Stability And Bifurcation In Ring Neural Networks With Multiple Delays

Ring networks have been found in a variety of neural structures such as neocortex, cerebellum,hippocampus and even in chemistry and electrical engineering.By studying ring networks,we can gain insight into the mechanisms underlying the behavior of re-current networks.In this thesis,we first consider the stability of the trivial solution to a three-unit ring network with multiple delays and without self-feedback.Most impor-tantly,we know how the number of the eigenvalues with positive real parts changes,which is fundamental for the discussion of bifurcation.Regarding the connecting weight a as the bifurcating parameter,we discuss three types of codimension one bifurcations,i.e., pitchfork bifurcation,Hopf bifurcation,and equivariant pitchfork bifurcation due to the D₃ equivariance of the system;in the half parameter plane determined by two parameters, we discuss two types of codimension two bifurcations,i.e.,equivariant pitchfork-Hopf bi-furcation and Hopf-Hopf bifurcation.Then we carry out a more complicated analysis to the case with self-feedback and obtain similar results.Finally,for the n-unit ring network, by means of space decomposition,we establish sufficient conditions on the stability of the trivial solution and also consider the existence and stability of nontrivial equilibria.This thesis consists of five chapters.In Chapter 1,we briefly address the background and history of neural networks, followed by the rationalization of neural networks as models of dynamical systems.We also expound and explain the dynamics of delayed neural networks and their applications. This gives the motivation of the study of this thesis.This chapter is concluded with some notations,definitions,and lemmas.The subject of Chapter 2 is three-unit neural networks with/without self-feedback. By analyzing the distribution of the zeros to the transcendental characteristic equation associated with the trivial.solution,we obtain some- sufficient conditions on the stability and instability of the trivial solution.These results are generalized to n-unit neural networks by means of space decomposition in this Chapter.In Chapter 3,we focus on three-unit neural networks without self-feedback.Finding out the way the eigenvalues with positive real parts changing,we consider bifurcation near the trivial solution.More precisely,regarding the connecting weight a as the bifurcation parameter and fixing the other parameters,we discuss codimension one bifurcations like pitchfork bifurcation,Hopf bifurcation and equivariant pitchfork bifurcation.Moreover, in the parameter plane determined by some two parameters,we discuss codimension two bifurcations caused by the intersections of two bifurcating curves.Finally,we demonstrate the rich dynamics of codimension two bifurcations with numerical simulations.Chapter 4 deals with the three-unit ring network with self-feedback.We mainly focus on the bifurcation phenomena near the trivial solution.Using some more complicated analysis,we obtain similar results as those in Chapter 3.Chapter 5 is devoted to the study of nontrivial equilibria of n-unit ring networks.We briefly discuss the existence and stability of nonzero synchronous equilibria.Moreover, when 3|n,we also get nonzero asynchronous equilibria such as standing wave equilibria and mirror-reflecting equilibria.