Relaxed Research on Synchronization Problem of Fractional-Order Fuzzy Octonion-Valued BAM Neural Networks by the Non-Decomposition Method on the High-Dimension Oblique Field

This paper develops a direct analytical framework for synchronizing and controlling fractional-order octonion-valued fuzzy bidirectional associative memory neural networks (FOOVFBAMNNs). Octonion algebra is neither commutative nor associative, which limits the application of standard analytical tools. To address this challenge, we first propose a generalized Cauchy–Schwarz inequality tailored to the octonionic domain, which operates directly without relying on system decomposition. This inequality lays the groundwork for a Lyapunov-based stability analysis that retains the system’s inherent geometric structure to avoid decomposition into real-valued components. Based on this framework, we derive concise 2-norm inequality criteria, which are sufficient to guarantee Mittag-Leffler synchronization of the proposed model. We also employ a Particle Swarm Optimization (PSO) algorithm to systematically optimize the flexible parameters in the generalized inequality, enhancing the practical performance of the synchronization scheme. To validate the effectiveness of the proposed method, we apply it to a multi-domain image restoration task. Numerical experiments verify the performance of our method. In terms of Peak Signal-to-Noise Ratio (PSNR), the octonion-valued network with PSO-tuned parameters achieves better results than its non-optimized counterpart as well as models constructed in complex or quaternion domains.