A point-wise criterion for quasi-periodic motions in the KAM theory

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Abstract

We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency vector, without any nondegeneracy of the integrable part. We reconstruct the KAM theorem under Rssmann's nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition. Our point-wise version is of the form analogous to the corresponding problems for the integrable case. We compare our framework with the standard KAM theorem through a brief review of the KAM theory.

Original languageEnglish
Pages (from-to)3151-3161
Number of pages11
JournalNonlinear Analysis, Theory, Methods and Applications
Volume73
Issue number10
DOIs
Publication statusPublished - 2010 Nov 15
Externally publishedYes

Keywords

  • KAM theory
  • Nearly integrable Hamiltonian systems
  • Quasi-periodic motions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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