Structural Analysis of General Fuzzy Machines via Fixed-Point Theory: Stability and Sensitivity

Articles in Press

Document Type : Research article

Authors

1 Department of Mathematics and Statistics, Faculty of Energy and Data Science, Behbahan Khatam Alanbia University of Technology, Behbahan, Iran

2 Department of Mathematics, Shi.C., Islamic Azad University, Shiraz, Iran

Abstract

This paper introduces the general fuzzy machine (GFM) as a principled framework for modeling and analyzing fuzzy automata through the lens of fixed-point theory. The system dynamics are formulated as a fixed-point equation \( \mu = T_\theta(\mu) \) on the compact membership space \( \mathcal{F}(Q) = [0,1]^n \), where \( T_\theta \) is a parameterized fuzzy transition operator constructed from the aggregation functions \( F_1 \) and \( F_2 \), and the transition membership function \( \delta \). We investigate the existence, uniqueness, and parametric sensitivity of the fixed point \( \mu^*(\theta) \) using the Banach contraction principle and the implicit function theorem. The contraction condition guarantees not only a unique steady-state fuzzy vector but also exponential convergence independent of the initial configuration. Analytical results include Lipschitz-type stability bounds that quantify the robustness of the equilibrium state under operator perturbations, and a closed-form expression for the sensitivity Jacobian \( \frac{d\mu^*}{d\theta} \), which provides a precise measure of how the fixed point responds to small changes in the tunable parameters. Numerical examples confirm that actual sensitivities are often significantly smaller than the worst-case bounds, indicating that GFMs can exhibit robust behavior under parameter perturbations. We also identify structural limitations arising from non-contractive transition operators, where multiple fixed points may exist and Banach's theorem becomes inapplicable, underscoring the need for principled operator design in fuzzy systems when robustness and predictability are required.

Keywords

Main Subjects

Analytical and Numerical Solutions for Nonlinear Equations

Articles in Press, Accepted Manuscript
Available Online from 06 September 2026

Files

History

  • Receive Date: 06 July 2026
  • Revise Date: 29 August 2026
  • Accept Date: 05 September 2026
  • Publish Date: 06 September 2026

Share

How to cite

Statistics