Numerical Study Of Hopf Bifurcation And Dissipativity Of Delay Differential Equations

In the past several decades, delay differential equations (DDEs) as a class of impor-tant mathematic models have been increasingly applied in many scientific fields, such asdemography, biology, modern physics, medicine, automatic control systems etc. As weall known, only some special DDEs can be solved explicitly, so it is practically valuableto develop some appropriate numerical methods. In practical, we find that, it is of greatsignificance to study whether the numerical methods can preserve the dynamical behaviorof the original system.In this dissertation, we construct some numerical methods for certain kinds of DDEs.Meanwhile, whether the numerical methods could preserve the Hopf bifurcation and thedissipativity of the original systems is researched, respectively.Firstly, a summary about the Hopf bifurcation and dissipativity of DDEs and theirnumerical discrete systems which have been studied by a lot of researchers is made.Secondly, we brie?y introduce two kinds of numerical methods for solving DDEs.On this basis, we present essential definitions and theorems for studying the Neimark-Sacker bifurcation and dissipativity of discrete systems. At the same time, we illuminatesome symbols used frequently in this paper.Thirdly, for the Nicholson blow?ies equation with delay, the dynamics of the nu-merical discrete system which obtained by a nonstandard finite-difference scheme is con-sidered. First of all, we try to show the stability of positive fixed point. The sufficientconditions under which the Neimark-Sacker bifurcation exists are derived by analyzingthe moving of characteristic roots and using the Neimark-Sacker bifurcation theorem.According to the normal form theory and the center manifold theorem, the explicit ex-pressions for determining the direction of the bifurcation and the stability of the closedinvariant curve are given. Through comparing the bifurcation of the origin system withthe numerical discrete system, it is showed that the solutions of the discretized modelshave the same properties as the original continuous models. Moreover, some numericalexamples are given which illustrate the correctness of the theoretical results.At last, we apply the Runge-Kutta method to a special class nonlinear neutral delaydifferential equations who have the dissipativity. The sufficient conditions under which the discrete system undergoes the dissipativity are obtained.