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How to perturb Bernoulli shift map

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  • Sukegawa, Noriyoshi
  • Ikeguchi, Tohru

Abstract

The Bernoulli shift map is a fundamental example of a chaotic map with applications in algorithm design, data analysis, and numerical simulation. When implementing the Bernoulli shift map in the binary system, some sorts of perturbation methods are employed to make its outputs have long periods for practical reasons. We here look at one of such methods that perturbs underlying state space, and apply modular arithmetic to analyze the behavior of periods attained by this method, which reveals a close relationship with Artin’s conjecture on primitive roots. As a consequence, we obtain an exhaustive list of values for a dominant parameter of this method that are best possible in a theoretical sense.

Suggested Citation

  • Sukegawa, Noriyoshi & Ikeguchi, Tohru, 2022. "How to perturb Bernoulli shift map," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    • Handle: RePEc:eee:chsofr:v:165:y:2022:i:p1:s0960077922009729
    DOI: 10.1016/j.chaos.2022.112793
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    References listed on IDEAS

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    1. Yi-Jen Sun & Fan Bai & An-Chi Luo & Xiang-Yu Zhuang & Tsai-Shun Lin & Yu-Cheng Sung & Yu-Ling Shih & Chien-Jung Lo, 2021. "Probing bacterial cell wall growth by tracing wall-anchored protein complexes," Nature Communications, Nature, vol. 12(1), pages 1-9, December.
    2. 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).
    3. Kanso, Ali & Smaoui, Nejib, 2009. "Logistic chaotic maps for binary numbers generations," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2557-2568.
    4. Bocheng Liu & Hongyue Xiang & Lingfeng Liu, 2020. "Reducing the Dynamical Degradation of Digital Chaotic Maps with Time-Delay Linear Feedback and Parameter Perturbation," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-12, February.
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    Cited by:

    1. García-Grimaldo, Claudio & Campos-Cantón, Eric, 2023. "Exploring a family of Bernoulli-like shift chaotic maps and its amplitude control," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).

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