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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2017.tde-12092017-080613
Document
Author
Full name
Wilker Thiago Resende Fernandes
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2017
Supervisor
Committee
Oliveira, Regilene Delazari dos Santos (President)
Angles, Claudia Valls
Arús, Joan Torregrosa I
Pessoa, Claudio Gomes
Title in English
Centers and isochronicity of some polynomial differential systems
Keywords in English
Darboux integrability
Differential systems with symmetry
Invariant surfaces and curves
Isochronous centers
Primary decompositions of ideals
Abstract in English
The center-focus and isochronicity problems are two classic problem in the qualitative theory of ordinary differential equations (ODEs). Although such problems have been studied during more than hundred years a complete understanding of them is far from be reached. Recently the computational algebra tools have been contributing significantly with the development of such problems. The aim of this thesis is to contribute with the studies of the center-focus and isochronicity problem. Using computational algebra tools we find conditions for the existence of two simultaneous centers for a family of quintic systems possessing symmetry. The studies of the simultaneous existence of two centers in differential systems is known as the bi-center problem. We investigate conditions for the isochronicity of centers for families of cubic and quintic systems and we study its global behaviour in the Poincaré disk. Finally, we study the existence of invariant surfaces and first integrals in a family of 3-dimensional systems. Such family is known as the May-Leonard asymmetric system and it appears in modelling, for instance it is a model for the competition of three species.
Title in Portuguese
Centros e isocronicidade de alguns sistemas diferenciais polinomiais
Keywords in Portuguese
Centros isócronos
Curvas e superfícies invariantes
Decomposição primária de ideais
Integrabilidade Darbouxiana
Sistemas diferenciais com simetria
Abstract in Portuguese
Os problemas do foco-centro e da isocronicidade são dois problemas clássicos da teoria qualitativa das equações diferenciais ordinárias (EDOs). Apesar de tais problemas serem investigados a mais de cem anos ainda pouco se sabe sobre eles. Recentemente o uso e desenvolvimento de ferramentas algebro-computacionais tem contribuído significativamente em seu avanço. O objetivo desta tese é colaborar com o estudo do problema do foco-centro e da isocronicidade. Utilizando ferramentas algebro-computacionais encontramos condições para a existência simultânea de dois centros em famílias de sistemas diferenciais quínticos com simetria. O estudo sobre a existência simultânea de dois centros é também conhecido como problema do bi-centro. Investigamos condições para a isocronicidade de centros para famílias de sistemas cubicos e quínticos e estudamos o comportamento global de suas órbitas no disco de Poincaré. Finalmente, tratamos da existência de superfícies invariantes e integrais primeiras para uma familia de sistemas 3-dimensionais encontrado entre outras situações na modelagem da competição entre três espécies e conhecido como sistema de May-Leonard.
 
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Publishing Date
2017-09-12
 
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