A note on the reducibility of linear differential equations with quasiperiodic coefficients
The paper deals with the \(n\)-dimensional linear differential system \(dx/dt=(A +\varepsilon Q(t))x\), where \(A\) is a constant matrix, \(Q(t)\) an analytic quasiperiodic matrix, and \(\varepsilon>0\) is sufficiently small. In analogy to the Floquet theory, the question is whether there exists a quasiperiodic transformation \(x=P(t)y\) such that \(y\) satisfies an equation \((*)\) \(dy/dt=By\) with constant coefficients. The authors refer to a previous paper by Jorba and Simo in which this question was answered affirmatively for an \(\varepsilon\)-Cantor set, if the frequencies of \(Q (t)\) and the eigenvalues of \(A\) satisfy some nonresonance condition, and if, moreover, the eigenvalues of the time average of \(A+\varepsilon Q(t)\) have some additional property. In the present paper, a complementary result yielding \((*)\) is proved: The matrix \(A\) and \(\varepsilon\) can be arbitrarily, while the set of admissible frequencies of \(Q(t)\) has to be slightly restricted in order to avoid small divisors during the iteration process applied.
scientific article; zbMATH DE number 2012666