Effective Reducibility of Quasi-periodic Linear Equations Close to Constant Coefficients∗

Let us consider the differential equation ẋ = (A+ εQ(t, ε))x, |ε| ≤ ε0, 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 ẏ = (A∗(ε) + εR∗(t, ε))y, |ε| ≤ ε0, where R∗ is exponentially small in ε, 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.