On the reducibility of a class of finitely differentiable quasi-periodic linear systems

In this paper, we consider the following systemx˙=(A+εQ˜(t))x, where A   is a constant matrix with different eigenvalues, and Q˜(t) is quasi-periodic with frequencies ω1,ω2,…,ωrω1,ω2,…,ωr. Moreover, Q(θ)=Q(ωt)=Q˜(t) has continuous partial derivatives ∂bQ∂θjb for j=1,2,…,rj=1,2,…,r, where b>94r+1∈Z, and the moduli of continuity of ∂bQ∂θjb satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. Under suitable hypothesis of non-resonance conditions and non-degeneracy conditions, we prove that for most sufficiently small ε, the system can be reducible to a constant coefficient differentiable equation by means of a quasi-periodic homeomorphism.