Lower dimensional invariant tori in the regions of instability for nearly integrable Hamiltonian systems
type: Item
instance of: scholarly article
title: Lower dimensional invariant tori in the regions of instability for nearly integrable Hamiltonian systems
zbMATH Open document ID: 0935.70013
author: Chongqing Cheng
published in: Communications in Mathematical Physics
publication date: 2000-04-26T00:00:00Z
review text: The author considers Hamiltonian systems of more than two degrees of freedom, consisting of a non-degenerate integrable part and a perturbation. The frequencies of the unperturbed part satisfy a simple resonant condition on an \((n-1)\)-dimensional submanifold \(\mathbb{M}\) in the action space. Almost every point of this manifold corresponds to a resonant \(n\)-dimensional torus of the unperturbed system, which is foliated by \((n-1)\)-dimensional ergodic components. It is proved that for each resonant \(n\)-dimensional torus of a subset of \(\mathbb{M}\) with positive measure, at least two \((n-1)\)-dimensional tori are continued under the perturbation, one being hyperbolic and the other elliptic. This important result fills a gap between the Poincaré-Birkhoff fixed point theory on the continuation of isolated periodic orbits from the non-isolated ones of the fully resonant tori of the integrable part, and the KAM theory, which guarantees the conservation under the perturbation of the non-resonant tori with frequencies which satisfy certain diophantine condition.
reviewed by: Simos Ichtiaroglou
Mathematics Subject Classification ID: 70H14
Mathematics Subject Classification ID: 70H08
Mathematics Subject Classification ID: 70H09
Mathematics Subject Classification ID: 37N05
zbMATH DE Number: 1334929
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: invariant tori
zbMATH Keywords: perturbation
zbMATH Keywords: diophantine condition for frequencies
zbMATH Keywords: non-degenerate integrable part
zbMATH Keywords: Poincaré-Birkhoff fixed point theory
zbMATH Keywords: continuation of isolated periodic orbits
zbMATH Keywords: KAM theory
MaRDI profile type: Publication
full work available at URL: s002200050618
OpenAlex ID: W2032280651
DOI: 10.1007/S002200050618
description: scientific article; zbMATH DE number 1334929