How a swing behaves
type: Item
instance of: scholarly article
title: How a swing behaves
zbMATH Open document ID: 0604.70045
DOI: 10.1016/0167-2789(86)90212-5
author: John M. Greene
published in: Physica D
publication date: 1986-01-01T00:00:00Z
review text: This paper is an advertisement for \textit{R. MacKay}'s thesis [Princeton University, Univ. Microfilm Intern., Ann Arbor, MI (order {\#}8301411)]. The present state of art of the theory of asymptotic behaviour of two conservative, coupled, nonlinear oscillators is dealt with. The author gives first a historical introduction, then ''the general nature'' of the problem is described. The role of area preserving maps in this theory is emphasized, for this purpose the swing with oscillating length is introduced as a model for explanation; the notion of KAM (Kolmogorov- Arnold-Moser) curves appears here too. The text shows further possibility in physical terms how the parametric instability (\(\equiv\) period doubling) the stability of periodic orbits destroys, and island structures disintegrate into chaos with increasing perturbation. Surface of the critical maps (which represents the transition from isolated islands to chaotic motion) is practically characterized. Notion of winding number and using of continued fractions is recapitulated. Importance of renormalization is described. Paper is a large collection of very interesting physical interpretations for some mathematical facts and methods and also a stimulus for mathematicians.
Mathematics Subject Classification ID: 70K20
Mathematics Subject Classification ID: 70K05
zbMATH DE Number: 3976441
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: conservative, coupled, nonlinear oscillators
zbMATH Keywords: oscillating length
zbMATH Keywords: parametric instability
zbMATH Keywords: period doubling
zbMATH Keywords: stability of periodic orbits
zbMATH Keywords: island structures
zbMATH Keywords: chaos
zbMATH Keywords: increasing perturbation
zbMATH Keywords: critical maps
zbMATH Keywords: transition from isolated islands to chaotic motion
zbMATH Keywords: winding number
zbMATH Keywords: continued fractions
zbMATH Keywords: renormalization
reviewed by: Á. P. Bosznay
MaRDI profile type: Publication
full work available at URL: 0167-2789(86)90212-5
OpenAlex ID: W1986368086
cites work: The universal metric properties of nonlinear transformations
cites work: Scaling for a Critical Kolmogorov-Arnold-Moser Trajectory
cites work: Q5524581
cites work: Regular and stochastic motion
cites work: Q3908994
description: scientific article; zbMATH DE number 3976441