The Numerical Analysis Of Hopf Bifurcation For Differential Equations With Two Delays

If the solution of a differential dynamical system depends not only the current state, but also the state of some time or some period ago, then we will call it to be a delay differential dynamical system and which is an important part of functional differential equations. Because delay differential dynamic system appears in many areas of application, there are theoretical and practical meanings for researches on delay differential dynamical system.There are lots of difficulties to obtain the analytic solution because of the complexity of delay differential equations, so it is necessary to carry out the numerical research and the character of the dynamical system. Since bifurcation is a character of the delay differential dynamical systems, Hopf bifurcation is a kind of bifurcation phenomena which should be cared specially. So it is more important on researching Hopf bifurcation of the delay differential systems. Because delay differential dynamical system is more complicated, we will use the numerical method to study Hopf bifurcation.In this thesis, we study two models with two delays by using the method of numerical discrete and also analyze the Hopf bifurcation in discrete formal.First, we will use the linear multistep methods with strictly zeros stability to numerical discrete on the predator-prey system with two delays. When the Hopf bifurcation of this system exists, we will analysis discrete Hopf bifurcation preservation for the analytical Hopf bifurcation solution. We will get that the bifurcation direction and stability of the period solution for the numerical Hopf bifurcation solution is the same as the analytical solution. At last, the computational example is given to demonstrate the conclusions.Then, the numerical analysis of Hopf bifurcation for a kind of van der Pol equation with two delays is considered. Here, we use linear multistep methods with a strict zero-stablility and Runge-Kutta methods to discretize the system numerically, respectively. We prove that under the numerical discretization scheme system, Hopf bifurcation of the original equation is preserved. Furthermore, some numerical examples and simulations are made to demonstrate the conclusions.