Fractal attractors of iterated function systems in extended fuzzy $ \mathscr{F} $-metric spaces

Abhishikta Das; Dipti Barman; Mohammad Sajid; Tarapada Bag; Abhishikta Das; Dipti Barman; Mohammad Sajid; Tarapada Bag

doi:10.3934/math.20261262

AIMS Mathematics

2026, Volume 11, Issue 9: 32109-32156. doi: 10.3934/math.20261262

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Research article

Fractal attractors of iterated function systems in extended fuzzy $ \mathscr{F} $-metric spaces

1.
Department of Mathematics, Siksha-Bhavana, Visva-Bharati, Santiniketan-731235, West-Bengal, India
2.
Department of Mechanical Engineering, College of Engineering, Qassim University, Buraydah, Saudi Arabia

Received: 02 July 2026 Revised: 10 September 2026 Accepted: 18 September 2026 Published: 29 September 2026
MSC : 03E72, 28A80, 47H09, 47H10, 54A40, 54E50

Certain fractal and fuzzy dynamical systems involve auxiliary control functions that exhibit irregular features such as non-monotonicity or discontinuities, placing them outside the scope of classical fuzzy metric frameworks. To analyze these dynamics, we introduce the concept of an extended fuzzy $ \mathscr{F} $-metric space and establish basic and fixed-point results in this setting. As an application of the fixed-point results, we use the framework to analyze Hutchinson–Barnsley operators arising in iterated function systems, providing a convergence analysis, a numerical simulation, and visualizations of fractal attractors on the Cantor set. This perspective aligns fixed-point techniques more closely with the analytical demands of fractal and fuzzy dynamic modeling. The proposed framework may also be a flexible mathematical setting for studying nonlinear systems involving uncertainty, including potential applications in environmental modeling, energy systems, and resource management.
- extended fuzzy $ \mathscr{F} $-metric space,
- fuzzy Hutchinson–Barnsley operator,
- fuzzy contractive mapping,
- fuzzy fractal attractor,
- simulation,
- contraction inequality
Citation: Abhishikta Das, Dipti Barman, Mohammad Sajid, Tarapada Bag. Fractal attractors of iterated function systems in extended fuzzy $ \mathscr{F} $-metric spaces[J]. AIMS Mathematics, 2026, 11(9): 32109-32156. doi: 10.3934/math.20261262

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Abstract

Certain fractal and fuzzy dynamical systems involve auxiliary control functions that exhibit irregular features such as non-monotonicity or discontinuities, placing them outside the scope of classical fuzzy metric frameworks. To analyze these dynamics, we introduce the concept of an extended fuzzy $ \mathscr{F} $-metric space and establish basic and fixed-point results in this setting. As an application of the fixed-point results, we use the framework to analyze Hutchinson–Barnsley operators arising in iterated function systems, providing a convergence analysis, a numerical simulation, and visualizations of fractal attractors on the Cantor set. This perspective aligns fixed-point techniques more closely with the analytical demands of fractal and fuzzy dynamic modeling. The proposed framework may also be a flexible mathematical setting for studying nonlinear systems involving uncertainty, including potential applications in environmental modeling, energy systems, and resource management.

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