Bifurcation Analysis Of FHN And BAM Neural Networks With Time Delay And Codimension Two

Rensently,various classes of neural networks such as Hopfied?Cohen-Grossberg?Bidirectional associative memory(BAM)?Fitzhugh-Nagumo(FHN)?cellular neural networks have been well applied in solving some optimization?signal processing?automatic control?image processing and other complex problems.These applications are mainly based on rich dynamic behaviors of neural networks.Therefore,the neural networks dynamics gradually developed into an important forefront topic in the research field of life sciences now,and have been investigated by many international scholars.In particular,the stability and bifurcation of the FHN and BAM neural networks systems have been paid more and more attention.Owing to the inevitability of the time delay in the signal transmission and the variability of the synaptic connection strength of the neurons of the biological nervous system,which makes the neural networks system produce more complicated dynamic behaviors.Thus,bifurcation nanlysis of neural networks with time delay and codimension two,which helps improve neural networks system and expand associated application in more fields.In this paper,Hopf-pitchfork bifurcation of a coupled FHN neural networks model with delay and Bogdanov-Takens bifurcation of a BAM neural networks model with neutral time delay are studied in depth.The main work and innovation are summarized as follows:First,for the coupled FHN neural networks model with time delay.First of all,by analyzing the distribution of the root of the characteristic equation of the linearized system at the equilibrium,the existence conditions on codimension two Hopf-pitchfork bifurcation are given.Then,by selecting the coupling strength and time delay as bifurcation parameters and using the center manifold reduction and normal form theory,the normal form for this singularity was given and used to analyze the bifurcation behaviors of the system.Finally,the bifurcation diagram of the Hopf-pitchfork bifurcation is analyzed in detail,and numerical simulation are performed to verify the correctness of theoretical analysis.Through the analysis we find that with the change of the two bifurcation parameters,there may be exist a stable fixed point,a stable periodic solution,a pair of stable fixed points,or the coexistence of a pair of stable fixed points and a stable periodic solution near the Hopf-pitchfork bifurcation critical point.It means that the neurons are not only in the resting or periodic spiking state,but also multistability coexistence with resting state and periodic spiking in the nervous system.Furthermore,with the change of bifurcation parameters,the state of a neuron system can transform from resting state to periodic spiking as well as from periodic spiking to resting state.Second,for the BAM neural network model with neutral time delay.Using the similar method as before.First by analyzing the distribution of the root of the characteristic equation of the linearized system at the equilibrium,we drive the critical conditions where codimension two Bogdanov-Takens and codimension three Triple-zero bifurcation occur.Then,regard the two connection weights as bifurcation parameters,the second-order and the third-order normal form of the Bogdanov-Takens bifurcation in the center are calculated respectively to analyze the bifurcation behaviors of the system.Finally,the bifurcation diagrams of the second-order and third-order normal form of the Bogdanov-Takens bifurcation are analyzed in detail,and numerical simulation are performed to verify the correctness of theoretical analysis.Based on the analysis,we find that with the change of the two bifurcation parameters,there may be exist some interesting phenomenon such as a pair of stable points,a stable periodic solutions or homoclinic orbits and so on near the B-T bifurcation critical point.