Narcis Miguel

Gran Via 585, 08007, Barcelona, Catalunya
Departament de Matematica Aplicada i Analisi, Universitat de Barcelona


Miguel N., Simó C., Vieiro A.
Escape Times Across the Golden Cantorus of the Standard Map
2022, vol. 27, no. 3, pp.  281-306
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.
Keywords: standard map, diffusion through a Cantor set, escape times
Citation: Miguel N., Simó C., Vieiro A.,  Escape Times Across the Golden Cantorus of the Standard Map, Regular and Chaotic Dynamics, 2022, vol. 27, no. 3, pp. 281-306
Miguel N., Simó C., Vieiro A.
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.
Keywords: Hénon maps, measure of regular and chaotic dynamics domains, islands in the standard map for large parameter, accelerator modes
Citation: Miguel N., Simó C., Vieiro A.,  From the Hénon Conservative Map to the Chirikov Standard Map for Large Parameter Values, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 469-489

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