On the Stability of a Periodic Solution of a Differential Delay Equation

This paper considers the class of scalar, first order, differential delay equations $y'(t) = - f(y(t - 1))$. It is shown that under certain restrictions there exists an annulus A in the $(y(t),y(t - 1))$-plane whose boundary is a pair of slowly oscillating periodic orbits and A is asymptotically stable. These results are applied to the frequently studied equation $x'(t) = - \alpha x(t - 1)[1 + x(t)]$. The techniques used are related to the Poincaré–Bendixson method, used in the $(y(t),y(t - 1))$-plane.