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Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 6 / June 2021
- Published online by Cambridge University Press:
- 09 March 2020, pp. 1883-1920
- Print publication:
- June 2021
-
- Article
- Export citation
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In this paper we consider the following nonlinear quasi-periodic system:
where
$$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$
$A$ 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: where
$$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$
$h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$ 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.
