On the global attractor of delay differential equations with unimodal feedback

We give bounds for the global attractor of the delay differential equation ẋ(t) = −μx(t) + f(x(t−τ)), where f is unimodal and has negative Schwarzian derivative. If f and μ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where f is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In this situation we determine the sharpest interval that contains the global attractor for any delay. In the absence of that condition, improving earlier results, we show that if the delay is sufficiently small, then all solution enter the domain where f ′ is negative. Our theorems then are illustrated by numerical examples using Nicholson’s blowflies equation and the Mackey-Glass equation.