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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Normal form and linearization for quasiperiodic systems

Authors: Shui-Nee Chow, Kening Lu and Yun Qiu Shen
Journal: Trans. Amer. Math. Soc. 331 (1992), 361-376
MSC: Primary 34C20; Secondary 58F36, 70H05
MathSciNet review: 1076612
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Abstract: In this paper, we consider the following system of differential equations:

$\displaystyle \dot \theta = \omega + \Theta (\theta,z), \quad \dot z = Az + f(\theta,z),$

where $ \theta \in {C^m}$, $ \omega = ({\omega _1}, \ldots,{\omega _m}) \in {R^m}$, $ z \in {C^n}$, $ A$ is a diagonalizable matrix, $ f$ and $ \Theta $ are analytic functions in both variables and $ 2\pi $-periodic in each component of the vector $ \theta,\Theta = O(\vert z\vert)$ and $ f = O(\vert z{\vert^2})$ as $ z \to 0$. We study the normal form of this system of the equations and prove that this system can be transformed to a system of linear equations

$\displaystyle \dot \theta = \omega, \quad \dot z = Az$

by an analytic transformation provided that the eigenvalues of $ A$ and the frequency $ \omega $ satisfy certain small-divisor conditions.

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Article copyright: © Copyright 1992 American Mathematical Society