一类非自治微分方程的$C^0$共轭性
Sharpness of $C^0$ conjugacy for the non-autonomous differential equations with Lipschitzian perturbation
摘要：
经典的非自治微分方程的$C^0$线性化定理说明了非线性系统与其线性部分之间存在$C^0$拓扑共轭。也就是说，存在将非线性系统的解发送到其线性部分的解上的同胚（等价函数）$H$。证明了当非线性扰动为Lipschitz时，等价函数$H$及其逆函数$G=H^{-1}$都是Hölder连续的.问题：是否有可能提高规律性？规律性强吗？为了回答这个问题，我们构造了一个反例来表明等价函数$H$完全是Lipschitzian，但逆函数$G=H^{-1}$仅仅是Hölder连续的。此外，我们还提出了一个猜想，即这种同胚的正则性是尖锐的（它不能再被改进了）。证明了该猜想对线性压缩系统成立。此外，我们提出了特殊情况下的线性扰动，这是密切相关的频谱。
英文摘要：
The classical $C^0$ linearization theorem for the non-autonomous differential equations states the existence of a $C^0$ topological conjugacy between the nonlinear system and its linear part. That is, there exists a homeomorphism (equivalent function) $H$ sending the solutions of the nonlinear system onto those of its linear part. It is proved in the previous literature that the equivalent function $H$ and its inverse $G=H^{-1}$ are both Hölder continuous if the nonlinear perturbation is Lipschitzian. Questions: is it possible to improve the regularity? Is the regularity sharp? To answer this question, we construct a counterexample to show that the equivalent function $H$ is exactly Lipschitzian, but the inverse $G=H^{-1}$ is merely Hölder continuous. Furthermore, we propose a conjecture that such regularity of the homeomorphisms is sharp (it could not be improved anymore). We prove that the conjecture is true for the systems with linear contraction. Furthermore, we present the special cases of linear perturbation, which are closely related to the spectrum.
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