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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Erratic solutions of simple delay equations
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by Bernhard Lani-Wayda PDF
Trans. Amer. Math. Soc. 351 (1999), 901-945 Request permission

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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Additional Information
  • Bernhard Lani-Wayda
  • Affiliation: Mathematisches Institut der Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany
  • Email: Bernhard.Lani-Wayda@math.uni-giessen.de
  • Received by editor(s): September 4, 1996
  • Additional Notes: Supported by the Deutsche Forschungsgemeinschaft within the Schwerpunkt Analysis, Ergodentheorie und Effiziente Simulation Dynamischer Systeme.
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 901-945
  • MSC (1991): Primary 34K15, 58F13, 70K50
  • DOI: https://doi.org/10.1090/S0002-9947-99-02351-X
  • MathSciNet review: 1615995