The $2$-Dimensional Attractor of $x’(t)=-𝜇 x(t) + f(x(t-1))$

The equation $x'(t) = - \mu x(t) + f(x(t-1))$, with $\mu \geq 0$ and $xf(x) \le 0$ for $0\neq x\in {\mathbb R}$, is a prototype for delayed negative feedback combined with friction. Its semiflow on $C=C([-1,0],{\mathbb R})$ leaves a set $S$ invariant, which also plays a major role for the dynamics on the full space $C$. The main result determines the attractor of the semiflow restricted to the closure of $S$ for monotone, bounded, smooth $f$. In the course of the proof, Walther derives Poincaré-Bendixson theorems for differential-delay equations. The method used here is unique in its use of winding numbers and homotopies in nonconvex sets.