| Extension of Floquet's theory to nonlinear quasiperiodic differential equations |
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| 引用本文： | WU Hao & LI Weigu School of Mathematical Sciences,Peking University,Beijing 100871,China. Extension of Floquet's theory to nonlinear quasiperiodic differential equations[J]. 中国科学A辑(英文版), 2005, 48(12): 1670-1682. DOI: 10.1360/04ys0248 |
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| 作者姓名： | WU Hao & LI Weigu School of Mathematical Sciences Peking University Beijing 100871 China |
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| 作者单位： | WU Hao & LI Weigu School of Mathematical Sciences,Peking University,Beijing 100871,China |
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| 摘 要： | In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.
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Extension of floquet’s theory to nonlinear quasiperiodic differential equations |
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| Wu?HaoEmail author,Li?Weigu. Extension of floquet’s theory to nonlinear quasiperiodic differential equations[J]. Science in China(Mathematics), 2005, 48(12): 1670-1682. DOI: 10.1360/04ys0248 |
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| Authors: | Wu?Hao author-information" > author-information__contact u-icon-before" > mailto:wuhao@math.pku.edu.cn" title=" wuhao@math.pku.edu.cn" itemprop=" email" data-track=" click" data-track-action=" Email author" data-track-label=" " >Email author,Li?Weigu |
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| Affiliation: | School of Mathematical Sciences,Peking University,Beijing 100871,China |
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| Abstract: | In this paper, we consider the following autonomous system of differential equations:  . where θ ∈ ℝm, ω=(ω1...,ωm) ∈ ℝm, x ∈ ℝn, A ∈ ℝn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector θ which satisfies f-O(‖x‖2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system:  Additionally, the proof of this paper naturally implies the extension of Chen’s theory in the quasiperiodic case. |
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| Keywords: | quasiperiodic system normal form. |
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