In this paper we show that first order differential delay equations with negative feedback can possess asymptotically stable rapidly oscillating solutions. We construct an analytically tractable example in which the feedback is piecewise constant. In this case, the continuous-time dynamics on a proper subset of the phase space can be reduced exactly to a three-dimensional discrete-time map. The existence and stability properties of the delay equation's rapidly oscillating periodic solutions are given by the existence and stability of one of the fixed points of the corresponding map. When the feedback is smoothed appropriately, the stable rapidly oscillating periodic solution is shown to persist.
"Stable rapidly oscillating solutions in delay differential equations with negative feedback." Differential Integral Equations 12 (6) 811 - 832, 1999. https://doi.org/10.57262/die/1367241477