On the Mandelbrot set of zp+logct via the Mann and Picard–Mann iterations

Muhammad Tanveer, Waqas Nazeer and Krzysztof Gdawiec

Mathematics and Computers in Simulation (MATCOM), 2023, vol. 209, issue C, 184-204

Abstract: Since its introduction, the Mandelbrot set has been studied and generalized in various directions. Some authors generalized it by using iterations from fixed point theory, whereas others characterized it by using different complex functions or polynomials. In this paper, we replace the constant c in the classical zp+c function with logct, where t∈R and t≥1. Moreover, we prove escape criteria for the Mann and Picard–Mann iterations in which we use the modified function. Then, we present graphical and numerical examples showing the behaviour of the generated sets depending on the parameters of the iterations and the parameter t. Using the proposed approach, we can generate a great variety of fascinating fractal patterns, and when t∈N the sets form rosette patterns.

Keywords: Mandelbrot set; Escape criterion; Mann iteration; Picard–Mann iteration; Rosette patterns (search for similar items in EconPapers)
Date: 2023
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:209:y:2023:i:c:p:184-204

DOI: 10.1016/j.matcom.2023.02.012

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