Turing-Hopf bifurcation of reaction-diffusion neural networks with leakage delay

Abstract

In artificial neural networks, the diffusion phenomenon of electrons exists inevitably, due to the electromagnetic field of neural networks is heterogeneous. In this paper, we study the spatio-temporal dynamical behaviors of a reaction-diffusion neural network with leakage delay. By analyzing the corresponding characteristic equation, the sufficient and necessary conditions of Turing instability are obtained and the existence of Turing, Hopf, and Turing-Hopf bifurcations is also established. Furthermore, the truncated normal form up to third order is derived to understand and classify the spatio-temporal dynamics close to the Turing-Hopf bifurcation point. By numerical simulations, we find a pair of spatially inhomogeneous periodic solutions and illustrate the effects of time delays and spatial diffusion on the spatio-temporal dynamics of the model.