Normal Forms Of A Ring Modeled By A System Of Neutral Functional Differential Equations

This thesis is devoted to stability, normal forms and codimension one bifur-cation in a ring of n identical neurons of modeled by neutral functional differential equations, which has delays and some kind of symmetry. And this symmetry can be identified by the Lie group Dn (?) Z2. This thesis is organized as follows:Firstly, linear stability is investigated by analyzing the characteristic equation of linearized system. By means of space decomposition, we subtly discuss the dis-tribution of zeros of the characteristic equation, and then we derive some sufficient conditions ensuring that all the characteristic roots have negative real parts, such that the trivial solution is asymptotically stable.Secondly, according to the distribution of zeros of the characteristic equation, we distinguish six cases to discuss normal forms and codimension one bifurcation (including Hopf bifurcation and Fold bifurcation) of the system near the bifurcation values. Spatio-patterns of Hopf bifurcating periodic solutions are investigated by means of isotropy subgroups. The concrete coefficients of normal forms for each cases are obtained as well. Based on the normal forms, we discuss all of the codimension one bifurcations (including Hopf bifurcation and Fold bifurcation), and obtain some dynamics of these solutions, such as stability, bifurcation direction and so on.Finally, we use numerical simulations to illustrate the main results of the paper.