On the reducibility of analytic quasi-periodic systems with Liouvillean basic frequencies

Abstract

<p style='text-indent:20px;'>In this paper we consider the linear quasi-periodic system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \dot{x} = (A+\epsilon P(t)) x, x\in \mathbb{R}^{d}, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ A $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M2">\begin{document}$ d\times d $\end{document}</tex-math></inline-formula> constant matrix with elliptic type, <inline-formula><tex-math id="M3">\begin{document}$ P(t) $\end{document}</tex-math></inline-formula> is analytic quasi-periodic with respect to <inline-formula><tex-math id="M4">\begin{document}$ t $\end{document}</tex-math></inline-formula> with basic frequencies <inline-formula><tex-math id="M5">\begin{document}$ \omega = (1, \alpha), $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> being irrational, and <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is a small perturbation parameter. If some suitable non-resonant conditions and non-degeneracy conditions hold, and the basic frequencies satisfy that <inline-formula><tex-math id="M8">\begin{document}$ 0\leq \beta(\alpha) &lt; r, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M9">\begin{document}$ \beta(\alpha) = \limsup\limits_{n\rightarrow \infty}\frac{\ln q_{n+1}}{q_{n}}, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M10">\begin{document}$ q_{n} $\end{document}</tex-math></inline-formula> is the sequence of denominations of the best rational approximations for <inline-formula><tex-math id="M11">\begin{document}$ \alpha \in \mathbb{R} \setminus\mathbb{Q}, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M12">\begin{document}$ r $\end{document}</tex-math></inline-formula> is the initial radius of analytic domain, it is proved that for most sufficiently small <inline-formula><tex-math id="M13">\begin{document}$ \epsilon, $\end{document}</tex-math></inline-formula> this system can be reduced to a constant system <inline-formula><tex-math id="M14">\begin{document}$ \dot{x} = A^{*}x, x\in \mathbb{R}^{d}, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M15">\begin{document}$ A^{*} $\end{document}</tex-math></inline-formula> is a constant matrix close to <inline-formula><tex-math id="M16">\begin{document}$ A. $\end{document}</tex-math></inline-formula> As some applications, we apply our results to quasi-periodic Schrödinger equations with an external parameter to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.</p>