Abstract:
We survey several recent achievements in KAM theory. The achievements chosen pertain to Hamiltonian systems only and are closely connected with the content of Kolmogorov's original theorem of 1954. They include weak non-degeneracy conditions, Gevrey smoothness of families of perturbed invariant tori, “exponential condensation” of perturbed tori, destruction mechanisms of resonant unperturbed tori, excitation of the elliptic normal modes of the unperturbed tori, and “atropic” invariant tori (i.e., tori that are neither isotropic nor coisotropic). The exposition is informal and nontechnical, and, as a rule, the methods of proofs are not discussed.
Key words and phrases:
KAM theory, Hamiltonian systems, invariant tori, quasi-periodic motions.
Received:June 22, 2002; in revised form October 22, 2002
\Bibitem{Sev03}
\by M.~B.~Sevryuk
\paper The classical KAM theory at the dawn of the twenty-first century
\jour Mosc. Math.~J.
\yr 2003
\vol 3
\issue 3
\pages 1113--1144
\mathnet{http://mi.mathnet.ru/mmj124}
\crossref{https://doi.org/10.17323/1609-4514-2003-3-3-1113-1144}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2078576}
\zmath{https://zbmath.org/?q=an:1121.37045}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000208594300017}
Linking options:
https://www.mathnet.ru/eng/mmj124
https://www.mathnet.ru/eng/mmj/v3/i3/p1113
This publication is cited in the following 47 articles:
Weichao Qian, Xue Yang, Yong Li, “Partial Frequency and Frequency Ratio in Multiscale KAM Formulism”, J Dyn Diff Equat, 2025
Weichao Qian, “KAM THEOREM AND ISO-ENERGETIC KAM THEOREM ON POISSON MANIFOLD”, jaac, 13:2 (2023), 1088
Luigi Chierchia, Michela Procesi, Encyclopedia of Complexity and Systems Science Series, Perturbation Theory, 2022, 247
Francesco Fassò, Encyclopedia of Complexity and Systems Science Series, Perturbation Theory, 2022, 307
Francesco Fassò, Encyclopedia of Complexity and Systems Science, 2022, 1
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Luigi Chierchia, Michela Procesi, Encyclopedia of Complexity and Systems Science, 2018, 1
Mikhail B. Sevryuk, “Families of Invariant Tori in KAM Theory: Interplay of Integer Characteristics”, Regul. Chaotic Dyn., 22:6 (2017), 603–615
Bounemoura A., “Non-Degenerate Liouville Tori Are Kam Stable”, Adv. Math., 292 (2016), 42–51
Abed Bounemoura, “Generic Perturbations of Linear Integrable Hamiltonian Systems”, Regul. Chaotic Dyn., 21:6 (2016), 665–681
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Heinz Hanßmann, Encyclopedia of Complexity and Systems Science, 2015, 1
Abed Bounemoura, “A KAM theorem through Dirichlet's box and Khintchine's transference principles”, Mosc. Math. J., 14:4 (2014), 697–709
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Wiggins S., Mancho A.M., “Barriers To Transport in Aperiodically Time-Dependent Two-Dimensional Velocity Fields: Nekhoroshev'S Theorem and “Nearly Invariant” Tori”, Nonlinear Process Geophys., 21:1 (2014), 165–185
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