This paper introduces a novel class of fuzzy neural network interpolation operators designed for the approximation andsynchronization of fuzzy-valued functions under weak hybrid continuity. To address structural irregularities and topologicalimprecision in fuzzy data, we propose operators that act level-wise on α-cuts and are compatible with Kuratowski uppersemicontinuity and generalized Hausdorff-type metrics. The proposed framework generalizes classical models, such as theCardaliaguet-Euvrard approach, by establishing a Jackson-type fuzzy approximation theorem under hybrid moduli of continuity.To the best of our knowledge, this is the first framework that rigorously integrates hybrid fuzzy interpolation into fuzzy-valued reservoir computing (FVRC) for the synchronization of chaotic systems under fuzzy uncertainty. Extensive numericalsimulations—including fuzzy versions of the Kuramoto–Sivashinsky equation, the complex Ginzburg–Landau system, and afuzzy chemo-fluidic insulin oscillator—demonstrate improved accuracy, robustness to noise, and preservation of fuzzy structuralproperties. These results provide a solid theoretical and computational foundation for next-generation fuzzy inference systemsin biomedical and uncertain environments.The proposed framework unifies classical and modern approaches by rigorously establishing a Jackson-type theorem underhybrid fuzzy metrics. The proposed operators are therefore suitable for intelligent systems and biomedical applications thatrequire structurally faithful fuzzy approximations. Notably, the operator guarantees convergence even in the presence ofstructural discontinuities that are not addressed by classical fuzzy interpolation schemes. Mathematics Subject Classification: 41A25; 46S40; 68T05; 93C41; 35Q93

This paper presents a finite element scheme with P1 and P2 basis functions, using an Euler-Taylor-Galerkin method described in [1], for a nonlinear model describing the behavior of a new chemo-fluidic oscillator[2]. This model is expressed by coupling an ordinary differential equation describing the hydrogel dynamics, the equation of nonlinear transport and an auxiliary equation that determines the flow volume. The numerical solution is constructed by taking a semi-discretization in time of the transport equation, employing Taylor series expansions in time forward that includes second and third order derivatives in time, avoiding instability problems. In this semi-discrete equation, the spatial variable is approximated by a finite element formulation according to Galerkin. Some simulations are performed taking different initial conditions for the hydrogel concentration. The numerical results describe the oscillatory behavior of the system as in [2], where the Matlab tools are used as a black box.

This study examines second-order dynamical systems incorporating Tikhonov regularization. It focuses on how nonlinearities induce bifurcations and chaotic dynamics. By using Lyapunov functions, bifurcation theory, and numerical simulations, we identify critical transitions that lead to complex behaviors like strange attractors and chaos. The findings provide a theoretical framework for applications in optimization, machine learning, and biological modeling. Key contributions include stability conditions, characterization of chaotic regimes, and methods for managing nonlinear instabilities in interdisciplinary systems.

We introduce fuzzy skew maps as a levelwise ($\alpha$-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 $\alpha \in [0,1]$, the induced crisp family $\{F(·,q):q\in {\left[\tilde{q}\right]}_{\alpha }\}$ 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 $\alpha$-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.

This study redefines the analysis of Devaney chaos in multiple mappings from a set-valued perspective and introduces new conditions to characterize their chaotic behavior. As an innovative advancement, we develop computational algorithms to detect and visualize chaotic features such as transitivity and sensitivity. These algorithms provide tools to explore complex dynamics in higher-dimensional systems, validating theoretical concepts and opening new research avenues in chaos theory.