  This work concerns the dynamics of nonlinear systems that are subjected to
delayed self-feedback. Perturbation methods applied to such systems give rise
to slow flows which characteristically contain delayed variables. We consider
two approaches to analyzing Hopf bifurcations in such slow flows. In one
approach, which we refer to as approach I, we follow many researchers in
replacing the delayed variables in the slow flow with non-delayed variables,
thereby reducing the DDE slow flow to an ODE. In a second approach, which we
refer to as approach II, we keep the delayed variables in the slow flow. By
comparing these two approaches we are able to assess the accuracy of making the
simplifying assumption which replaces the DDE slow flow by an ODE. We apply
this comparison to two examples, Duffing and van der Pol equations with
self-feedback.
