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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Erratic solutions of simple delay equations

Author: Bernhard Lani-Wayda
Journal: Trans. Amer. Math. Soc. 351 (1999), 901-945
MSC (1991): Primary 34K15, 58F13, 70K50
MathSciNet review: 1615995
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Abstract: 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.

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Additional Information

Bernhard Lani-Wayda
Affiliation: Mathematisches Institut der Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany

Received by editor(s): September 4, 1996
Additional Notes: Supported by the Deutsche Forschungsgemeinschaft within the Schwerpunkt Analysis, Ergodentheorie und Effiziente Simulation Dynamischer Systeme.
Article copyright: © Copyright 1999 American Mathematical Society