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Doctoral Thesis
DOI
Document
Author
Full name
Mário César Monteiro do Prado
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2019
Supervisor
Committee
Gameiro, Márcio Fuzeto (President)
Bonotto, Everaldo de Mello
Castelo Filho, Antonio
Rodrigues, Savio Brochini
Title in Portuguese
Multiplicidade exata de soluções de equações diferenciais via um método assistido por computador
Keywords in Portuguese
Demonstrações assistidas por computador
Equações diferenciais
Método de Newton
Órbitas periódicas
Polinômios radiais
Abstract in Portuguese
Neste trabalho, apresentamos um método computacional rigoroso para a demonstração de existência de órbitas periódicas de alguns sistemas de equações diferenciais ordinárias com campo autônomo do tipo polinomial. Mostraremos que o problema de encontrar órbitas periódicas para esses sistemas de equações é equivalente a buscar por raízes de certas funções definidas no espaço de Banach das sequências com decaimento algébrico. O método pode ser dividido em duas etapas. Na primeira, buscamos numericamente por soluções periódicas aproximadas. Na segunda, mostraremos a existência de uma órbita periódica numa vizinhança da curva encontrada numericamente. O rigor das verificações computacionais é garantido pelo uso de aritimética intervalar.
Title in English
Computer assisted proof for ordinary differential equations
Keywords in English
Computer assisted proof
Differential equations
Newton's method
Periodic orbits
Radii Polinomials
Abstract in English
In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic.
 
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Publishing Date
2019-08-12
 
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