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A novel approach to generate Mandelbrot sets, Julia sets and biomorphs via viscosity approximation method

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  • Kumari, Sudesh
  • Gdawiec, Krzysztof
  • Nandal, Ashish
  • Postolache, Mihai
  • Chugh, Renu

Abstract

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a new approach to visualize Mandelbrot and Julia sets for complex polynomials of the form W(z)=zn+mz+r; n≥2 where m,r∈ℂ, and biomorphs for any complex function through a viscosity approximation method which is among the most widely used iterative methods for finding fixed points of non-linear operators. We derive novel escape criterion for generating Julia and Mandelbrot sets via proposed viscosity approximation method. Moreover, we visualize the sets using the escape time algorithm and the proposed iteration. Then, we discuss the shape change of the obtained sets depending on the parameters of the iteration using graphical and numerical experiments. The presented examples reveal that this change can be very complex, and we are able to obtain a great variety of shapes.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:chsofr:v:163:y:2022:i:c:s0960077922007366
    DOI: 10.1016/j.chaos.2022.112540
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    References listed on IDEAS

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    1. Mihai Postolache & Ashish Nandal & Renu Chugh, 2019. "Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space," Mathematics, MDPI, vol. 7(9), pages 1-24, August.
    2. Sudesh Kumari & Renu Chugh & Jinde Cao & Chuangxia Huang, 2019. "Multi Fractals of Generalized Multivalued Iterated Function Systems in b -Metric Spaces with Applications," Mathematics, MDPI, vol. 7(10), pages 1-17, October.
    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. Jakubska-Busse, A. & Janowicz, M.W. & Ochnio, L. & Ashbourn, J.M.A., 2018. "Pickover biomorphs and non-standard complex numbers," Chaos, Solitons & Fractals, Elsevier, vol. 113(C), pages 46-52.
    5. Kaboudian, Abouzar & Cherry, Elizabeth M. & Fenton, Flavio H., 2019. "Large-scale interactive numerical experiments of chaos, solitons and fractals in real time via GPU in a web browser," Chaos, Solitons & Fractals, Elsevier, vol. 121(C), pages 6-29.
    6. 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.
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    1. 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.
    2. 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.
    3. 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).
    4. 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.

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