Escape times across the golden cantorus of the standard map. (English) Zbl 1502.37048
The authors study the escape rates across a Cantorus (i.e., a Cantor structure with gaps) for the Chirikov standard map \(M_k : \mathbb{S}^1 \times \mathbb{R} \rightarrow \mathbb{S}^1 \times \mathbb{R}\) where
\[
\begin{bmatrix} x\\
y\\
\end{bmatrix} \mapsto \begin{bmatrix} x+y+\frac{k}{2 \pi} \sin(2 \pi x)\\
y+\frac{k}{2 \pi} \sin(2 \pi x)\\
\end{bmatrix}.
\]
They investigate the connection between the escape rates for parameters \(k\) greater than but near Greene’s parameter \(k_G \approx 0.971635406\) and the fine structure of the phase space close to the destruction of the golden invariant curve. For related results about the Cantorus as \(k \rightarrow k_G\), see also R. S. Mackay [Physica 7D, 283–300 (1983; Zbl 0568.70023); Renormalisation in area-preserving maps. Singapore: World Scientific (1993; Zbl 0791.58002)].
Reviewer: Steve Pederson (Atlanta)
MSC:
| 37E05 | Dynamical systems involving maps of the interval |
| 37E20 | Universality and renormalization of dynamical systems |
| 37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |
Software:
PARI/GPReferences:
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