Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies

In this paper we consider the following nonlinear quasi-periodic system: (x) over dot = (A + epsilon P(t, epsilon))x + epsilon g(t, epsilon) + h(x, t, epsilon), x is an element of R-d; where A is a d x d constant matrix of elliptic type, epsilon g(t, epsilon) is a small perturbation with epsilon as a small parameter, h(x, t, epsilon) = O(x(2)) as x -> 0, and P, g and h are all analytic quasi-periodic in t with basic frequencies omega = (1, alpha), where alpha is irrational. It is proved that for most sufficiently small epsilon, the system is reducible to the following form: (x) over dot = (A + B-*(t))x + h(*)(x, t, epsilon) , x is an element of R-d, where h(*)(x, t, epsilon) = O(x(2)) (x -> 0) is a high-order term. Therefore, the system has a quasi-periodic solution with basic frequencies omega = (1, alpha), such that it goes to zero when epsilon does.