Qualitative Analysis Of Two Kinds Of Delay Differential Equations

Delay differential equations with after effect or dead time has important applications in many fields, such as physics, engineering, information, life sciences and so on. With further development of the study of functional differential equations, the qualitative properties of some delay differential equations can be obtained. So it is more and more common to study the dynamic systems with time delay. In this paper, we study the qualitative analysis of two kinds of delay differential equations.Firstly, a class of delay differential equations with scalar delay is studied. Using the method which was deduced by Faria and Magalhaes in 1995 to calculate the normal form of the functional differential equations, we get the canonical basis of Im(M12)c, Ker(M12)cand Im(M13)cunder the critical case that the corresponding linear system has a simple zero characteristic root, and derive the normal form of the delay differential equation on its center manifolds, then analyze some bifurcation that have occurred. Then we give a concrete differential equation which satisfies all assumptions of this paper and get the corresponding bifurcations.Secondly, we establish a epidemic model with some delay, which is on the basis of SEIRS model proposed by K.L.Cooke and P.van.den Driessche in 1996, and discuss the effect of vaccination to susceptible people. By analyzing, the existence, local and global asymptotic stability of equilibrium points are studied, and our results obtained are verified by some numerical simulation.