Extension of floquet’s theory to nonlinear quasiperiodic differential equations

Abstract

In this paper, we consider the following autonomous system of differential equations: $\dot x = Ax + f(x,\theta ), \dot \theta = \omega$. where θ ∈ ℝ^m, ω=(ω₁...,ω_m) ∈ ℝ^m, x ∈ ℝⁿ, A ∈ ℝ^n×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: $\dot x = Ax + g(x), \dot \theta = \omega$ Additionally, the proof of this paper naturally implies the extension of Chen’s theory in the quasiperiodic case.