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From the Hénon conservative map to the Chirikov standard map for large parameter values

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Abstract

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 orientation-reversing 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→∞. Elementary considerations about diffusion properties of the standard map are also presented.

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Authors and Affiliations

  1. Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Catalunya, Spain

    Narcís Miguel, Carles Simó & Arturo Vieiro

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  1. Narcís Miguel
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  2. Carles Simó
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  3. Arturo Vieiro
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Correspondence to Narcís Miguel.

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Dedicated to our friend Sergey Gonchenko on his 60th birthday

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Miguel, N., Simó, C. & Vieiro, A. From the Hénon conservative map to the Chirikov standard map for large parameter values. Regul. Chaot. Dyn. 18, 469–489 (2013). https://doi.org/10.1134/S1560354713050018

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  • DOI: https://doi.org/10.1134/S1560354713050018

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