<p>Research on the Mandelbrot set has been ongoing for decades and occupies a significant place in the study of fractal geometry. It is obtained by employing a function in the complex plane in an iterative procedure. This method is further developed in existing literature in two ways: either by employing complicated functions of various kinds or by using iterative methods beyond the standard Picard iteration. In both cases, we use a single function to obtain the Mandelbrot set. In this paper, we propose an approach in which we use two polynomial functions instead of one. For this aim, we have utilized the Das–Debata iteration, which combines two operators into a single iterative process. Polynomials of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2217_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">\(z^m+pz+r\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2217_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(z, p, r\in \mathbb {C}\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2217_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(m \in \mathbb {N}\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2217_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(m \ge 2\)</EquationSource> </InlineEquation>, are used to derive the escape criterion. By employing the escape time algorithm, we have provided interesting graphical representations of Mandelbrot sets that exhibit notable variations in patterns compared to those obtained by the Picard iteration. To create a comparison analysis of the resulting sets based on the iteration parameters, we merged polynomials of the same degree and different degrees and changed their ordering in the iterative process. Finally, we examined two numerical measures: the average escape time and the non-escaping area index, to determine how these fractal sets rely on the iteration parameters, which turns out that the dependency of both iterative approaches is nonlinear.</p>

Complex dynamics of Mandelbrot sets induced by iterative composition of two polynomials under the Das–Debata framework

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2025年9月23日

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Subhadip Roy,
Krzysztof Gdawiec

摘要

Research on the Mandelbrot set has been ongoing for decades and occupies a significant place in the study of fractal geometry. It is obtained by employing a function in the complex plane in an iterative procedure. This method is further developed in existing literature in two ways: either by employing complicated functions of various kinds or by using iterative methods beyond the standard Picard iteration. In both cases, we use a single function to obtain the Mandelbrot set. In this paper, we propose an approach in which we use two polynomial functions instead of one. For this aim, we have utilized the Das–Debata iteration, which combines two operators into a single iterative process. Polynomials of the form \(z^m+pz+r\) , where \(z, p, r\in \mathbb {C}\) , \(m \in \mathbb {N}\) , and \(m \ge 2\) , are used to derive the escape criterion. By employing the escape time algorithm, we have provided interesting graphical representations of Mandelbrot sets that exhibit notable variations in patterns compared to those obtained by the Picard iteration. To create a comparison analysis of the resulting sets based on the iteration parameters, we merged polynomials of the same degree and different degrees and changed their ordering in the iterative process. Finally, we examined two numerical measures: the average escape time and the non-escaping area index, to determine how these fractal sets rely on the iteration parameters, which turns out that the dependency of both iterative approaches is nonlinear.