Abstract:
In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno–Rüssmann condition, in real-analytic non-degenerate Hamiltonian systems close to integrable. The proof, which uses rational approximations instead of small divisors estimates, is an adaptation to the Hamiltonian setting of the method we introduced in [4] for perturbations of constant vector fields on the torus.
Keywords:
perturbation of integrable Hamiltonian systems, KAM theory, Diophantine duality, periodic approximations.
Citation:
Abed Bounemoura, Stéphane Fischler, “The Classical KAM Theorem for Hamiltonian Systems via Rational Approximations”, Regul. Chaotic Dyn., 19:2 (2014), 251–265
\Bibitem{BouFis14}
\by Abed~Bounemoura, St\'ephane~Fischler
\paper The Classical KAM Theorem for Hamiltonian Systems via Rational Approximations
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 2
\pages 251--265
\mathnet{http://mi.mathnet.ru/rcd134}
\crossref{https://doi.org/10.1134/S1560354714020087}
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\zmath{https://zbmath.org/?q=an:1339.37042}
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Linking options:
https://www.mathnet.ru/eng/rcd134
https://www.mathnet.ru/eng/rcd/v19/i2/p251
This publication is cited in the following 6 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)
M.S. Santhanam, Sanku Paul, J. Bharathi Kannan, “Quantum kicked rotor and its variants: Chaos, localization and beyond”, Physics Reports, 956 (2022), 1
Bounemoura A., Fejoz J., “Hamiltonian Perturbation Theory For Ultra-Differentiable Functions”, Mem. Am. Math. Soc., 270:1319 (2021), 1+
D. Zhang, X. Xu, “On the linearization of vector fields on a torus with prescribed frequency”, Topol. Methods Nonlinear Anal., 54:2 (2019), 649–663
D. Zhang, J. Xu, H. Wu, “On invariant tori with prescribed frequency in Hamiltonian systems”, Adv. Nonlinear Stud., 16:4 (2016), 719–735
F. Cong, J. Hong, H. Li, “Quasi-effective stability for nearly integrable Hamiltonian systems”, Discrete Contin. Dyn. Syst.-Ser. B, 21:1 (2016), 67–80
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