KAM理论中光滑性问题的研究

KAM theory is the mathematical theory and methods founded and developed by the mathematicians Kolmogorov, Arnold and Moser. It deals with the problems of small divisors and the small perturbation of conservative systems, such as Hamiltonian systems and reversible systems and so on. KAM theory is related with astronomy and physics. So there are important theoretical and practical value for the study of KAM theory. Applying some techniques and methods in KAM theory, in this item, we discuss the following problems： the reducibility of finitely differentiable nonlinear quasi-periodic systems; the lower dimensional KAM tori of finitely differentiable reversible systems and Hamiltonian systems; the effective reducibility of a class of linear quasi-periodic Hamiltonian systems. We will expand KAM theory, get the new KAM iteration methods, and obtain the valuable results.

KAM 理论是由数学家Kolmogorov，Arnold 和Moser建立并发展起来的数学理论方法，它解决了小分母问题以及哈密顿系统和可逆系统等保守系统的小扰动问题。KAM理论和天文，物理等领域密切相关。因此，KAM理论的研究具有重要的理论和应用意义。本课题主要利用KAM理论方法，研究有限光滑的非线性拟周期系统的约化性；研究有限光滑的可逆系统和哈密顿系统的低维KAM环面；研究线性拟周期哈密顿系统的有效约化性，取得有意义的结论。

本项目主要利用KAM理论方法，研究有限光滑的可逆系统的低维双曲不变环面；研究哈密顿系统关于固定频率的低维双曲不变环面的保持性；研究具有重特征值的有限光滑的线性拟周期系统的约化性。