The Numerical Analysis Of Hopf Bifurcation For Delay Differential Equation

In this paper, a class of Van der Pol equation with delay is considered. The property of the analytical and numerical Hopf bifurcation of the system is investigated, including the direction of Hopf bifurcation and the stability of periodic solution.Firstly, the Hopf bifurcation theories of the analytical solution and the numerical solution are listed, which include the direction of Hopf bifurcation, and the stability of bifurcation periodic solution. Then, the numerical methods for delay differential equations are introduced.Secondly, a class of Van der Pol equation with delay is introduced. We prove that a sequence of Hopf bifurcations occur at the equilibrium as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using the theory of normal form and center manifold. Numerical simulations are given to support the theoretical results.Finally, we use Runge-Kutta method (trapezoidal method) to the Van der Pol equation, and prove the existence of numerical Hopf bifurcation, that is, for h = 1m, it is proved that when the Van der Pol equation has a Hopf bifurcation atτ*, then the discrete system also undergoing a Hopf bifurcation in the neighborhood ofτ*. By discussing the roots of characteristic equation, we give the stability domain of numerical solutions, and then according to the theories that we already know, we give the direction of numerical Hopf bifurcation and stability of periodic solution. We also give some numerical examples of the system to support the theories above.