Stability And Hopf Bifurcation Analysis On Delay Differential Equations

System as a common differential equation with delay is closely related to daily life. The differential equation with delay research plays a very important role for us to development of society. In recent years, differential equation with delay has been an important subject for researchers, and more and more scholars begin to pay close attention to the field of it.Differential equation with delay has instability problem caused by non-linear. The instability problem makes the dynamic properties of differential equation with complex forms.Considering the system response lagging, delay is introduced, which makes the study of differential equation more and more complicated. Considering the influence of different factors sometimes, effect equation should been introduced into the system, which makes the system more complicated. Therefore some new theories suitable to system with delay and effect equation should be explored, by which the dynamics properties of stability, bifurcation,periodic solutions at balance point and direction of periodicity can be analyzed. Therefore, in the theoretical research the differential equation with delay is more difficult than the ordinary differential equation.In this paper, first of all, non-linear system of with two delays is analyzed, especially the related system characteristics. This paper describes the difference between the new system and the original one, calculates mathematically the existence condition of the nonnegative equilibrium, discusses the locally asymptotic stability and instability at nonnegative equilibrium point by knowledge of the characteristic equation of Jacobi matrix, proves the existence of the Hopf bifurcation, and obtains Hopf bifurcation point by the mathematical calculation. Understand center manifold theorem and normative theory, and combine them to further calculate the direction of the Hopf bifurcation, orbit period and the transformation of period according to delay. To solve the equation above, first, expand the described model by Taylor formula in order to obtain the expression of correlation coefficient between the linear part and nonlinear part. Second, solve the expression deciding the direction of the Hopf bifurcation and orbit period by relevant knowledge. Finally, simulate the systems model with time-delay under the obtained conditions by delay differential equations. We can observe changes in system movement characteristics along with the parameters changing under different conditions. As the time-delay increasing, gradual stability at system balance point is converted into instability, and the Hopf bifurcation occurs at a certain point. According to the obtained theory it is derived that delay can change the dynamic characteristics of the differential equation.Then on, delay is introduced to obtain the system, which is the second differential equation with delay in this paper. By the method of the same, discuss the stability of theequilibrium point and the existence of the Hopf bifurcation, find out the bifurcation point, and obtain the direction of the Hopf bifurcation at bifurcation point, orbit period and the transformation of period according to delay by Taylor formula expansion, center manifold theorem and normative theory.Finally, transform the original system adding in a delay, we get a new system with delay,which is the third dynamic system model analyzed below. The stability of the equilibrium point and the existence of Hopf bifurcation at positive equilibrium point are discussed, and the parameters at bifurcation points are determined by basically the same research method and the lemma. Using Taylor formula, center manifold theorem and normative theory to obtain the direction of the Hopf bifurcation at bifurcation point, orbit period and the transformation of period according to delay.