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<p>In this paper, we study the dynamics of Mandelbrot and Julia sets generated by a generalized rational map under the M-iteration scheme. An explicit escape criterion is established, providing a rigorous foundation for the application of escape-time algorithms within this iterative framework. Using this criterion, Mandelbrot and Julia sets are generated, and the influence of the iteration parameter on their geometry and dynamics is systematically investigated. Moreover, quantitative analysis is performed based on the average escape time, non-escaping area index, and fractal dimension. The results demonstrate that the colour, size, and shape of these fractals are highly sensitive to changes in the M-iteration’s parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. Even slight modifications to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> can produce substantial structural differences, emphasizing the intricate dependencies governing fractal formation.</p>

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Visual dynamics of Julia and Mandelbrot sets via M-iteration under generalized rational mapping

  • 期刊论文

  • 发表日期：

    2026年3月17日

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  • Bashir Nawaz,
  • Krzysztof Gdawiec,
  • Kifayat Ullah

摘要

In this paper, we study the dynamics of Mandelbrot and Julia sets generated by a generalized rational map under the M-iteration scheme. An explicit escape criterion is established, providing a rigorous foundation for the application of escape-time algorithms within this iterative framework. Using this criterion, Mandelbrot and Julia sets are generated, and the influence of the iteration parameter on their geometry and dynamics is systematically investigated. Moreover, quantitative analysis is performed based on the average escape time, non-escaping area index, and fractal dimension. The results demonstrate that the colour, size, and shape of these fractals are highly sensitive to changes in the M-iteration’s parameter \(\alpha \) α . Even slight modifications to \(\alpha \) α can produce substantial structural differences, emphasizing the intricate dependencies governing fractal formation.