The Classical KAM Theory at the Dawn of the Twenty-First Century
The Classical KAM Theory at the Dawn of the Twenty-First Century
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DOI：
10.17323/1609-4514-2003-3-3-1113-1144
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发表时间：
2003
影响因子：
0.8
通讯作者：
M. Sevryuk;M. Sevryuk
中科院分区：
文献类型：
--
作者：
M. Sevryuk;M. Sevryuk
本文综述了近年来在KAM理论方面取得的一些成果。所选的成果只与哈密顿系有关，并且与1954年柯尔莫哥洛夫原定理的内容密切相关。它们包括弱非简并条件、摄动不变环面族的Gevrey平滑性、摄动不变环面的“指数凝聚”、共振不变环面的破坏机制、不变环面椭圆正规模态的激发和“热带”不变环面（即既非各向同性也非共向同性的环面）。论述是非正式的和非技术的，而且，作为一个规则，证明的方法不讨论。2000年数学。主题。类。主要37 j40;二级26E10, 58A10, 70H08。
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. 2000 Math. Subj. Class. Primary 37J40; Secondary 26E10, 58A10, 70H08.