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Generation of Mandelbrot and Julia sets by using M-iteration process

Author

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  • Nawaz, Bashir
  • Ullah, Kifayat
  • Gdawiec, Krzysztof

Abstract

In this manuscript, we introduce the M-iteration process for generating Mandelbrot and Julia sets. We establish an escape criterion for a polynomial of the form xk+1+c in the complex plane corresponding to the M-iteration process. Next, we present some graphical examples of Mandelbrot and Julia sets generated using the proven escape criterion and the escape-time algorithm. We also compare the images generated with the M, Mann, and Picard–Mann iterations. Moreover, we study the dependency between the iterations’ parameters and three numerical measures (the average escape time, non-escaping area index, and box-counting dimension) used in the literature. The results show that fractal images generated using the M-iteration are entirely different from those generated using the other two analysed iteration schemes. Moreover, the dependencies are highly non-linear and vary between the iterations.

Suggested Citation

  • Nawaz, Bashir & Ullah, Kifayat & Gdawiec, Krzysztof, 2024. "Generation of Mandelbrot and Julia sets by using M-iteration process," Chaos, Solitons & Fractals, Elsevier, vol. 188(C).
    • Handle: RePEc:eee:chsofr:v:188:y:2024:i:c:s0960077924010683
    DOI: 10.1016/j.chaos.2024.115516
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    References listed on IDEAS

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    1. 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).
    2. 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.
    3. Negi, Ashish & Rani, Mamta, 2008. "A new approach to dynamic noise on superior Mandelbrot set," Chaos, Solitons & Fractals, Elsevier, vol. 36(4), pages 1089-1096.
    4. Adhikari, Nabaraj & Sintunavarat, Wutiphol, 2024. "Exploring the Julia and Mandelbrot sets of zp+logct using a four-step iteration scheme extended with s-convexity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 357-381.
    5. Atangana, Abdon & Mekkaoui, Toufik, 2019. "Trinition the complex number with two imaginary parts: Fractal, chaos and fractional calculus," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 366-381.
    6. 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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