Stability and Hopf Bifurcation of Fractional-Order Quaternary Numerical Three-Neuron Neural Networks with Different Types of Delays - Qiankun Wang; Tianzeng Li; Yu Wang; Xiaowen Tan (Axioms)
In this paper, the stability and Hopf bifurcation of fractional-order quaternion-valued neural networks (FOQVNNs) with various types of time delays are studied. The fractional-order quaternion neural networks with time delays are decomposed into an equivalent complex-valued system through the Cayley–Dickson construction. The existence and uniqueness of the solution for the considered fractional-order delayed quaternion neural networks are proven by using the compression mapping theorem. It is demonstrated that the solutions of the involved fractional delayed quaternion neural networks are bounded by constructing appropriate functions. Some sufficient conditions for the stability and Hopf bifurcation of the considered fractional-order delayed quaternion neural networks are established by utilizing the stability theory of fractional differential equations and basic bifurcation knowledge. To validate the rationality of the theoretical results, corresponding simulation results and bifurcation diagrams are provided. The relationship between the order of appearance of bifurcation phenomena and the order is also studied, revealing that bifurcation phenomena occur later as the order increases. The theoretical results established in this paper are of significant guidance for the design and improvement of neural networks.