On The Numerical Hopf Bifurcations For Delay Differential Equations

The delay differential systems are widely applied in various fields that existed in from natural world to human being society, from natural science and engineering technique to the Soci-Science. It is meaningful both for the systems themselves and to improve the studies on other science to carry out a deeply research on the delay differential systems. The theoretical and numerical researches on the systems are very important.Hopf bifurcation, as a kind of important bifurcation behavior, is generally occurred since the singular behavior of the systems which depend on the parameters, which describes that the phenomenon of the periodic solution derived from the equilibrium when the parameter passes through some critical value. There are three questions will be discussed, the existence of the bifurcation, the direction of the bifurcation and the bifurcation parameter value.Consider the following delay differential system with parameter y′( t ) = f ( y (t ), y (t ? 1),α), y∈n,α∈(1) and the corresponding Runge-Kutta method. In this dissertation we will consider that whether the numerical solution defined by the Runge-Kutta method could preserve the Hopf bifurcation of the original system (1) or not.We assume that f ( x , y ,α)∈Cp +1( n×n×, n), and there exists anα* and its some neighborhoodδ(α*), there holds f ( 0,0,α)=0, ?α∈δ(α*). We use Kronecker product to give out the general form of the Runge-Kutta method since it is implicit, and by taking advantages of the smoothness of function f ( x,y,α) and the skills of matrix computing, we finally obtain the explicit form of the characteristic equation. Furthermore, by the direct comparison and analysis between the auxiliary matrix we introduced and the matrix which forms the characteristic equation of the original system, we obtain the characteristic structure of the Runge-Kutta discretization. Finally, we, by using of the Hopf bifurcation theory of map, proved that if the system (1) has a Hopf bifurcation atα*, then there will be a Hopf bifurcation occurs in the Runge-Kutta method when the step size h 1 ( m)= m∈+ is small enough, moreover the bifurcation parameterαh =α* +O(hp), with p is the order of the Runge-Kutta method.