Abstract
This paper is a continuation to our work (Xu et al. in Ann Henri Poincaré 18(1):53–83, 2017) concerning the persistence of lower-dimensional tori on resonant surfaces of a multi-scale, nearly integrable Hamiltonian system. This type of systems, being properly degenerate, arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem ties closely to the stability of the systems. For such a system, under certain non-degenerate conditions of Rüssmann type, the majority persistence of non-resonant tori and the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on a resonant surface corresponding to the highest order of scale is proved in Han et al. (Ann Henri Poincaré 10(8):1419–1436, 2010) and Xu et al. (2017), respectively. In this work, we consider a resonant surface corresponding to any intermediate order of scale and show the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on the resonant surface. The proof is based on a normal form reduction which consists of a finite step of KAM iterations in pushing the non-integrable perturbation to a sufficiently high order and the splitting of resonant tori on the resonant surface according to the Poincaré–Treshchev mechanism.
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Communicated by Paul Newton.
Lu Xu was partially supported by NSFC Grant 11401251 and a visiting scholar program from Jilin University and CSC. Yong Li was partially supported by National Basic Research Program of China Grant 2013CB834100 and NSFC Grant 11171132, 11571065. Yingfei Yi was partially supported by 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. Poincaré–Treshchev Mechanism in Multi-scale, Nearly Integrable Hamiltonian Systems. J Nonlinear Sci 28, 337–369 (2018). https://doi.org/10.1007/s00332-017-9410-5
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DOI: https://doi.org/10.1007/s00332-017-9410-5
Keywords
- Multi-scale Hamiltonian systems
- High-order proper degeneracy
- Resonant tori
- Lower-dimensional tori
- KAM theory