Research Of Hopf Bifurcation And Hopf-Pitchfork Bifurcation In Coupled Neural Oscillator System With Delay

Recently,oscillation,as one of the ubiquitous dynamic behaviors of oscillatory neural networks,has received attractive attentions because it plays a key role in sensory,cognitive,motor and so on.Many researchers suggest that oscillation is caused by neuronal connections interacted by electrical or chemical synapses.In order to describe the neuronal interactions between inhibitory and excitatory populations,many oscillatory neural networks are proposed such as integrate and fire model,McGregor model and Wilson–Cowan model.Based on these models,the dynamic behaviors including stability,synchronization,Hopf bifurcation,pitchfork bifurcation and chaos are widely studied.The main content and innovation points in this paper are shown as following:(1)Hopf bifurcation analysis of reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delayA reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay is proposed,and its dynamic behaviors under the Neumann boundary conditions are investigated.By constructing a basis of phase space based on the eigenvectors of Laplace operator,the characteristic equation of this system is obtained.Then,the local stability of zero solution and the occurrence of Hopf bifurcation are established by regarding the time delay as the bifurcation parameter.In particular,by using the normal form theory and center manifold theorem of the partial differential equation,the normal forms are obtained,which determines the bifurcation direction and the stability of the periodic solutions.Finally,two examples are given to verify the correctness of the theoretical results.(2)Hopf-pitchfork bifurcation analysis of a coupled neural oscillator system with excitatory-to-excitatory connection and time delayFirst,the pitchfork bifurcation and the patterns of equilibria are investigated.It is found that the system has one,three and five equilibria as the coupled weight increases.Then,by choosing the connection weight and time delay as the bifurcation parameters,the conditions where Hopf–pitchfork bifurcation occurs are obtained.Moreover,the normal form of Hopf-pitchfork bifurcation and the bifurcation diagrams are obtained by using the normal forum and center manifold.Analyzing the bifurcation diagram,we find that the system exhibits many complex phenomena such as co-existence of two synchronous equilibria or two anti-synchronous equilibria,two stable periodic solutions,and two quasi-periodic attractors,the correctness of which is validated by numerically simulations.