Hopf Bifurcation Of Several Kinds Of Reaction-diffusion Neural Networks With Time Delay

As we all know,the research about the bifurcation behavior of neural networks has always been a hot topic,and it is also an important and difficult problems in the study of dynamic behavior of neural network.The time-delay reaction-diffusion neural network,as an extension of the common neural network,has become a key research field for scholars because it is more in line with the characteristics of real biological neural networks and has more dynamic behavior,is more suitable for industrial development and application.In this paper,we study stability and Hopf Bifurcation for a reaction-diffusion neutral neuron system with time delay and Turing instability and bifurcation analysis in a reaction-diffusion neural network with time delay.The main contents and innovations of this paper are as follows:(1)Stability and Hopf Bifurcation for a reaction-diffusion neutral neuron system with time delay.In this paper,a type of reaction–diffusion neutral neuron system with time delay under homogeneous Neumann boundary conditions is considered.By constructing a basis of phase space based on the eigenvectors of the corresponding Laplace operator,the characteristic equation of this system is obtained.Then,by selecting time delay and self-feedback strength as the bifurcating parameters respectively,the dynamicbehaviors including local stability and Hopf bifurcation near the zero equilibrium point are investigated when the time delay and self-feedback strength vary.Furthermore,the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the normal form and the center manifold theorem for the corresponding partial differential equation.Finally,two simulation examples are given to verify the theory.(2)Turing instability and bifurcation analysis in a reaction-diffusion neural network with time delay.In this paper,a class of delayed reaction-diffusion neural network under homogeneous Neumann boundary conditions is considered.First,the conditions of Turing instability of the zero equilibrium are obtained.Our conditions show that the diffusion coefficients can lead to the diffusion-driven instability and the occurrence of stripe spatial pattern.Then,based on the Turing instability conditions,some sufficient conditions of Hopf bifurcations are obtained.Comparing with the existing works,our results contain the parameter space of the diffusion coefficients,which are more exact.Finally,numerical results not only validate the obtained theorems,but also show that the diffusion coefficients have a marked impact o in pattern formation.With the diffusion coefficients increasing,different patterns appear.