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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 ω 1 ,,ω r if Q(t)=F(ω 1 t,,ω r t), where F=F(v 1 ,,v r ) is 2π periodic in all its arguments. The author considers the system (1) x ' =(A+εQ(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 𝒮(0,ε 0 ) (ε 0 >0) with positive Lebesgue measure such that for ε𝒮 (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:
34C20Transformation and reduction of ODE and systems, normal forms
34A30Linear ODE and systems, general
34C27Almost and pseudo-almost periodic solutions of ODE