Abstract
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.
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Communicated by Dmitry Dolgopyat.
The first author was supported by NSFC Grant 11401251. The second author was partially supported by the National Basic Research Program of China Grant 2013CB834100 and NSFC Grant 11171132. The third author was partially supported by NSF Grant DMS1109201, NSERC discovery Grant 1257749, a faculty development grant from the University of Alberta, and a Scholarship from Jilin University.
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Xu, L., Li, Y. & Yi, Y. Lower-Dimensional Tori in Multi-Scale, Nearly Integrable Hamiltonian Systems. Ann. Henri Poincaré 18, 53–83 (2017). https://doi.org/10.1007/s00023-016-0516-3
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DOI: https://doi.org/10.1007/s00023-016-0516-3