Effective stability and KAM theory
type: Item
instance of: scholarly article
title: Effective stability and KAM theory
zbMATH Open document ID: 0858.58012
author: Amadeu Delshams
author: Pere Gutiérrez
published in: Journal of Differential Equations
publication date: 1996-09-18T00:00:00Z
full work available at URL: http://hdl.handle.net/2117/785
review text: This paper revisits two main stability results for nearly-integrable Hamiltonian systems. Nekhoroshev's theorem provides exponential lower bounds for the stability time. The theorem of Kolmogorov, Arnold and Moser (KAM), on the other hand, establishes conditions for the preservation of a majority of the nonresonant invariant tori. The former stability may be called ``effective stability'', and the latter ``perpetual stability''. To stress the relationships and interconnections between these two theorems, the authors provide proofs for both theorems using a common approach. The core of this approach consists of bringing the system to a normal form via Lie series methods. The authors' proof of Nekhoroshev's theorem provides an optimal stability exponent for quasi-convex systems. The authors' method provides a direct proof of the isoenergetic KAM theorem. In addition, a modification of this proof leads to the concept of the nearly-invariant torus, which connects the KAM and Nekhoroshev results more directly.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37C75
zbMATH DE Number: 909999
zbMATH Keywords: KAM theory
zbMATH Keywords: invariant tori
zbMATH Keywords: Lie series methods
zbMATH Keywords: Nekhoroshev's theorem
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
OpenAlex ID: W2088516761
DOI: 10.1006/JDEQ.1996.0102
description: scientific article; zbMATH DE number 909999