On the reducibility of a class of finitely differentiable quasi-periodic linear systems

In this paper, we consider the following systemx˙=(A+εQ˜(t))x, where A is a constant matrix with different eigenvalues, and Q˜(t) is quasi-periodic with frequencies ω1,ω2,…,ωr. Moreover, Q(θ)=Q(ωt)=Q˜(t) has continuous partial derivatives ∂bQ∂θjb for j=1,2,…,r, where b>94r+1∈Z, and the moduli of...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 413; no. 1; pp. 69 - 83
Main Authors Li, Jia, Zhu, Chunpeng
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.05.2014
Subjects
Finitely differentiable
KAM theory
Quasi-periodic
Reducibility
Reducibility
Finitely differentiable
Quasi-periodic
KAM theory
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Summary:In this paper, we consider the following systemx˙=(A+εQ˜(t))x, where A is a constant matrix with different eigenvalues, and Q˜(t) is quasi-periodic with frequencies ω1,ω2,…,ωr. Moreover, Q(θ)=Q(ωt)=Q˜(t) has continuous partial derivatives ∂bQ∂θjb for j=1,2,…,r, where b>94r+1∈Z, and the moduli of continuity of ∂bQ∂θjb satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. Under suitable hypothesis of non-resonance conditions and non-degeneracy conditions, we prove that for most sufficiently small ε, the system can be reducible to a constant coefficient differentiable equation by means of a quasi-periodic homeomorphism.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2013.10.077