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On the reducibility of linear quasi-periodic systems with Liouvillean basic frequencies and multiple eigenvalues
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2020-12-01 , DOI: 10.1016/j.jde.2020.07.025
Dongfeng Zhang , Junxiang Xu , Hao Wu , Xindong Xu

Abstract In this paper we consider the linear quasi-periodic system x ˙ = ( A + ϵ P ( t ) ) x , x ∈ R d , where A is a d × d constant matrix of elliptic type and has multiple eigenvalues, P ( t ) is analytic quasi-periodic with respect to t with basic frequencies ω = ( 1 , α ) , where α is irrational, and ϵ is a small perturbation parameter. Under suitable non-resonant condition, non-degeneracy condition and 0 ≤ β ( α ) r ⁎ , where β ( α ) = lim sup n → ∞ ln ⁡ q n + 1 q n , q n is the sequence of denominations of the best rational approximations for α ∈ R ∖ Q , 0 r ⁎ r , r is the initial radius of analytic strip, it is proved that for most sufficiently small ϵ, this system can be reduced to a constant system x ˙ = A ⁎ x , where A ⁎ is a constant matrix close to A. As some applications, we apply our results to quasi-periodic Hill's equations, three dimensional skew symmetric systems and n coupled Schrodinger equations to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.

中文翻译:

关于具有刘维尔基本频率和多个特征值的线性准周期系统的可约性

摘要 本文考虑线性拟周期系统 x ˙ = ( A + ϵ P ( t ) ) x , x ∈ R d ,其中 A 为 ad × d 椭圆型常数矩阵,具有多个特征值,P ( t ) 是关于 t 的解析准周期,基本频率 ω = ( 1 , α ) ,其中 α 是无理数, ϵ 是一个小的扰动参数。在合适的非共振条件、非简并条件和 0 ≤ β ( α ) r ⁎ ,其中 β ( α ) = lim sup n → ∞ ln ⁡ qn + 1 qn , qn 是最佳有理近似的面额序列对于 α ∈ R ∖ Q , 0 r ⁎ r , r 是解析带的初始半径,证明对于最足够小的 ϵ,该系统可以简化为常数系统 x ˙ = A ⁎ x ,其中 A ⁎是接近 A 的常数矩阵。 作为一些应用,我们将我们的结果应用于准周期希尔方程,
更新日期:2020-12-01
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