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Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies
Ergodic Theory and Dynamical Systems ( IF 0.8 ) Pub Date : 2020-03-09 , DOI: 10.1017/etds.2020.23 DONGFENG ZHANG , JUNXIANG XU
Ergodic Theory and Dynamical Systems ( IF 0.8 ) Pub Date : 2020-03-09 , DOI: 10.1017/etds.2020.23 DONGFENG ZHANG , JUNXIANG XU
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.
中文翻译:
一类具有刘维尔基本频率的非线性准周期系统的可约性
在本文中,我们考虑以下非线性准周期系统:$$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g( t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ 在哪里$澳元 是一个$d\倍 d$ 椭圆型常数矩阵,$\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$ 是一个小扰动$\unicode[STIX]{x1D716}$ 作为一个小参数,$h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$ 作为$x\右箭头 0$ , 和$P,g$ 和$h$ 都是解析准周期的$t$ 基本频率$\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$ , 在哪里$\unicode[STIX]{x1D6FC}$ 是不合理的。证明对于大多数足够小的$\unicode[STIX]{x1D716}$ ,系统可简化为以下形式:$$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\四元 x\in \mathbb{R}^{d},\end{eqnarray}$$ 在哪里$h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$ 是高阶项。因此,系统有一个具有基本频率的准周期解$\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$ , 这样当它变为零时$\unicode[STIX]{x1D716}$ 做。
更新日期:2020-03-09
中文翻译:
一类具有刘维尔基本频率的非线性准周期系统的可约性
在本文中,我们考虑以下非线性准周期系统: