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Zbl 0761.34026
Jorba, Àngel; Simó, Carles
On the reducibility of linear differential equations with quasiperiodic coefficients.
(English)
[J] J. Differ. Equations 98, No.1, 111-124 (1992). ISSN 0022-0396

We say that a matrix $Q(t)$ is a quasiperiodic matrix of time with basic frequencies $\omega\sb 1,\dots,\omega\sb r$ if $Q(t)=F(\omega\sb 1 t,\dots,\omega\sb r t)$, where $F=F(v\sb 1,\dots,v\sb r)$ is $2\pi$ periodic in all its arguments. The author considers the system (1) $x'=(A+\varepsilon Q(t))x$, where $A$ is a constant matrix and $Q(t)$ is a quasiperiodic analytic matrix with $r$ basic frequencies. Suppose $A$ has different eigenvalues (including the purely imaginary case) and the set formed by the eigenvalues of $A$ and the basic frequencies of $Q(t)$ satisfies a nonresonant condition. It is proved under a nondegeneracy condition that there exists a Cantorian set ${\cal S}\subset(0,\varepsilon\sb 0)$ ($\varepsilon\sb 0>0$) with positive Lebesgue measure such that for $\varepsilon\in{\cal S}$ (1) is reducible (i.e. there exists a nonsingular quasiperiodic matrix $P(t)$ such that $P(t)$, $P\sp{-1}(t)$ and $P'(t)$ are bounded on $R$ and the change of variables $x=P(t)y$ transforms (1) to $y'=By$ with a constant matrix $B$).
[S.Staněk (Olomouc)]
MSC 2000:
*34C20 Transformation of ODE and systems
34A30 Linear ODE and systems
34C27 Almost periodic solutions of ODE

Keywords: quasiperiodic function; reducible system; basic frequencies

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