Research On Hopf Bifurcation Of Delayed Complex-valued Neural Network

Bifurcation research of neural networks is one of the key and difficult points in the study of the behavior of neural network dynamics.It has very important theoretical and practical significance.As an extension of the ordinary real-valued neural networks,the complex-valued neural networks have gradually become the focus of researchers due to its irreplaceable natures and advantages,such as dealing with complex signal related problems,small network structure,strong computing power,and strong generalization ability.In this paper,the Hopf bifurcations of two common types of delayed complex-valued neural networks are researched in mathematical proofs and numerical simulations.The research contents and innovations in this thesis are as follows:(1)A complex-valued tri-neuron bidirectional associative memory neural network with multiple time delays is considered.In complex-valued neural networks,the activation function could be separated into its real and imaginary parts.By taking the multiple time delays as the bifurcation parameter,the dynamical behaviors including local stability and local Hopf bifurcation are investigated.Using the Jacobian matrix and analyzing the distribution of the eigenvalues of the associated characteristic equation,the critical value of bifurcation parameter is obtained and the Hopf bifurcation occurs.Furthermore,the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are also investigated by the normal form method and center manifold theorem.Finally,some numerical simulations are performed to verify the theoretical results.(2)A class of complex-valued reaction-diffusion neural network with time delays under Dirichlet boundary conditions is considered.By using the properties of the Laplacian operator and separating the neural network into real and imaginary parts,the corresponding characteristic equation of neural network is obtained.Then,the dynamical behaviors including the local stability,the existence of Hopf bifurcation of zero equilibrium are investigated.Furthermore,by using the normal form theory and the center manifold reduction theorem,the explicit formulae which determine the direction of bifurcations and stability of bifurcating periodic solutions are obtained.Finally,a numerical simulation is carried out to illustrate the results.