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Journal of Dynamics and Differential Equations

, Volume 31, Issue 3, pp 1495–1523 | Cite as

A Delay Differential Equation with a Solution Whose Shortened Segments are Dense

  • Hans-Otto WaltherEmail author
Article

Abstract

We construct a delay functional \(d_Y\) on an infinite-dimensional subset \(Y\subset C^1([-r,0],\mathbb {R})\), \(r>1\), so that the delay differential equation
$$\begin{aligned}x'(t)=-\alpha \,x(t-d_Y(x_t)),\quad \alpha >0,\end{aligned}$$
has a solution whose short segments\(x_t|[-1,0]\) are dense in \(C^1([-1,0],\mathbb {R})\). This implies complicated behaviour of the trajectory \(t\mapsto x_t\) in \( C^1([-r,0],\mathbb {R})\).

Keywords

Delay differential equation State-dependent delay Dense orbit 

Mathematics Subject Classification

34 K 23 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität GießenGiessenGermany

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