Stability and Hopf bifurcation of fractional-order complex-valued neural networks with time-delay

Integer-order complex-valued neural networks (CVNNs) have drawn much attention, and which have already been well discussed. Fractional complex-valued neural networks (FCVNNs) are more suitable for describing the dynamical properties of neural networks, but have rarely been discussed. This paper deals with the dynamics of a class of nonlinear delayed FCVNNs and its main focus is on stability and Hopf bifurcation. Firstly, the concerned fractional-order complex-valued neural networks with time delay are transformed into corresponding a class of nonlinear delayed fractional-order real-valued neural networks (FRVNNs). Then, with the help of the linearization method, Laplace transform and fractional calculus theory, some sufficient conditions that make sure the considered system are asymptotic stability are established. Take the time delay as the critical values, which Hopf bifurcation can occur. At the same time, the quantitative relationship between the order of the system and the bifurcation point is given. In addition, it shows that the onset of the bifurcation point is delayed as the fractional order decreases. At last, one numerical example is described to illustrate our obtained theoretical results are correct and valuable.