The period function for second-order quadratic ODEs is monotone
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Tipus de documentArticle
Data publicació2003
Condicions d'accésAccés obert
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Reconeixement-NoComercial-SenseObraDerivada 2.5 Espanya
Abstract
Very little is known about the period function for large families of centers. In one of
the pioneering works on this problem, Chicone [?] conjectured that all the centers encountered
in the family of second-order differential equations ¨x = V (x, ˙ x), being V a quadratic polynomial,
should have a monotone period function. Chicone solved some of the cases but some others
remain still unsolved. In this paper we fill up these gaps by using a new technique based on
the existence of Lie symmetries and presented in [?]. This technique can be used as well to
reprove all the cases that were already solved, providing in this way a compact proof for all the
quadratic second-order differential equations. We also prove that this property on the period
function is no longer true when V is a polynomial which nonlinear part is homogeneous of
degree n > 2.
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