Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies

Abstract

In this paper we consider the following nonlinear quasi-periodic system:$$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $A$ is a $d\times d$ constant matrix of elliptic type, $\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$ is a small perturbation with $\unicode[STIX]{x1D716}$ as a small parameter, $h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$ as $x\rightarrow 0$, and $P,g$ and $h$ are all analytic quasi-periodic in $t$ with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$, where $\unicode[STIX]{x1D6FC}$ is irrational. It is proved that for most sufficiently small $\unicode[STIX]{x1D716}$, the system is reducible to the following form: $$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$ is a high-order term. Therefore, the system has a quasi-periodic solution with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$, such that it goes to zero when $\unicode[STIX]{x1D716}$ does.