Effective Reducibility of Quasi-Periodic Linear Equations close to Constant Coefficients

Abstract

Let us consider the differential equation $$ \dot{x}=(A+\varepsilon Q(t,\varepsilon))x, \;\;\;\; |\varepsilon|\le\varepsilon_0, $$ 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 $$ \dot{y}=(A^{*}(\varepsilon)+\varepsilon R^{*}(t,\varepsilon))y, \;\;\;\; |\varepsilon|\le\varepsilon_0, $$ where $R^{*}$ is exponentially small in $\varepsilon$, 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.