurwitz o único ponto de equilíbrio na reta de separação é globalmente assintoticamente estável. Por outro lado, provamos que nesta classe de sistemas, podemos ter um ponto de equilíbrio instável na origem quando uma curva poligonal separa as zonas, levando a um resultado contra-intuitivo do comportamento dinâmico de sistemas lineares por partes no plano. Além disso, estudamos os ciclos limites em perturbações suaves por partes de centros Hamiltonianos. Neste cenário, é comum adaptar resultados clássicos de sistemas suaves, como funções de Melnikov, para sistemas não-suaves. No entanto, existe pouca justificativa para este procedimento na literatura. Ao utilizar o método de regularização damos uma prova que suporta o uso de funções de Melnikov diretamente do problema não-suave original.
Title in English
Global asymptotic stability and continuation of periodic solutions in piecewise smooth systems with two zones in the plane
Keywords in English
Limit cycle
Melnikov function
Piecewise differential system
Regularization method
Abstract in English
In this thesis we study one of the main problems in the qualitative theory of planar differential equations: the problem of determining the basin of attraction of an equilibrium point. We give a rigorous proof that for planar sewing piecewise linear systems with two zones, defined by Hurwitz matrices the unique equilibrium point in the separation straight line is globally asymptotically stable. On the other hand, we prove that sewing piecewise linear systems with two zones in the plane, defined by Hurwitz matrices can have one unstable equilibrium point at the origin allowing a broken line to separate the zones, leading to counterintuitive dynamical behaviors of simple piecewise linear systems in the plane. Furthermore, we study limit cycles in piecewise smooth perturbations of Hamiltonians centers. In this setting it is common to adapt classical results for smooth systems, like Melnikov functions, to non-smooth ones. However, there is little justification for this procedure in the literature. By using the regularization method we give a proof that supports the use of Melnikov functions directly from the original non-smooth problem.
 
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TESEALEXANDER.pdf (1.07 Mbytes)
Publishing Date
2016-09-09
 
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