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Escape Times Across the Golden Cantorus of the Standard Map

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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 \(\left<N_{k}\right>\) to cross the Cantorus as \(k\to k_{G}\) 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.

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Notes

  1. For a \(q\)-periodic point of (1.1), the symbol tr refers to the trace of the differential matrix of \(M_{k}^{q}\) evaluated at the periodic point.

  2. We will slightly change this notation in Section 4.

  3. This point has to be chosen on different sides of the symmetry line depending on the parity due to the fact that two periodic orbits with consecutive approximants as rotation number lie on different sides of the invariant curve or Cantorus.

  4. In the case of the standard map (1.1), it can be found either on the lines \(\{y=2x\}\) and \(\{y=2x-1\}\) or in one of them, depending on \(j\).

  5. For iterates \(\geqslant 14\) small chaotic zones are visible by magnifying the plot in the electronic version.

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ACKNOWLEDGMENTS

We thank Jaume Timoneda for maintaining the computing facilities of the Dynamical Systems Group of the Universitat de Barcelona, which have been widely used in this work.

Funding

This work has been supported by grants PID2019-104851GB-I00 (Spain) and 2017-SGR-1374 (Catalonia). AV also acknowledges the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M).

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Authors and Affiliations

  1. Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585 , 08007, Barcelona, Spain

    Narcís Miguel, Carles Simó & Arturo Vieiro

  2. Centre de Recerca Matemàtica, Campus Bellaterra, 08193, Bellaterra, Spain

    Arturo Vieiro

Authors
  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, Carles Simó or Arturo Vieiro.

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The authors declare that they have no conflicts of interest.

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Dedicated to the memory of Alexey V. Borisov

MSC2010

37E40, 37E20, 37C05

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Miguel, N., Simó, C. & Vieiro, A. Escape Times Across the Golden Cantorus of the Standard Map. Regul. Chaot. Dyn. 27, 281–306 (2022). https://doi.org/10.1134/S1560354722030029

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