Fractal Attractors in Random Nonlinear Iterated Function Systems: Existence, Stability, and Dimensional Properties

TL;DR

This paper introduces a comprehensive framework for Random Nonlinear Iterated Function Systems (RNIFS), establishing their existence, stability, and analyzing the fractal dimensions of their attractors through theoretical proofs and high-resolution simulations.

Contribution

It generalizes classical IFS models to include nonlinearity and stochasticity, providing new theoretical guarantees and computational methods for analyzing complex random fractals.

Findings

Existence and stability of RNIFS attractors proven mathematically.

Fractal dimensions of attractors range from 1.4 to 1.89.

RNIFS can model complex structures beyond traditional deterministic systems.

Abstract

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 Sierpi\'nski 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.