On the persistence of lower dimensional invariant tori under quasi-periodic perturbations
type: Item
instance of: scholarly article
title: On the persistence of lower dimensional invariant tori under quasi-periodic perturbations
zbMATH Open document ID: 0898.58044
published in: Journal of Nonlinear Science
publication date: 1998-10-25T00:00:00Z
full work available at URL: http://hdl.handle.net/2117/1218
review text: A class of perturbed Hamiltonian systems is studied which can roughly be characterised as follows. The integrable, unperturbed Hamiltonian gives rise to invariant tori of dimension lower than the number of degrees of freedom and these are not contained in a resonant higher-dimensional torus. The (analytic) perturbation is time-dependent, but quasi-periodic. The issue, of course, is the study of survival of the lower dimensional tori under suitable Diophantine-type conditions. Two ways of estimating the measure of destroyed tori are presented. The first one corresponds to the study of persistence of a single torus, when the size of the perturbation is used as a parameter. In the second approach, with fixed size of the perturbation parameter, existence of invariant tori is proved for every vector of frequencies close to the unperturbed one, except for a set of frequencies of measure exponentially small with respect to the distance to the unperturbed torus. One of the extra assumptions is that the linear variational equations around the initial torus can be reduced to equations with constant coefficients (as in Floquet's theorem when the torus is a periodic orbit). A couple of applications are discussed to specific situations in the restricted three-body problem of celestial mechanics. As is customary with KAM-theory, the proofs are quite technical, but the first part of the paper extensively discusses the general ideas.
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37C55
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 34C99
zbMATH DE Number: 1129284
zbMATH Keywords: KAM theory
zbMATH Keywords: lower-dimensional tori
zbMATH Keywords: quasi-periodic perturbations
author: Àngel Jorba
author: Jordi Villanueva
reviewed by: W. Sarlet
MaRDI profile type: Publication
OpenAlex ID: W1963992485
DOI: 10.1007/S003329900036
description: scientific article; zbMATH DE number 1129284