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Numerical and Theoretical Studies on the Rational Standard Map at Moderate-to-Large Values of the Amplitude Parameter

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Abstract

In this work an exhaustive numerical and analytical investigation of the dynamics of a bi-parametric symplectic map, the so-called rational standard map, at moderate-to-large values of the amplitude parameter is addressed. After reviewing the model, a discussion concerning an analytical determination of the maximum Lyapunov exponent is provided together with thorough numerical experiments. The theoretical results are obtained in the limit of a nearly uniform distribution of the phase values. Correlations among phases lead to departures from the expected estimates. In this direction, a detailed study of the role of stable periodic islands of periods 1, 2 and 4 is included. Finally, an experimental relationship between the Lyapunov and instability times is shown, while an analytical one applies when correlations are irrelevant, which is the case, in general, for large values of the amplitude parameter.

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ACKNOWLEDGMENTS

The authors are grateful for the useful suggestions of two anonymous reviewers. C. S. thanks J. Timoneda, A. Jorba and A. Vieiro for maintaining the computing facilities of the Group of Dynamical Systems of the Universitat de Barcelona, which have been largely used in this work.

Funding

P.M.C. and C.M.G. were supported by grants from Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina PIP11220170100569, Universidad Nacional de La Plata 11G173. C.S. was supported by grants PID2019-104851GB-I00 and PID2021-125535NB-I00 (Spain).

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Authors and Affiliations

  1. Grupo de Caos en Sistemas Hamiltonianos, Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata and Instituto de Astrofísica de La Plata (CONICET), Paseo del Bosque S/N, B1900FWA, La Plata, Argentina

    Pablo M. Cincotta & Claudia M. Giordano

  2. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007, Barcelona, Spain

    Carles Simó

Authors
  1. Pablo M. Cincotta
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  2. Claudia M. Giordano
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  3. Carles Simó
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Correspondence to Pablo M. Cincotta, Claudia M. Giordano or Carles Simó.

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MSC2010

37J25

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Cincotta, P.M., Giordano, C.M. & Simó, C. Numerical and Theoretical Studies on the Rational Standard Map at Moderate-to-Large Values of the Amplitude Parameter. Regul. Chaot. Dyn. 28, 265–294 (2023). https://doi.org/10.1134/S1560354723030024

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