Escape Times Across the Golden Cantorus of the Standard Map

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 study escape rates across it. Mackay $[17,19]$ described the behaviour of the mean of the number of iterates $\left\langle N_k\right\rangle$ to cross the Cantorus as $k \rightarrow k_G$ and showed that there exists $B<0$ so that $\left\langle N_k\right\rangle\left(k-k_G\right)^B$ 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.