Break-up of the spiral mean torus in a volume-preserving map
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.
scientific article; zbMATH DE number 64959