A note on the reducibility of linear differential equations with quasiperiodic coefficients
type: Item
instance of: scholarly article
title: A note on the reducibility of linear differential equations with quasiperiodic coefficients
zbMATH Open document ID: 1048.34081
DOI: 10.1155/S0161171203206025
author: D. Massart
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2003-12-03T00:00:00Z
full work available at URL: https://eudml.org/doc/50977
review text: The paper deals with the \(n\)-dimensional linear differential system \(dx/dt=(A +\varepsilon Q(t))x\), where \(A\) is a constant matrix, \(Q(t)\) an analytic quasiperiodic matrix, and \(\varepsilon>0\) is sufficiently small. In analogy to the Floquet theory, the question is whether there exists a quasiperiodic transformation \(x=P(t)y\) such that \(y\) satisfies an equation \((*)\) \(dy/dt=By\) with constant coefficients. The authors refer to a previous paper by Jorba and Simo in which this question was answered affirmatively for an \(\varepsilon\)-Cantor set, if the frequencies of \(Q (t)\) and the eigenvalues of \(A\) satisfy some nonresonance condition, and if, moreover, the eigenvalues of the time average of \(A+\varepsilon Q(t)\) have some additional property. In the present paper, a complementary result yielding \((*)\) is proved: The matrix \(A\) and \(\varepsilon\) can be arbitrarily, while the set of admissible frequencies of \(Q(t)\) has to be slightly restricted in order to avoid small divisors during the iteration process applied.
reviewed by: Josef Hainzl
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 37C55
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 2012666
MaRDI profile type: Publication
OpenAlex ID: W2166493308
description: scientific article; zbMATH DE number 2012666