Abstract:
In this paper we develop a new KAM technique to prove two general KAM theorems for nearly integrable Hamiltonian systems without assuming any nondegeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some nondegeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies.
Keywords:
Hamiltonian system, KAM iteration, invariant tori, nondegeneracy condition.
Citation:
Junxiang Xu, Xuezhu Lu, “General KAM Theorems and their Applications to Invariant Tori with Prescribed Frequencies”, Regul. Chaotic Dyn., 21:1 (2016), 107–125
\Bibitem{XuLu16}
\by Junxiang Xu, Xuezhu Lu
\paper General KAM Theorems and their Applications to Invariant Tori with Prescribed Frequencies
\jour Regul. Chaotic Dyn.
\yr 2016
\vol 21
\issue 1
\pages 107--125
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\crossref{https://doi.org/10.1134/S1560354716010068}
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Linking options:
https://www.mathnet.ru/eng/rcd69
https://www.mathnet.ru/eng/rcd/v21/i1/p107
This publication is cited in the following 9 articles:
Qi Li, Junxiang Xu, “A Formal KAM Theorem for Hamiltonian Systems and Its Application to Hyperbolic Lower Dimensional Invariant Tori”, Qual. Theory Dyn. Syst., 23:2 (2024)
Jiayin Du, Yong Li, Hongkun Zhang, “Kolmogorov's theorem for degenerate Hamiltonian systems with Hölder continuous parameters”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2024, 1
Zhichao Ma, Junxiang Xu, “A KAM theorem for quasi-periodic non-twist mappings and its application”, DCDS, 42:7 (2022), 3169
J. Xu, “Persistence of lower dimensional degenerate invariant tori with prescribed frequencies in Hamiltonian systems with small parameter”, Nonlinearity, 34:12 (2021), 8192–8247
M. B. Sevryuk, “Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2”, J Math Sci, 253:5 (2021), 730
K. Wang, J. Xu, “Existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions”, Front. Math. China, 15:3 (2020), 571–591
Sh. Jiang, “A KAM theorem for lower dimensional elliptic invariant tori of nearly integrable symplectic mappings”, J. Funct. space, 2017, 3719395
M. B. Sevryuk, “Chastichnoe sokhranenie chastot i pokazatelei Floke invariantnykh torov v obratimom kontekste 2 teorii KAM”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 63, no. 3, Rossiiskii universitet druzhby narodov, M., 2017, 516–541
J. Xu, W. Kun, Zh. Min, “On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters”, Proc. Amer. Math. Soc., 144:11 (2016), 4793–4805
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