Bifurcation Analysis In Several Types Of Differential Systems With Delay

Bifurcation is an important subject in the field of delay differential equations. Socalled bifurcation means that, when the parameters cross through certain critical values,the phenomenon of the change of some structural properties in the system. The study ofbifurcation in delay differential equation is not only related to the theories of classicaldynamical system, but also related to the knowledge in topology, algebra and functionalanalysis. It is of great theoretical significance and practical backgrounds.Bifurcation is made up of three parts: local bifurcation, semi-local bifurcation andglobal bifurcation. Hopf Bifurcation is a kind of familiar local bifurcation, and it studiesthe changes of stability for equilibrium point as the parameters vary. In this way, smallamplitude oscillatory periodic solutions come out near the equilibrium point.In this thesis we mainly deal with Hopf bifurcation of several kinds of delay differ-ential equations. The main work of the thesis is summarized as follows.(1). The dynamics of the Mayer model with delays has been studied. Firstly, thelinear stabilities with one delay and two delays are investigated, respectively. It is foundthat there are stability switches for time delays, and Hopf bifurcations when time delayscross through some critical values. Then, using the normal form method and the centermanifold theorem, the direction and stability of the Hopf bifurcations are determined.(2). The bifurcation of a gene expression model with time delay have been investi-gated. Using the method of analyzing the distribution of the roots of associated charac-teristic equation, we can investigate the stability of zero equilibrium and the existence ofHopf bifurcation. An explicit algorithm for determining the direction of the Hopf bifur-cations and the stability of the bifurcating periodic solutions has been derived by usingthe theory of the center manifold and the normal forms method. We can conclude that a8-dimensional ordinary differential equation has no periodic solution via the Bendixsoncriterion for higher dimensional ordinary differential equations due to Li and Muldowney,and then the global existence of periodic solutions has been established using the globalHopf bifurcation result of Wu.(3). An age-structured model of a single species living in two identical patches with delay is considered. We discuss the characteristic equation of the system linearized nearthe homogeneous equilibria due to the conclusion of Beretta and Kuang, and investigatethe stability and Hopf bifurcations by analyzing the distribution of the roots of associatedcharacteristic equation. By the theory founded by Hassard and Kazarinoff, an explicitalgorithm for determining the direction of the Hopf bifurcation and stability of the bifur-cating periodic solutions are derived.(4). From the point of bifurcation, we investigate the cross-coupled laser model withdelay. The investigation confirms that a Hopf bifurcation occurs when the product of thecoupling strengths varies. As a result, the modulation of the coupling strengths would bea efficient and easily implementable method to control these undesirable oscillations.Besides, some numerical simulations are carried out for each model to illustrate theanalytic results.