Synchronization of Hypercomplex Neural Networks with Mixed Time-Varying Delays

This article discusses the fixed-time synchronization (FTS) of hypercomplex neural networks (HCNNs) with mixed time-varying delays. Unlike finite-time synchronization (FNTS) based on initial conditions, the settling time of FTS can be adjusted to meet the needs. The state vector, weight matrices, activation functions, and input vectors of HCNNs are all hypercomplex numbers. The techniques used in complex-valued neural networks (CVNNs) and quaternion-valued neural networks (QVNNs) cannot be used directly with HCNNs because they do not work with eight or more dimensions. To begin with, the decomposition method is used to split the HCNNs into \((n+1)\) $(n+1)$ real-valued neural networks (RVNNs) applying distributive law to handle non-commutativity and non-associativity. A nonlinear controller is constructed to synchronize the master-response systems of the HCNNs. Lyapunov-based method is used to prove the stability of an error system. The FTS of mixed time-varying delayed HCNNs is achieved using a suitable lemma, Lipschitz condition, appropriate Lyapunov functional construction, and designing suitable controllers. Two different algebraic criteria for settling time have been achieved by employing two distinct lemmas. It is demonstrated that the settling time derived from Lemma 1 produces a more precise result than that obtained from Lemma 2. Three numerical examples for CVNNs, QVNNs, and octonions-valued neural networks (OVNNs) are provided to demonstrate the efficacy and effectiveness of the proposed theoretical results.