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Fractal attractors in random nonlinear iterated function systems: Existence, stability, and dimensional properties

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  • Bouke, Mohamed Aly

Abstract

This study develops a theoretical and computational framework for Random Nonlinear Iterated Function Systems (RNIFS), extending classical IFS by combining stochastic selection with nonlinear transformations. We provide sufficient conditions for the existence of a unique invariant measure and for statistical stability of trajectories under contractive assumptions and a Lyapunov-type criterion. Numerically, we conduct eight RNIFS experiments spanning diverse nonlinear function families and probability schemes, and quantify geometric complexity primarily via box-counting dimension estimates, yielding non-integer dimensions in the range 1.43–1.89. To assess reliability, we include an uncertainty analysis based on repeated stochastic trials and bootstrap resampling, and a measure-theoretic cross-check using the correlation dimension (D2 ≈ 1.228), indicating heterogeneous measure concentration. Finally, a baseline structural comparison with the classical Sierpin'ski triangle illustrates how deterministic IFS arise as a special case of RNIFS and how a minimal nonlinear perturbation increases geometric complexity (from dimH ≈ 1.585 to dimB ≈ 1.787).

Suggested Citation

  • Bouke, Mohamed Aly, 2026. "Fractal attractors in random nonlinear iterated function systems: Existence, stability, and dimensional properties," Chaos, Solitons & Fractals, Elsevier, vol. 207(C).
    • Handle: RePEc:eee:chsofr:v:207:y:2026:i:c:s0960077926001372
    DOI: 10.1016/j.chaos.2026.117996
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