How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation?

In this paper we consider the numerical solution of delay differential equations (DDEs) undergoing a Hopf bifurcation. Some authors use special methods to calculate bifurcating periodic solutions. We investigate what will happen when simple standard numerical methods (based on ODE methods) are used to obtain an approximate solution to the DDE. We want to establish whether the method will predict the true behaviour of the solution. We present three distinctive and complementary approaches to the analysis which together provide us with the result that ϑ-methods applied to a DDE will retain Hopf bifurcations and preserve their type, for sufficiently small h>0.