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<p>The study of fractal structures, particularly Julia sets, is pivotal in complex dynamics due to their intricate self-similar patterns and sensitivity to initial conditions. This paper applies a viscosity approximation-based iteration process to a newly introduced generalized logistic map for generating and analyzing associated Julia sets. Viscosity approximation methods, rooted in fixed-point theory, improve iterative scheme stability and convergence through controlled perturbations. Through numerical simulations, we visualize these fractals and examine their geometric properties under varying parameters. To quantify behavior, we compute the Average Escape Time (AET) and Non-Escaping Area Index (NAI), providing insights into divergence rates, stability, and computational complexity. Unlike prior applications of viscosity methods to standard quadratic or logistic maps, this work extends the approach to a generalized logistic map, enabling enhanced control over fractal stability and convergence. Our viscosity-based approach outperforms classical iterations in stability and fractal diversity, as quantified by AET and NAI metrics.</p>

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Analysis of Julia sets using viscosity approximation-based iteration processes

  • 期刊论文

  • 发表日期：

    2026年1月20日

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  • Shivam Rawat,
  • Darshana J. Prajapati,
  • Anita Tomar,
  • Divyanshu Chamoli,
  • Monika Bisht

摘要

The study of fractal structures, particularly Julia sets, is pivotal in complex dynamics due to their intricate self-similar patterns and sensitivity to initial conditions. This paper applies a viscosity approximation-based iteration process to a newly introduced generalized logistic map for generating and analyzing associated Julia sets. Viscosity approximation methods, rooted in fixed-point theory, improve iterative scheme stability and convergence through controlled perturbations. Through numerical simulations, we visualize these fractals and examine their geometric properties under varying parameters. To quantify behavior, we compute the Average Escape Time (AET) and Non-Escaping Area Index (NAI), providing insights into divergence rates, stability, and computational complexity. Unlike prior applications of viscosity methods to standard quadratic or logistic maps, this work extends the approach to a generalized logistic map, enabling enhanced control over fractal stability and convergence. Our viscosity-based approach outperforms classical iterations in stability and fractal diversity, as quantified by AET and NAI metrics.