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Comparative Study of Some Fixed-Point Methods in the Generation of Julia and Mandelbrot Sets

Author

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  • Hao Zhou
  • Muhammad Tanveer
  • Jingjng Li
  • Basil K. Papadopoulos

Abstract

Fractal is a geometrical shape with property that each point of the shape represents the whole. Having this property, fractals procured the attention in computer graphics, engineering, biology, mathematics, physics, art, and design. The fractals generated on highest priorities are the Julia and Mandelbrot sets. So, in this paper, we develop some necessary conditions for the convergence of sequences established for the orbits of M, M∗, and K-iterative methods to generate these fractals. We adjust algorithms according to the develop conditions and draw some attractive Julia and Mandelbrot sets with sequences of iterates from proposed fixed-point iterative methods. Moreover, we discuss the self-similarities with input parameters in each graph and present the comparison of images with proposed methods.

Suggested Citation

  • Hao Zhou & Muhammad Tanveer & Jingjng Li & Basil K. Papadopoulos, 2020. "Comparative Study of Some Fixed-Point Methods in the Generation of Julia and Mandelbrot Sets," Journal of Mathematics, Hindawi, vol. 2020, pages 1-15, July.
    • Handle: RePEc:hin:jjmath:7020921
    DOI: 10.1155/2020/7020921
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    Cited by:

    1. Munir, Noor & Khan, Majid & Jamal, Sajjad Shaukat & Hazzazi, Mohammad Mazyad & Hussain, Iqtadar, 2021. "Cryptanalysis of hybrid secure image encryption based on Julia set fractals and three-dimensional Lorenz chaotic map," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 826-836.

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