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On the Mandelbrot set of zp+logct via the Mann and Picard–Mann iterations

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  • Tanveer, Muhammad
  • Nazeer, Waqas
  • Gdawiec, Krzysztof

Abstract

Since its introduction, the Mandelbrot set has been studied and generalized in various directions. Some authors generalized it by using iterations from fixed point theory, whereas others characterized it by using different complex functions or polynomials. In this paper, we replace the constant c in the classical zp+c function with logct, where t∈R and t≥1. Moreover, we prove escape criteria for the Mann and Picard–Mann iterations in which we use the modified function. Then, we present graphical and numerical examples showing the behaviour of the generated sets depending on the parameters of the iterations and the parameter t. Using the proposed approach, we can generate a great variety of fascinating fractal patterns, and when t∈N the sets form rosette patterns.

Suggested Citation

  • Tanveer, Muhammad & Nazeer, Waqas & Gdawiec, Krzysztof, 2023. "On the Mandelbrot set of zp+logct via the Mann and Picard–Mann iterations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 209(C), pages 184-204.
    • Handle: RePEc:eee:matcom:v:209:y:2023:i:c:p:184-204
    DOI: 10.1016/j.matcom.2023.02.012
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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. Kumari, Sudesh & Gdawiec, Krzysztof & Nandal, Ashish & Postolache, Mihai & Chugh, Renu, 2022. "A novel approach to generate Mandelbrot sets, Julia sets and biomorphs via viscosity approximation method," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    3. Tassaddiq, Asifa, 2022. "General escape criteria for the generation of fractals in extended Jungck–Noor orbit," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 1-14.
    4. Lateef Olakunle Jolaoso & Safeer Hussain Khan, 2020. "Some Escape Time Results for General Complex Polynomials and Biomorphs Generation by a New Iteration Process," Mathematics, MDPI, vol. 8(12), pages 1-18, December.
    5. Lateef Olakunle Jolaoso & Safeer Hussain Khan & Kazeem Olalekan Aremu, 2022. "Dynamics of RK Iteration and Basic Family of Iterations for Polynomiography," Mathematics, MDPI, vol. 10(18), pages 1-16, September.
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    Cited by:

    1. Adhikari, Nabaraj & Sintunavarat, Wutiphol, 2024. "The Julia and Mandelbrot sets for the function zp−qz2+rz+sincw exhibit Mann and Picard–Mann orbits along with s-convexity," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).

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