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The Julia and Mandelbrot sets for the function zp−qz2+rz+sincw exhibit Mann and Picard–Mann orbits along with s-convexity

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  • Adhikari, Nabaraj
  • Sintunavarat, Wutiphol

Abstract

This research paper introduces a novel approach to visualize Julia and Mandelbrot sets by employing iterative techniques, which play a crucial role in creating fractals. The primary focus is on complex functions of the form F(z)=zp−qz2+rz+sincw for all z∈ℂ, where p∈N∖{1}, q∈ℂ, r,c∈ℂ∖{0} and w∈[1,∞). The Mann and Picard–Mann iteration schemes with s-convexity are utilized throughout the study. Innovative escape criteria are developed to generate Julia and Mandelbrot sets using these iterative methods. These criteria serve as guidelines for determining when the iterative process should terminate, leading to the creation of captivating fractal patterns. The research investigates the impact of parameter variations within the iteration schemes on the resulting fractal’s shape, size, and color. By manipulating these parameters, a wide range of captivating fractal patterns can be generated and visualized, encompassing various aesthetic possibilities. Additionally, we discuss the numerical examples related to Julia and Mandelbrot sets generated through the proposed iteration. We also delve into discussions concerning execution time and the average number of iterations.

Suggested Citation

  • 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).
  • Handle: RePEc:eee:chsofr:v:181:y:2024:i:c:s0960077924001516
    DOI: 10.1016/j.chaos.2024.114600
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    References listed on IDEAS

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    1. 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.
    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. Pastor, G. & Romera, M. & Álvarez, G. & Arroyo, D. & Montoya, F., 2007. "On periodic and chaotic regions in the Mandelbrot set," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 15-25.
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