Erratic solutions of simple delay equations

We give an example of a smooth function $g:\mathbb{R}\longrightarrow \mathbb{R}$ with only one extremum, with $\mathrm{ sign } g(x) = - \mathrm{ sign } g(-x)$ for $x \neq 0$, and the following properties: The delay equation $\dot x (t) = g(x(t-1))$ has an unstable periodic solution and a solution with phase curve transversally homoclinic to the orbit of the periodic solution. The complicated motion arising from this structure, and its robustness under perturbation of $g$, are described in terms of a Poincaré map. The example is minimal in the sense that the condition $g' < 0$ (under which there would be no extremum) excludes complex solution behavior. Based on numerical observations, we discuss the role of the erratic solutions in the set of all solutions.