02-297 Sevryuk M.B.

The classical KAM theory at the dawn of the twenty-first century (347K, PostScript) Jul 8, 02

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Abstract. We survey several recent achievements in the 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 the weak nondegeneracy conditions, Gevrey smoothness of families of perturbed invariant tori, the ``exponential condensation'' of perturbed tori, destruction mechanisms of the resonant unperturbed tori, the 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. The paper contains 152 references.

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