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On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization

Author

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  • A. A. Elsadany

    (Department of Mathematics, Faculty of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
    Basic Science Department, Faculty of Computers and Information, Suez Canal University, New Campus, Ismailia 41522, Egypt)

  • A. Aldurayhim

    (Department of Mathematics, Faculty of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia)

  • H. N. Agiza

    (Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt)

  • Amr Elsonbaty

    (Department of Mathematics, Faculty of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
    Mathematics & Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt)

Abstract

In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map’s parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work.

Suggested Citation

  • A. A. Elsadany & A. Aldurayhim & H. N. Agiza & Amr Elsonbaty, 2023. "On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization," Mathematics, MDPI, vol. 11(3), pages 1-21, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:727-:d:1053612
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    References listed on IDEAS

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    1. Wang, Yupin & Li, Xiaodi & Wang, Da & Liu, Shutang, 2022. "A brief note on fractal dynamics of fractional Mandelbrot sets," Applied Mathematics and Computation, Elsevier, vol. 432(C).
    2. Amr Elsonbaty & A. Elsadany & Fatma Kamal & Amin Jajarmi, 2022. "On Discrete Fractional Complex Gaussian Map: Fractal Analysis, Julia Sets Control, and Encryption Application," Mathematical Problems in Engineering, Hindawi, vol. 2022, pages 1-18, April.
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