https://doi.org/10.1140/epjs/s11734-025-01579-y
Regular Article
Analyzing complex dynamics of Mandelbrot and Julia sets generated using Picard–SP iteration scheme
1
Department of Mathematics, University of Lakki Marwat, 28420, Lakki Marwat, Khyber Pakhtunkhwa, Pakistan
2
Institute of Computer Science, University of Silesia in Katowice, Bedzinska 39, 41-200, Sosnowiec, Poland
Received:
8
October
2024
Accepted:
14
March
2025
Published online:
29
March
2025
One of the generalizations of Mandelbrot and Julia sets introduced in the literature involves using iteration schemes from fixed-point theory to generate new classes of these sets. In this article, we propose employing the Picard–SP iteration to define a new class of Mandelbrot and Julia sets for polynomials of the form 
, where 
and 
. We formulate an escape criterion to generate complex graphics of the considered class of sets. Our analysis covers dynamic variations, generation timelines, shapes and colours of the resulting visuals. Furthermore, we inspect the impact of iteration parameters on Mandelbrot and Julia sets using established numerical measures from the existing literature. The results show that the sets obtained with the Picard–SP iteration are very complex, and the variation of the iteration’s parameters has a significant impact on the sets. Moreover, the dependency between the parameters and the numerical measures is non-trivial and non-linear.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
