Some Studies About The Hopf Bifurcation Problems Of Degenerate Differential Systems With Delay

In many practical fields, delay and degenerate phenomena are existing extensively, such as economy, engineering, biology and so on.About these phenomena many results have been obtained. But, as we know, delay and degenerate phenomena appear in many systems together, these systems are called degenerate differential system with delay. And some systems have two delay, we call these systems system with multiple delay.Some parameters'variation of models will cause the variation of stability of solutions, which can produce periodic solutions(or limit circle). Bifurcaion phenomenon is kind of it. This phenomenon is common in the degenerate differential system and ecolgical system This paper discusses the problems of Hopf bifurcation for three-neuron degenerate differential equations with delay and a neutral predator- prey model with two delay. We take delayτas a bifurcation parameter to investigate the hopf bifucation phenomena of thes two systems. By analyzing the associated characteristic transcendental equations of systems, it is found that Hopf bifurcation occurs when delay pass through a value, and using Hopf bifucation theorem, we obtain one condition for the existence of Hopf bifucation in systems. Furthermore, based on the center manifold theorem and the method of normal form, some results about properties of Hopf bifucation are obtained including the direction of Hopf bifucation and stablity of Hopf bifucating periodic solutions.In chapter one, the backgrounds and significance of the problems are given. The main work done in this paper is introduced.In chapter two, discusses the existence of Hopf bifurcation in three-neuron degenerate differential equations with delay. In chapter three, discusses the existence of Hopf bifurcation in a neutral predatorprdy model with two delay.In chapter four, studies the characters of Hopf bifucation.