This work concerns the dynamics of nonlinear systems that are subjected to delayed self-feedback. An example would be Duffing's equation:

$x'' + x + a x' + b x^3 = 0$

in which the state variable $x(t)$ is delayed by a time $T$ and fed back into the differential equation:

$x'' + x + a x' + b x^3 = k x(t-T)$

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, we follow an approximation made by many researchers in replacing the delayed variables in the slow flow with non-delayed variables, thereby reducing the DDE (delay-differential equation) slow flow to an ODE.

In a second approach, 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.

[Paper by Simo Sah and Richard Rand]