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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 \(B\)).

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
Full Text: DOI
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