2025/05/24 by Mohamed Aly Bouke, Bouke, Mohamed Aly · 2 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Mathematical Dynamics and Fractals #Symbolic Computation (cs.SC)
paper · doi:10.48550/arxiv.2505.18849
openalex publication_date 2025/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This study develops a comprehensive theoretical and computational framework for Random Nonlinear Iterated Function Systems (RNIFS), a generalization of classical IFS models that incorporates both nonlinearity and stochasticity. We establish mathematical guarantees for the existence and stability of invariant fractal attractors by leveraging contractivity conditions, Lyapunov-type criteria, and measure-theoretic arguments. Empirically, we design a set of high-resolution simulations across diverse nonlinear functions and probabilistic schemes to analyze the emergent attractors geometry and dimensionality. A box-counting method is used to estimate the fractal dimension, revealing attractors with rich internal structure and dimensions ranging from 1.4 to 1.89. Additionally, we present a case study comparing RNIFS to the classical Sierpiński triangle, demonstrating the generalization's ability to preserve global shape while enhancing geometric complexity. These findings affirm the capacity of RNIFS to model intricate, self-similar structures beyond the reach of traditional deterministic systems, offering new directions for the study of random fractals in both theory and applications.