A study of Mandelbrot and Julia Sets via Picard–Thakur iteration with s-convexity
Bashir Nawaz, Krzysztof Gdawiec, Kifayat Ullah, Maggie Aphane
TL;DR
This paper uses a new iterative method to study fractal patterns like Mandelbrot and Julia sets, exploring how their shapes depend on iteration parameters.
Contribution
The novelty lies in applying the Picard–Thakur iteration with s-convexity to generate and analyze Mandelbrot and Julia sets.
Findings
An escape criterion was established using a complex polynomial for generating fractal sets.
Graphical illustrations of Mandelbrot and Julia sets were produced using the proposed algorithms.
The relationship between fractal set sizes and iteration parameters was analyzed.
Abstract
Nowadays, many researchers are employing various iterative techniques to analyse the dynamics of fractal patterns. In this paper, we explore the formation of Mandelbrot and Julia sets using the Picard–Thakur iteration process, extended with s-convexity. To achieve this, we establish an escape criterion using a complex polynomial of the form xk+1+c, where k ≥ 1 and x, c ∈ ℂ. Based on our proposed algorithms, we provide graphical illustrations of the Mandelbrot and Julia sets. Additionally, we extend our research to examine the relationship between the sizes of Mandelbrot and Julia sets and the iteration parameters, utilising some well-known methods from the literature.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Analytic and geometric function theory · Point processes and geometric inequalities