On the reducibility of two-dimensional quasi-periodic systems with Liouvillean basic frequencies and without non-degeneracy condition

Abstract

In this paper we consider the linear quasi-periodic systemx˙=(A+P(t,ϵ))x,x∈R2, where A is a 2×2 constant matrix with different eigenvalues, P(t,ϵ) is Cm(m=0,1)-smooth in ϵ and analytic quasi-periodic with respect to t with basic frequencies ω=(1,α), with α being irrational. Under some non-resonant conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy condition, and 0≤β(α)<r, where β(α)=limsupn→∞ln⁡qn+1qn, qn is the sequence of denominations of the best rational approximations for α∈R∖Q, r is the initial radius of analytic domain, it is proved that for many sufficiently small ϵ, this system can be reduced to a constant system x˙=A⁎x,x∈R2, where A⁎ is a constant matrix close to A. As some applications, we apply our results to quasi-periodic Schrödinger equations with an external parameter to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.