Lower-Dimensional Tori in Multi-Scale, Nearly Integrable Hamiltonian Systems

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• 作者：Lu Xu ; Yong Li ; Yingfei Yi
• 关键词：Mathematics Subject ClassificationPrimary 37J40 ; 70H08
• 刊名：Annales Henri Poincaré
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：18
• 期：1
• 页码：53-83
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Theoretical, Mathematical and Computational Physics; Dynamical Systems and Ergodic Theory; Quantum Physics; Mathematical Methods in Physics; Classical and Quantum Gravitation, Relativity Theory; Eleme
• 出版者：Springer International Publishing
• ISSN：1424-0661
• 卷排序：18

文摘

We consider a multi-scale, nearly integrable Hamiltonian system. With proper degeneracy involved, such a Hamiltonian system arises naturally in problems of celestial mechanics such as Kepler problems. Under suitable non-degenerate conditions of Bruno–Rüssmann type, the persistence of the majority of non-resonant, quasi-periodic invariant tori has been shown in Han et al. (Ann. Henri Poincaré 10(8):1419–1436, 2010). This paper is devoted to the study of splitting of resonant invariant tori and the persistence of certain class of lower-dimensional tori in the resonance zone. Similar to the case of standard nearly integrable Hamiltonian systems (Li and Yi in Math. Ann. 326:649–690, 2003, Proceedings of Equadiff 2003, World Scientific, 2005, pp 136–151, 2005), we show the persistence of the majority of Poincaré–Treschev non-degenerate, lower-dimensional invariant tori on a the given resonant surface corresponding to the highest order of scale. The proof uses normal form reductions and KAM method in a non-standard way. More precisely, due to the involvement of multi-scales, finite steps of KAM iterations need to be firstly performed to the normal form to raise the non-integrable perturbation to a sufficiently high order for the standard KAM scheme to carry over.