We consider a non-smooth second order delay differential equation (DDE) that was previously studied as a model of the pupil light reflex. It can also be viewed as a prototype model for a system operated under delayed relay control. We use the explicit construction of solutions of the non-smooth DDE hand-in-hand with a numerical continuation study of a related smoothed system. This allows us to produce a comprehensive global picture of the dynamics and bifurcations, which extends and completes previous results. Specifically, we find a rich combinatorial structure consisting of solution branches connected at resonance points. All new solutions of the smoothed system were subsequently constructed as solutions of the non-smooth system. Furthermore, we show an example of the unfolding in the smoothed system of a non-smooth bifurcation point, from which infinitely many solution branches emanate. This shows that smoothing of the DDE may provide insight even into bifurcations that can only occur in non-smooth systems.
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