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

It is shown that if the delay differential equation undergoes a Hopf bifurcation at τ=τ∗, then the discrete scheme undergoes a Hopf bifurcation at τ(h)=τ∗+O(hp) for sufficiently small step size h, where 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.