Reducibility of ultra-differentiable quasi-periodic linear systems

Abstract

In ultra-differentiable classes, this paper studies the reducibility of the quasi-periodic linear system $\dot{x}=(A+\epsilon Q\left(t\right))x,x\in R^d$, where $A$ is a constant matrix with different eigenvalues $\lambda =({\lambda }_1,{\lambda }_2,\text{…},{\lambda }_d)$, $Q\left(t\right)$ is a ultra-differentiable quasi-periodic matrix with $r$ basic frequencies $\omega =({\omega }_1,{\omega }_2,\text{…},{\omega }_r)$ and $\epsilon$ is a small perturbation parameter. Suppose that the set formed by the eigenvalues of A and the basic frequencies of Q satisfies a non-resonant condition. Then, it is proved that the linear system can be conjugated to a constant system by a quasi-periodic change of variables.