拟周期系统的Floquet理论

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名

邮箱

手机号码

标题

留言内容

验证码

The Floquet Theory for Quasi-Periodic Linear System

摘要: 在这篇文章,我们对拟周期系统dx/dt=A(ω₁t,ω₂t.…,ω_mt)x (0.1)建立了Floquet理论.其中n×n方阵A(u₁,u₂,…,u_m)是u₁,u₂,…,u_m以2π为周期的周期方阵,同时假定A(u₁,u₂,…,u_m)∈C^τ,τ=(N+1)τ₀,τ₀=2(m+1),N=1/2n(n+1).我们定义了(0.1)的特征指数根β₁,β₂,…,β_n,假设下式成立:其中K(ω),K(ω,β)>0,k_μ,i_v是整数,k₁,k₂…,k_m不全为零:i²=-1.那末有拟周期线性变换,把(0.1)化为常系数的线性系统.

Abstract: In this paper we establish the Floguet theory for the quasi-periodic system where A(u₁,u₂...u_m) is an nxn periodic matrix function ofwith period 2π, and it is of C^τ, τ=(N+1)τ₀, τ₀=2(m+1), N = (1/2)n(n+l).Meanwhile, we define the characteristic exponential roots β₁,β₂,…,β_n of (0.1), and assume that where K(ω), K(ω,β)->0. k_u, i_v, are integers, all the integers k₁,k₂,…,k_m,k_m are not zero, ,rhen therequasi-periodic linear transformation, which carries (0.1) into a linear system with constant coefficients.

[1]	Arnol'd,V.I.,Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics,Uspekni Mat.Nauk.USSR.18(1963),91-192.
[2]	Moser,J.,On the theory of quasi-periodic motions,SIAM.Rev.8(1966),145-172.
[3]	Moser,J.,Convergent series expansion of quasi-periodic motions,Math.Ann.,169(1967),136-176.
[4]	Giacaglia,G.E.O.,Perturbation Method in Nonlinear Systems,Springer,New York,Berlin,(1972).
[5]	Diliberto,S.P.,On Systems of Ordinary Differential Equations,Contributions to the Theory of Nonlinear Oscillations,Vol.I,(1950),1-48.