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Discrete chaotic maps obtained by symmetric integration

Author

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  • Butusov, Denis N.
  • Karimov, Artur I.
  • Pyko, Nikita S.
  • Pyko, Svetlana A.
  • Bogachev, Mikhail I.

Abstract

Chaotic return maps are widely used to model various dynamical systems such as charged particle movement, laser beam dynamic, celestial body orbiting and many others. Return maps are commonly obtained by discretization of continuous equations using the Euler–Cromer operator with the only motivation that it is the simplest symplectic operator. Recent progress in geometric integration raised considerable interest to symmetric operators due to their ability to preserve some geometric properties of continuous flows this way yielding better agreement between discrete and continuous dynamical systems. Here we compare symmetric and asymmetric discretization approaches applied to several examples of Hamiltonian systems. In particular, we suggest symmetric modifications of Chirikov and Hénon maps and show explicitly that the implied symmetric integration procedure yields reflectional symmetry in the phase space. For verification, we show that a smooth even perturbation function used instead of a discontinuous delta pulse provides asymptotically similar results. Numerical experiments using several statistical methods show that symmetric and asymmetric maps, while yielding similar asymptotic behavior, often exhibit considerably different statistical properties for intermediate regimes providing smoother transitions that are more reminiscent to those observed in various natural phenomena. We believe that the proposed approach may be useful for modeling empirical systems by preserving their keynote physical properties.

Suggested Citation

  • Butusov, Denis N. & Karimov, Artur I. & Pyko, Nikita S. & Pyko, Svetlana A. & Bogachev, Mikhail I., 2018. "Discrete chaotic maps obtained by symmetric integration," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 955-970.
  • Handle: RePEc:eee:phsmap:v:509:y:2018:i:c:p:955-970
    DOI: 10.1016/j.physa.2018.06.100
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    References listed on IDEAS

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    Cited by:

    1. Tutueva, Aleksandra V. & Nepomuceno, Erivelton G. & Karimov, Artur I. & Andreev, Valery S. & Butusov, Denis N., 2020. "Adaptive chaotic maps and their application to pseudo-random numbers generation," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    2. Tutueva, Aleksandra V. & Karimov, Artur I. & Moysis, Lazaros & Volos, Christos & Butusov, Denis N., 2020. "Construction of one-way hash functions with increased key space using adaptive chaotic maps," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    3. Tutueva, Aleksandra V. & Moysis, Lazaros & Rybin, Vyacheslav G. & Kopets, Ekaterina E. & Volos, Christos & Butusov, Denis N., 2022. "Fast synchronization of symmetric Hénon maps using adaptive symmetry control," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    4. Pyko, Nikita S. & Pyko, Svetlana A. & Markelov, Oleg A. & Karimov, Artur I. & Butusov, Denis N. & Zolotukhin, Yaroslav V. & Uljanitski, Yuri D. & Bogachev, Mikhail I., 2018. "Assessment of cooperativity in complex systems with non-periodical dynamics: Comparison of five mutual information metrics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 1054-1072.
    5. Jaishree Jain & Arpit Jain & Saurabh Kumar Srivastava & Chaman Verma & Maria Simona Raboaca & Zoltán Illés, 2022. "Improved Security of E-Healthcare Images Using Hybridized Robust Zero-Watermarking and Hyper-Chaotic System along with RSA," Mathematics, MDPI, vol. 10(7), pages 1-16, March.
    6. Liu, Xianggang & Ma, Li, 2020. "Chaotic vibration, bifurcation, stabilization and synchronization control for fractional discrete-time systems," Applied Mathematics and Computation, Elsevier, vol. 385(C).

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