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Some Escape Time Results for General Complex Polynomials and Biomorphs Generation by a New Iteration Process

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  • Lateef Olakunle Jolaoso

    (Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94, Medunsa 0204, South Africa)

  • Safeer Hussain Khan

    (Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar)

Abstract

Biomorphs are graphic objects with very interesting shapes resembling unicellular and microbial organisms such as bacteria. They have applications in different fields like medical science, art, painting, engineering and the textile industry. In this paper, we present for the first time escape criterion results for general complex polynomials containing quadratic, cubic and higher order polynomials. We do so by using a more general iteration method also used for the first time in this field. This also generalizes some previous results. Then, biomorphs are generated using an algorithm whose pseudocode is included. A visualization of the biomorphs for certain polynomials is presented and their graphical behaviour with respect to variation of parameters is examined.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2172-:d:457263
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    References listed on IDEAS

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    2. Negi, Ashish & Rani, Mamta & Mahanti, P.K., 2008. "Computer simulation of the behaviour of Julia sets using switching processes," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 1187-1192.
    3. Krzysztof Gdawiec & Wiesław Kotarski & Agnieszka Lisowska, 2015. "Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods," Abstract and Applied Analysis, Hindawi, vol. 2015, pages 1-19, February.
    4. Geum, Young Hee & Hare, Kevin G., 2009. "Groebner basis, resultants and the generalized Mandelbrot set," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1016-1023.
    5. Singh, S.L. & Jain, Sarika & Mishra, S.N., 2009. "A new approach to superfractals," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3110-3120.
    6. 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. 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. Lateef Olakunle Jolaoso & Safeer Hussain Khan & Kazeem Olalekan Aremu, 2022. "Dynamics of RK Iteration and Basic Family of Iterations for Polynomiography," Mathematics, MDPI, vol. 10(18), pages 1-16, September.
    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).

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