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

Mohamed Aly Bouke

TL;DR

This work formalizes Random Nonlinear Iterated Function Systems (RNIFS), integrating randomness and nonlinear maps to generate fractal attractors. It proves existence and stability results via a contracted Hutchinson operator on probability measures and a Lyapunov-type condition, and it empirically characterizes attractors using box-counting dimensions across a suite of nonlinear function families. The eight numerical experiments reveal a diverse zoo of self-similar structures with box-counting dimensions spanning roughly $1.43$ to $1.89$, and a case study shows RNIFS extend the classical Sierpiński triangle with increased geometric richness. The findings advance the theory of random fractals and offer a flexible framework for modeling complex stochastic self-similar structures in science and engineering.

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 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.

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

TL;DR

This work formalizes Random Nonlinear Iterated Function Systems (RNIFS), integrating randomness and nonlinear maps to generate fractal attractors. It proves existence and stability results via a contracted Hutchinson operator on probability measures and a Lyapunov-type condition, and it empirically characterizes attractors using box-counting dimensions across a suite of nonlinear function families. The eight numerical experiments reveal a diverse zoo of self-similar structures with box-counting dimensions spanning roughly to , and a case study shows RNIFS extend the classical Sierpiński triangle with increased geometric richness. The findings advance the theory of random fractals and offer a flexible framework for modeling complex stochastic self-similar structures in science and engineering.

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 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.

Paper Structure

This paper contains 32 sections, 2 theorems, 25 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Theoretical Framework
  4. Definition and Construction of RNIFS
  5. Metric and Functional Assumptions
  6. Existence of Attractors
  7. Stability and Lyapunov-Like Conditions
  8. Dimensional Complexity and Scaling Laws
  9. Mathematical Guarantees
  10. Existence of Invariant Measure
  11. Stability of Trajectories
  12. Numerical Methodology
  13. Simulation Procedure
  14. Function Families
  15. Experimental Design
  16. ...and 17 more sections

Key Result

Theorem 1

Under the assumptions above, the operator $W$ is a contraction on the space $(\mathcal{P}(X), W_1)$, where $W_1$ is the 1-Wasserstein metric. Hence, there exists a unique invariant measure $\mu^* \in \mathcal{P}(X)$ satisfying:

Figures (11)

  • Figure 1: Experiment 1 — Branching Structure. Top-left: hexbin density map; top-right: raw attractor plot; bottom: box-counting dimension log-log plot.
  • Figure 2: Experiment 2 — Chaotic Explosion. Top-left: hexbin density map; top-right: raw attractor plot; bottom: box-counting dimension log-log plot.
  • Figure 3: Experiment 3 — Concentric Energy. Top-left: hexbin density map; top-right: raw attractor plot; bottom: box-counting dimension log-log plot.
  • Figure 4: Experiment 4 — Disruptive Mixture. Top-left: hexbin density map; top-right: raw attractor plot; bottom: box-counting dimension log-log plot.
  • Figure 5: Experiment 5 — High-Frequency Disturbance. Top-left: hexbin density map; top-right: raw attractor plot; bottom: box-counting dimension log-log plot.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof : Proof Sketch