Ciclos limite para a equação de Abel generalizada
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Data
2009-10-30
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Universidade Federal de Goiás
Resumo
In this work we conducted a study on the equations of the type
dx
dt
=
nå
i=0
ai(t)xi; (A)
where ai 2 C1, i = 0; ;n and 0 t 1. An equation of the form (A) is called a
generalized Abel equation. Our study refers to the problem proposed by C. Pugh: There
is a natural number N depending only on n, such that the equation (A) has at most N limit
cycles?
Initially we study the problem of C. Pugh for n = 1 and n = 2, for which the equation
(A) has at most one and two limit cycles, respectively. For n = 3, A. Lins Neto shows
that if a3(t) does not change sign on [0;1], then the equation (A) has at most three limit
cycles. Also A. Lins Neto shows that, given a natural number l, it is possible to construct
an equation of the form (A) with n = 3 that has at least l limit cycles. Still for n = 3, A.
Gasull and J. Llibre study the problem of C. Pugh considering that a2(t) does not change
sign on [0;1], and M. J. Alvarez, A. Gasull and H. Giacomini also study the problem of
C. Pugh considering that there are real numbers a and b such that aa3(t)+ba2(t) does
not change sign on [0;1] and a1(t) = a0(t) = 0. Besides this, we study some more general
results studied by A. Gasull and A. Guillamon.
Descrição
Citação
BELISÁRIO, Hugo Leonardo da Silva. Ciclos limite para a equação de Abel generalizada. 2009. 39 f. Dissertação (Mestrado em Matemática) - Universidade Federal de Goiás, Goiânia, 2009.