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On the reducibility of linear differential equations with quasiperiodic coefficients. (English) Zbl 0761.34026
We say that a matrix $$Q(t)$$ is a quasiperiodic matrix of time with basic frequencies $$\omega_ 1,\dots,\omega_ r$$ if $$Q(t)=F(\omega_ 1 t,\dots,\omega_ r t)$$, where $$F=F(v_ 1,\dots,v_ r)$$ 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 $${\mathcal S}\subset(0,\varepsilon_ 0)$$ ($$\varepsilon_ 0>0$$) with positive Lebesgue measure such that for $$\varepsilon\in{\mathcal S}$$ (1) is reducible (i.e. there exists a nonsingular quasiperiodic matrix $$P(t)$$ such that $$P(t)$$, $$P^{-1}(t)$$ 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 $$B$$).

##### MSC:
 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34A30 Linear ordinary differential equations and systems 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
##### Keywords:
quasiperiodic function; reducible system; basic frequencies
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##### References:
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