Beyond Quaternions: Adaptive Fixed-Time Synchronization of High-Dimensional Fractional-Order Neural Networks Under Lévy Noise Disturbances

This paper develops a unified synchronization framework for octonion-valued fractional-order neural networks (FOOVNNs) subject to mixed delays, Lévy disturbances, and topology switching. A fractional sliding surface is constructed by combining I1−μeg with integral terms in powers of |eg|. The controller includes a nonsingular term −ρ2gsgc2sign(sg), a disturbance-compensation term −θ^gsign(sg), and a delay-feedback term −λgeg(t−τ), while dimension-aware adaptive laws ,CDtμρg=k1gN∥sg∥c2 and ,CDtμθ^g=k2gN∥sg∥ ensure scalability with network size. Fixed-time convergence is established via a fractional stochastic Lyapunov method, and predefined-time convergence follows by a time-scaling of the control channel. Markovian switching is treated through a mode-dependent Lyapunov construction and linear matrix inequality (LMI) conditions; non-Gaussian perturbations are handled using fractional Itô tools. The architecture admits observer-based variants and is implementation-friendly. Numerical results corroborate the theory: (i) Two-Node Baseline: The fixed-time design drives ∥e(t)∥1 to O(10−4) by t≈0.94s, while the predefined-time variant meets a user-set Tp=0.5s with convergence at t≈0.42s. (ii) Eight-Node Scalability: Sliding surfaces settle in an O(1) band, and adaptive parameter means saturate well below their ceilings. (iii) Hyperspectral (Synthetic): Reconstruction under Lévy contamination achieves a competitive PSNR consistent with hypercomplex modeling and fractional learning. (iv) Switching Robustness: under four modes and twelve random switches, the error satisfies maxt∥e(t)∥1≤0.15. The results support octonion-valued, fractionally damped controllers as practical, scalable mechanisms for robust synchronization under non-Gaussian noise, delays, and time-varying topologies.