Extension of Floquet's theory to nonlinear quasiperiodic differential equations

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.

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 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.