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

Part of: Qualitative theory Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  09 March 2020

DONGFENG ZHANG Open the ORCID record for DONGFENG ZHANG [Opens in a new window]  and
JUNXIANG XU
DONGFENG ZHANG
Affiliation:
School of Mathematics, Southeast University, Nanjing 210096, PR China email zhdf@seu.edu.cn, xujun@seu.edu.cn
JUNXIANG XU
Affiliation:
School of Mathematics, Southeast University, Nanjing 210096, PR China email zhdf@seu.edu.cn, xujun@seu.edu.cn
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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.

Keywords

reducibilityquasi-periodicKAM iterationLiouvillean frequencies

MSC classification

Primary: 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
Secondary: 34C27: Almost and pseudo-almost periodic solutions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1883 - 1920
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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