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 sta...
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| Format: | Article |
| Language: | en |
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Elsevier
2026
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| Online Access: | http://psasir.upm.edu.my/id/eprint/123009/1/123009.pdf http://psasir.upm.edu.my/id/eprint/123009/ https://www.sciencedirect.com/science/article/pii/S0960077926001372 |
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| Summary: | 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). |
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