Bernhard Lani-Wayda

Erratic solutions of simple delay equations

We give an example of a smooth function g : R −→ R with only one extremum, with sign g(x) = −sign g(−x) for x 6= 0, and the following properties: The delay equation ẋ(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.