Extension of Floquet's theory to nonlinear quasiperiodic differential equations

摘要

In this paper, we consider the following autonomous system of differential equations: x = Ax + f(x,θ), θ = ω, where θ ∈ R~m, ω = (ω_1,…,ω_m) ∈ R~m, x ∈ R~n, A ∈ R~(n x 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: 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.