On the reducibility of linear differential equations with quasiperiodic coefficients

Abstract

The system x ̇ = (A + εQ(t))x in R d is considered, where A is a constant matrix and Q a quasiperiodic analytic matrix with r basic frequencies. The eigenvalues of A are arbitrary including the purely imaginary case. Suppose that the set formed by the eigenvalues of A and the basic frequencies of Q satisfies a nonresonant condition. Then there is a positive measure cantorian set E such that for ε ϵ E the system is reducible to constant coefficients by means of a quasiperiodic change of variables, provided a nondegeneracy condition holds. This condition prevents locking at resonance.