Generation of Mandelbrot and Julia sets by using M-iteration process

Bashir Nawaz, Kifayat Ullah and Krzysztof Gdawiec

Chaos, Solitons & Fractals, 2024, vol. 188, issue C

Abstract: In this manuscript, we introduce the M-iteration process for generating Mandelbrot and Julia sets. We establish an escape criterion for a polynomial of the form xk+1+c in the complex plane corresponding to the M-iteration process. Next, we present some graphical examples of Mandelbrot and Julia sets generated using the proven escape criterion and the escape-time algorithm. We also compare the images generated with the M, Mann, and Picard–Mann iterations. Moreover, we study the dependency between the iterations’ parameters and three numerical measures (the average escape time, non-escaping area index, and box-counting dimension) used in the literature. The results show that fractal images generated using the M-iteration are entirely different from those generated using the other two analysed iteration schemes. Moreover, the dependencies are highly non-linear and vary between the iterations.

Keywords: Julia set; Mandelbrot set; M iteration; Escape criterion (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:188:y:2024:i:c:s0960077924010683

DOI: 10.1016/j.chaos.2024.115516

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