Stability and Hopf bifurcation of a complex-valued neural network with two time delays

Abstract

In this paper, a class of complex-valued neural networks with two time delays is considered. By considering that the activation function can be expressed by separating into its real and imaginary part and regarding the sum of time delays as a bifurcating parameter, the dynamical behaviors that include local asymptotical stability and local Hopf bifurcation are investigated. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the sum of time delays passes through a sequence of critical value. The linearized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.