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<p>This article examines a methodology for the dynamic visualization and analysis of Julia and Mandelbrot sets using iterative techniques, which are fundamental for fractal generation. The four-step iteration technique is introduced in this article. This iterative scheme represents a novel approach that has been utilized for the construction of Mandelbrot and Julia sets. A condition in the form of an escape criterion is derived for the complex polynomial map, which helps in the generation of the Julia and Mandelbrot sets. These fractals are generated and visualized using both the escape time algorithm and the proposed iteration scheme. The impact of iteration parameters on the shape, size, and color of fractals is investigated by using graphical exploration. The resulting images are compared with those produced by the well-known iteration schemes, specifically the Picard-Thakur and Noor iteration schemes. The study further explores the non-linear dependence of the iteration parameters on two key numerical measures: the Average Escape Time (AET) and the Non-Escaping Area Index (NAI). The box-counting dimension, a key tool in fractal geometry, is employed to assess how iteration parameters affect fractal shape, boundary roughness, and overall complexity. A comparative analysis with the Picard–Thakur and Noor iteration schemes reveals that the fractal patterns generated by the Picard <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(N^*\)</EquationSource> </InlineEquation>-algorithm display distinctive structural features not observed in the other methods. These findings underscore the unique characteristics of the Picard <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(N^*\)</EquationSource> </InlineEquation>-algorithm and highlight its potential applications in the study of fractal geometry. The proposed iteration scheme and technique may also facilitate the generation of other fractals, such as biomorphs, tricorns, for various functions.</p>

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Exploration of dynamics of Mandelbrot and Julia sets through a new four-step iteration algorithm

  • 期刊论文

  • 发表日期：

    2026年1月13日

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  • Manoj Kumar,
  • Syed Abbas

摘要

This article examines a methodology for the dynamic visualization and analysis of Julia and Mandelbrot sets using iterative techniques, which are fundamental for fractal generation. The four-step iteration technique is introduced in this article. This iterative scheme represents a novel approach that has been utilized for the construction of Mandelbrot and Julia sets. A condition in the form of an escape criterion is derived for the complex polynomial map, which helps in the generation of the Julia and Mandelbrot sets. These fractals are generated and visualized using both the escape time algorithm and the proposed iteration scheme. The impact of iteration parameters on the shape, size, and color of fractals is investigated by using graphical exploration. The resulting images are compared with those produced by the well-known iteration schemes, specifically the Picard-Thakur and Noor iteration schemes. The study further explores the non-linear dependence of the iteration parameters on two key numerical measures: the Average Escape Time (AET) and the Non-Escaping Area Index (NAI). The box-counting dimension, a key tool in fractal geometry, is employed to assess how iteration parameters affect fractal shape, boundary roughness, and overall complexity. A comparative analysis with the Picard–Thakur and Noor iteration schemes reveals that the fractal patterns generated by the Picard \(N^*\) -algorithm display distinctive structural features not observed in the other methods. These findings underscore the unique characteristics of the Picard \(N^*\) -algorithm and highlight its potential applications in the study of fractal geometry. The proposed iteration scheme and technique may also facilitate the generation of other fractals, such as biomorphs, tricorns, for various functions.