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Generation of Mandelbrot and Julia sets for generalized rational maps using SP-iteration process equipped with s-convexity

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Listed:
  • Rawat, Shivam
  • Prajapati, Darshana J.
  • Tomar, Anita
  • Gdawiec, Krzysztof

Abstract

In this paper, we introduce a generalized rational map to develop a theory of escape criterion via the SP-iteration process equipped with s-convexity. Furthermore, we develop algorithms for the exploration of unique kinds of Mandelbrot as well as Julia sets. We demonstrate graphically the change in colour, size, and shape of images with the change in values of the considered iteration’s parameters. The new fractals thus obtained are visually very pleasing and attractive. Most of these newly generated fractals resemble natural objects around us. Moreover, we numerically study the dependence between the iteration’s parameters and the set size. The experiments show that this dependency is non-linear. We believe that the obtained conclusions will motivate researchers who are interested in fractal geometry.

Suggested Citation

  • Rawat, Shivam & Prajapati, Darshana J. & Tomar, Anita & Gdawiec, Krzysztof, 2024. "Generation of Mandelbrot and Julia sets for generalized rational maps using SP-iteration process equipped with s-convexity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 148-169.
    • Handle: RePEc:eee:matcom:v:220:y:2024:i:c:p:148-169
    DOI: 10.1016/j.matcom.2023.12.040
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    References listed on IDEAS

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    1. 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.
    2. Muhammad Tanveer & Waqas Nazeer & Krzysztof Gdawiec, 2020. "New Escape Criteria for Complex Fractals Generation in Jungck-CR Orbit," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(4), pages 1285-1303, December.
    3. 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).
    4. 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.
    5. Zhihua Chen & Muhammad Tanveer & Waqas Nazeer & Jing Wu & Paolo Renna, 2021. "Fractals via Generalized Jungck–S Iterative Scheme," Discrete Dynamics in Nature and Society, Hindawi, vol. 2021, pages 1-12, March.
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