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New Escape Criteria for Complex Fractals Generation in Jungck-CR Orbit

Author

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  • Muhammad Tanveer

    (University of Lahore)

  • Waqas Nazeer

    (University of Education)

  • Krzysztof Gdawiec

    (Universityof Silesia)

Abstract

In recent years, researchers have studied the use of different iteration processes from fixed point theory for the generation of complex fractals. Examples are the Mann, the Ishikawa, the Noor, the Jungck-Mann and the Jungck-Ishikawa iterations. In this paper, we present a generalisation of complex fractals, namely Mandelbrot, Julia and multicorn sets, using the Jungck-CR implicit iteration scheme. This type of iteration does not reduce to any of the other iterations previously used in the study of complex fractals; thus, this generalisation gives rise to new fractal forms. We prove a new escape criterion for a polynomial of the following form zm − az + c, where a, c ∈ ℂ, and present some graphical examples of the obtained complex fractals.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:indpam:v:51:y:2020:i:4:d:10.1007_s13226-020-0466-9
    DOI: 10.1007/s13226-020-0466-9
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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. Yuanyuan Sun & Lina Chen & Rudan Xu & Ruiqing Kong, 2014. "An Image Encryption Algorithm Utilizing Julia Sets and Hilbert Curves," PLOS ONE, Public Library of Science, vol. 9(1), pages 1-9, January.
    3. S. L. Singh & Charu Bhatnagar & S. N. Mishra, 2005. "Stability of Jungck-type iterative procedures," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2005, pages 1-9, January.
    4. Gdawiec, Krzysztof & Kotarski, Wiesław, 2017. "Polynomiography for the polynomial infinity norm via Kalantari’s formula and nonstandard iterations," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 17-30.
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    Cited by:

    1. 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.
    2. 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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