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

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 ( D 2 ≈ 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 dim H ≈ 1 . 585 to dim B ≈ 1 . 787).