Effective reducibility of quasi-periodic linear equations close to constant coefficients

Let us consider the differential equation x = (A + epsilon Q(t,epsilon))x, less than or equal to epsilon(0), where A is an elliptic constant matrix and Q depends on time in a quasi-periodic (and analytic) way. It is also assumed that the eigenvalues of A and the basic frequencies of Q satisfy a diophantine condition. Then it is proved that this system can be reduced to y = (A*(epsilon) + epsilon R*(t,epsilon))y, less than or equal to epsilon(0), where R* is exponentially small in epsilon, and the Linear change of variables that performs such a reduction is also quasi-periodic with the same basic frequencies as Q. The results are illustrated and discussed in a practical example.