Escape times across the golden Cantorus of the standard map

N. Miguel, C. Sim\'o and A. Vieiro

Departament de Matem\`atiques i Inform\`atica, Universitat de Barcelona (UB), Gran Via, 585, 08007 Barcelona, Spain

Abstract:

We consider the Chirikov standard map for values of the parameter
larger than but close to Greene's $k_G$. We investigate the dynamics near the
golden Cantorus and we study escape rates across it. Mackay \cite{Mac83,Mac92t}
described the behaviour of the mean of the number of iterates
$\left<N_k\right>$ to cross the Cantorus when $k\to k_G$ and showed that there
exists $B<0$ so that $\left<N_k\right>(k-k_G)^B$ becomes 1-periodic in a
suitable logarithmic scale.  The numerical explorations here give evidence of
the shape of this periodic function and of the relation between the escape
rates and the evolution of the stability islands close to the Cantorus.