This article is cited in 48 papers

The classical KAM theory at the dawn of the twenty-first century

M. B. Sevryuk

Institute of Energy Problems of Chemical Physics, Russian Academy of Sciences

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.

MSC: Primary 37J40; Secondary 26E10, 58A10, 70H08

Received: June 22, 2002; in revised form October 22, 2002

Language: English

DOI: 10.17323/1609-4514-2003-3-3-1113-1144

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