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Fractal generation via Khan iteration: Escape criteria and dynamical analysis

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  • Tyagi, Akansha
  • Vashistha, Sachin

Abstract

We introduce a novel fractal generation technique based on an iterative scheme for locating common fixed points. We derive the escape criterion for Khan iteration scheme by employing complex polynomials and rational functions. Several examples illustrating the Mandelbrot and Julia sets are obtained via the proposed method. To investigate the influence of iteration parameters on the fractal sets, we examine two numerical indicators: the average escape time and the non-escaping area index. Our analysis reveals a nonlinear relationship between these fractal characteristics and the iteration parameters.

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  • Tyagi, Akansha & Vashistha, Sachin, 2025. "Fractal generation via Khan iteration: Escape criteria and dynamical analysis," Chaos, Solitons & Fractals, Elsevier, vol. 201(P1).
    • Handle: RePEc:eee:chsofr:v:201:y:2025:i:p1:s096007792501197x
    DOI: 10.1016/j.chaos.2025.117184
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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. 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.
    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. 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.
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