Chaos-Preserving Fuzzy Difference Operators: A Bridge Between Fuzzy Arithmetic and Fuzzy Dynamical Systems

Abstract

In this work, we propose a novel framework that links generalized fuzzy difference operators—defined through α-level set constructions—with the dynamical behavior of fuzzy systems. By revisiting the compatibility between fuzzy set operations and their α-level counterparts, we introduce the concept of chaos-preserving operators, i.e., binary fuzzy operations that maintain or amplify chaotic dynamics under the Zadeh extension. We demonstrate that, under specific structural conditions (such as upper semicontinuity and nestedness of level sets), certain generalized Hausdorff-type differences not only admit consistent fuzzy representations but also preserve Devaney chaos, Li–Yorke chaos, and distributional chaos in fuzzy dynamical systems. Our theoretical development is supported by explicit constructions involving triangular fuzzy numbers and set-valued dynamics. The proposed framework opens a new avenue for analyzing uncertainty-propagating chaos in fuzzy environments, with potential applications in nonlinear systems, decision theory, and complex modeling.