Effective Reducibility for a Class of Linear Almost Periodic Systems

This paper studies the effective reducibility of the linear almost periodic equation
\begin{equation*} \dot{x}=(A+\varepsilon Q(t,\varepsilon))x,~|\varepsilon|\le\varepsilon_0, \end{equation*}
where $Q(t)$ is analytic and almost periodic on $D_\rho$ and $A$ is a constant matrix. By an almost periodic transformation, without any nondegeneracy condition, under nonresonance conditions, the system is reduced to an almost periodic system
\begin{equation*} \dot{y}=(A^*(\varepsilon)+\varepsilon R^*(t,\varepsilon))y,~|\varepsilon|\le\varepsilon_0, \end{equation*}
where $R^*$ is small with respect to $\varepsilon$ (i.e., $\lim\limits_{\varepsilon\rightarrow 0} R^*(t,\varepsilon)=0$).