On reducibility for a class of n-dimensional quasi-periodic systems with a small parameter

In this paper we consider the reducibility of a class of n-dimensional real analytic quasi-periodic systems with a small parameter: x(over dot) - (A + epsilon Q(t,epsilon))x, x is an element of R-n We prove that if the basic frequencies of Q and the eigenvalues of Lambda satisfy some non-resonance conditions, then for most of the sufficiently small parameters in the sense of Lebesgue measure, the system is reducible without any non-degeneracy assumption with respect to the parameter. Moreover, under some assumptions, we obtain a similar result for nonlinear quasi-periodic systems. (C) 2015 Elsevier Inc. All rights reserved.