New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays

During the past decades, many works on Hopf bifurcation of fractional-order neural networks are mainly concerned with real-valued and complex-valued cases. However, few publications involve the quaternion-valued neural networks which is a generalization of real-valued and complex-valued neural networks. In this present study, we explorate the Hopf bifurcation problem for fractional-order quaternion-valued neural networks involving leakage delays. Taking advantage of the Hamilton rule of quaternion algebra, we decompose the addressed fractional-order quaternion-valued delayed neural networks into the equivalent eight real valued networks. Then the delay-inspired bifurcation condition of the eight real valued networks are derived by making use of the stability criterion and bifurcation theory of fractional-order differential dynamical systems. The impact of leakage delay on the bifurcation behavior of the involved fractional-order quaternion-valued delayed neural networks has been revealed. Software simulations are implemented to support the effectiveness of the derived fruits of this study. The research supplements the work of Huang et al. (Neural Netw 117:67–93, 2019).