Reducibility of ultra‐differentiable quasi‐periodic linear systems

Xiangyuan Zhang and Dongfeng Zhang

Mathematische Nachrichten, 2025, vol. 298, issue 5, 1482-1495

Abstract: In ultra‐differentiable classes, this paper studies the reducibility of the quasi‐periodic linear system ẋ=(A+εQ(t))x,x∈Rd$\dot{x}=(A+\varepsilon Q(t))x,x\in \mathbb {R}^{d}$, where A$A$ is a constant matrix with different eigenvalues λ=(λ1,λ2,…,λd)$\lambda =(\lambda _{1},\lambda _{2},\ldots,\lambda _{d})$, Q(t)$Q(t)$ is a ultra‐differentiable quasi‐periodic matrix with r$r$ basic frequencies ω=(ω1,ω2,…,ωr)$\omega =(\omega _{1},\omega _{2},\ldots,\omega _{r})$ and ε$\varepsilon$ 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.

Date: 2025
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