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On the reducibility of linear differential equations with quasiperiodic coefficients. (English) Zbl 0761.34026
We say that a matrix $Q\left(t\right)$ is a quasiperiodic matrix of time with basic frequencies ${\omega }_{1},\cdots ,{\omega }_{r}$ if $Q\left(t\right)=F\left({\omega }_{1}t,\cdots ,{\omega }_{r}t\right)$, where $F=F\left({v}_{1},\cdots ,{v}_{r}\right)$ is $2\pi$ periodic in all its arguments. The author considers the system (1) ${x}^{\text{'}}=\left(A+\epsilon Q\left(t\right)\right)x$, where $A$ is a constant matrix and $Q\left(t\right)$ 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 $Q\left(t\right)$ satisfies a nonresonant condition. It is proved under a nondegeneracy condition that there exists a Cantorian set $𝒮\subset \left(0,{\epsilon }_{0}\right)$ (${\epsilon }_{0}>0$) with positive Lebesgue measure such that for $\epsilon \in 𝒮$ (1) is reducible (i.e. there exists a nonsingular quasiperiodic matrix $P\left(t\right)$ such that $P\left(t\right)$, ${P}^{-1}\left(t\right)$ and ${P}^{\text{'}}\left(t\right)$ are bounded on $R$ and the change of variables $x=P\left(t\right)y$ transforms (1) to ${y}^{\text{'}}=By$ with a constant matrix $B$).

##### MSC:
 34C20 Transformation and reduction of ODE and systems, normal forms 34A30 Linear ODE and systems, general 34C27 Almost and pseudo-almost periodic solutions of ODE