Numerical Hopf bifurcation of Runge–Kutta methods for a class of delay differential equations

In this paper, we consider the discretization of parameter-dependent delay differential equation of the form y0ðtÞ ¼ f ðyðtÞ; yðt 1Þ; sÞ; s P 0; y 2 Rd: It is shown that if the delay differential equation undergoes a Hopf bifurcation at s ¼ s , then the discrete scheme undergoes a Hopf bifurcation at sðhÞ ¼ s þ Oðhp Þ for sufficiently small step size h, where p P 1 is the order of the Runge–Kutta method applied. The direction of numerical Hopf bifurcation and stability of bifurcating invariant curve are the same as that of delay differential equation.