On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point

This work focuses on the reducibility of the following real nonlinear analytical quasiperiodic system: ẋ=Ax+f(t,x,ϵ),x∈R2 where AA is a real 2×2 constant matrix, and f(t,0,ϵ)=O(ϵ)f(t,0,ϵ)=O(ϵ) and ∂xf(t,0,ϵ)=O(ϵ)∂xf(t,0,ϵ)=O(ϵ) as ϵ→0ϵ→0. With some non-resonant conditions of the frequencies with the eigenvalues of AA and without any nondegeneracy condition with respect to ϵϵ, by an affine analytic quasiperiodic transformation we change the system to a suitable norm form at the zero equilibrium for most of the sufficiently small perturbation parameter ϵϵ.