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Doctoral Thesis
DOI
10.11606/T.55.2018.tde-02072018-131725
Document
Author
Full name
Mara Sueli Simao Moraes
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 1994
Supervisor
Committee
Táboas, Plácido Zoega (President)
Barbanti, Luciano
Godoy, Sandra Maria Semensato de
Nowosad, Pedro
Reis, José Geraldo dos
Title in Portuguese
FORMA ASSINTÓTICA DE SOLUÇÕES PERIÓDICAS DE UMA EQUAÇÃO DIFERENCIAL NO PLANO COM RETARDAMENTO
Keywords in Portuguese
Não disponível
Abstract in Portuguese
A equação diferencial com retardamento perturbada singularmente ε x = -x(t) + F (x(t-1)) é estudada, com ε > 0, x = (x1, x2), F = (f1, f2), f1, f2 : R → R, diferenciáveis até ordem 2 na origem é ímpares. Para ε pequeno e f = -f1 = f2 monótona num intervalo [-A, A], A > 0, é provado que a solução periódica lentamente espiralante x(t) da equação (1) tem a forma de uma "onda quadrada". e está relacionada aos pontos periódicos da função F = (f1, f2). Como é destacado em [1], para o caso escalar, quando f não é monótona a convergência de x(t) para a "onda quadrada" é tipicamente não uniforme, e ocorre um fenômeno similar ao de Gibbs, da clássica série de Fourier.
Title in English
Not available
Keywords in English
Not available
Abstract in English
The singularly perturbed differential-delay equation ε x(t) = -x(t) + F(x(t-1)) is studied, with ε > 0, x = (x1, x2), F = (f1, f2), f1, f2 : R → R odd and differentiable up to order two at the origin. For small ε and f = -f1 = f2 monotone in some interval [-A, A], A > 0, the slowly spiralling periodic solutions x(t) of the equation (1) are proved to have square-wave shape, and are related to periodic points of the mapping F = (f1, f2). As it is pointed out in [1], for the scaler case, when f is not monotone the convergence of x(t) to the square-wave typically is not uniform, and a phenomenon similar to the Gibbs one of classical Fourier serie, must occur.
 
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Publishing Date
2018-07-02
 
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