Fuzzy Skew Maps: Preserving Robust Chaos Under Uncertainty with Applications to Cryptography

We introduce fuzzy skew maps as a levelwise ( α -cut) extension of robustly chaotic skew transformations of S -unimodal maps to epistemically uncertain environments. Our central hypothesis is that the robust-chaos mechanism of the underlying skew family transfers to fuzzy parameter uncertainty in a set-based (not probabilistic) sense is as follows: for every α ∈ [ 0 , 1 ] , the induced crisp family { F ( · , q ) : q ∈ [ q ˜ ] α } preserves the absence of periodic windows and maintains strictly positive Lyapunov exponents. This yields a precise notion of fuzzy robustness that is distinct from interval enclosures (pure bounds) and stochastic robustness (average-case guarantees). We also formalize fuzzy topological entropy via the extension principle and discuss its basic structural properties under mild continuity assumptions. For chaos-based image encryption, fuzzification provides an uncertainty-aware key representation and stabilizes cryptographic indicators across α -cuts as follows: in our experiments, NPCR remains within 99.58 – 99.64 % , UACI within 33.41 – 33.52 % , and the cipher entropy is near 8 bits, while pixel correlation stays close to zero. These results support fuzzy skew maps as a robust primitive for secure information systems operating under parametric uncertainty.