Break-up of the spiral mean torus in a volume-preserving map
type: Item
instance of: scholarly article
title: Break-up of the spiral mean torus in a volume-preserving map
zbMATH Open document ID: 0757.58024
DOI: 10.1016/0960-0779(92)90007-A
published in: Chaos, Solitons and Fractals
publication date: 1992-09-27T00:00:00Z
review text: This paper is a sketch of an investigation of the volume-preserving map \[ \begin{pmatrix} y_{n+1}\\x_{n+1}\\z_{n+1}\end{pmatrix}=T\begin{pmatrix} y_ n\\x_ n\\ z_ n\end{pmatrix}=\begin{pmatrix} y_ n-(K+\varepsilon\cos z_ n)\sin x_ n\\x_ n+y_{n+1}\\z_ n+z\pi r_ 2\end{pmatrix}. \] Each orbit of this map can be labelled by a pair of frequencies \(r_ 1=\lim_{n\to \infty}(x_ n-x_ 0)/2\pi n\) and \(r_ 2=\lim_{n\to\infty}(z_ n-z_ 0)/2\pi n\). The authors consider the map with \(r_ 1=1/\theta^ 2\) and \(r_ 2=1/\theta\) where \(\theta\) is the real root of \(\theta^ 3-\theta-1=0\). (This choice is known as the spiral mean.) The authors carry out a primarily numerical examination of the break-up of the spiral mean torus. They note that although the critical line seems to be smooth, the numerical behavior observed is not universal. They then discuss some scaling features of the model.
Mathematics Subject Classification ID: 37C70
zbMATH DE Number: 64959
zbMATH Keywords: spiral mean torus
zbMATH Keywords: universality
zbMATH Keywords: scaling
zbMATH Keywords: renormalization
author: R. Artuso
author: Giulio Casati
author: D. L. Shepelyansky
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
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full work available at URL: 0960-0779(92)90007-a
OpenAlex ID: W1963714346
description: scientific article; zbMATH DE number 64959