Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition
type: Item
instance of: scholarly article
title: Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition
zbMATH Open document ID: 1031.37053
DOI: 10.1016/S0021-7824(01)01221-1
author: Junxiang Xu
author: Jiangong You
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2002-07-24T00:00:00Z
review text: The dynamics of integrable Hamiltonian systems is simple, since all the compact energy surfaces are foliated by invariant tori which carry quasiperiodic motions. There are not many examples of this kind of systems (this is an active field of research); however, the KAM theory proves that for all Hamiltonian systems near a nondegenerate integrable Hamiltonian system the quasiperiodic motions in invariant tori are typical. In the middle of the 1960's, Melnikov founded a KAM persistence result for lower-dimensional tori, satisfying the now called first and second Melnikov's non-resonance conditions. In the present paper, the authors prove the persistence of lower-dimensional tori without assuming the second Melnikov's non-resonance condition. They also prove that this result is optimal in the sense that they are able to construct an example showing that the first Melnikov's non-resonance condition is necessary.
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 70H08
zbMATH DE Number: 1770825
zbMATH Keywords: KAM theory
zbMATH Keywords: integrable Hamiltonian systems
zbMATH Keywords: persistence of invariant tori
reviewed by: Manuel de León
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1770825