The Analysis Of Hopf Bifurcation And T-B Singularity For Delay Differential Equation

In this paper the formation and developments of delay differential equations bifurcation theories are presented. The direction of development of Hopf bifurcation in the present is studied. The existence problem and the numerical algorithm of delay differential equations numerical Hopf bifurcation are introduced in detail.The phenomenon of bifurcation exists in the nature extensively, such as physics, engineering, biology, medical science and economy and so on. The research of bifurcation is an important aspect for studies of the system dynamics property, which has important theories meaning and actual meaning. The research for numerical algorithm of the delay differential equations is important topic within the computation mathematics realm especially for the number analysis of continuous motive system dynamics property.At first the Hopf bifurcation theories of the analytical solution and the numerical solution are listed, which include the direction of bifurcation, the stability of bifurcation period solutions and so on. And then on, the numerical methods for delay differential equations are introduced。Based on these, the existence of numerical Hopf bifurcation of delay differential equations is investigated, that is, for , it is proved that when a class of biology model has a Hopf bifurcation ath = 1/mτ0, then the discrete system of the biology model also has a Hopf bifurcation in the neighborhood ofτ0. We use improved Euler method to the biology model, and then we obtain the characteristic equation. Discussing the roots of characteristic equation and according to the theories that have already know, we give the relations to the direction of Hopf bifurcation, bifurcation period solution stability and parameters.At last we study the theories about the singularity of the delay differential equations, and analysis the singularity of the biology model mentioned above. We study an example which has T-B singularity and prove that after used Euler forward method, the discrete system of the example still has T-B singularity.