Arturo Vieiro

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 $\langle N_k \rangle$ to cross the Cantorus as $k\to k_G$ and showed that there exists $B<0$ so that $\langle N_k\rangle (k-k_G)^B$ 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.

The effects of quasi-periodicity on the splitting of invariant manifolds are examined. We have found that some harmonics that could be expected to be dominant in some ranges of the perturbation parameter actually are nondominant. It is proved that, under reasonable conditions, this is due to the arithmetic properties of the frequencies.

In this paper we consider conservative quadratic Hénon maps and Chirikov’s standard map, and relate them in some sense.
First, we present a study of some dynamical properties of orientation-preserving and orientationreversing quadratic Hénon maps concerning the stability region, the size of the chaotic zones, its evolution with respect to parameters and the splitting of the separatrices of fixed and periodic points plus its role in the preceding aspects.
Then the phase space of the standard map, for large values of the parameter k, is studied. There are some stable orbits which appear periodically in $k$ and are scaled somehow. Using this scaling, we show that the dynamics around these stable orbits is one of the above Hénon maps plus some small error, which tends to vanish as $k \to \infty$. Elementary considerations about diffusion properties of the standard map are also presented.