Erratic solutions of simple delay equations

Lani-Wayda, Bernhard

doi:10.1090/S0002-9947-99-02351-X

Erratic solutions of simple delay equations
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by Bernhard Lani-Wayda

Trans. Amer. Math. Soc. 351 (1999), 901-945

DOI: https://doi.org/10.1090/S0002-9947-99-02351-X

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Abstract:

We give an example of a smooth function $g:\mathbb {R} \to \mathbb {R}$ with only one extremum, with $\operatorname {sign} g(x) = - \operatorname {sign} g(-x)$ 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.

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