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DESCRIPTION:Titre/Title : A delay differential equation with a solution who
 se shortened segments are dense\n	\n	Resume/Abstract :\n	Simple-looking auton
 omous delay differential equations $$x'(t)=f(x(t-r))$$ with a real functio
 n $f$ and single time lag $r>0$ can generate complicated (chaotic) solutio
 n behaviour\, depending on the shape of $f$. The same could be shown for e
 quations with a variable\, state-dependent delay $r=d(x_t)$\, even for the
  linear case $f(\xi)=-\alpha\\,\xi$ with $\alpha>0$. Here the argument $x_
 t$ of the {\it delay functional} $d$ is the history of the solution $x$ be
 tween $t-r$ and $t$ defined as the function $x_t:[-r\,0]\to\mathbb{R})$ gi
 ven by $x_t(s)=x(t+s)$. So the delay alone may be responsible for complica
 ted solution behaviour. In both cases the complicated behaviour which coul
 d be established occurs in a thin dust-like invariant subset of the infini
 te-dimensional Banach space or manifold of functions $[-r\,0]\to\mathbb{R}
 $ on which the delay equation defines a nice semiflow. The lecture present
 s a result which grew out of an attempt to obtain complicated motion on a 
 larger set with non-empty interior\, as certain numerical experiments seem
  to suggest. For some $r>1$ we construct a delay functional $d:Y\to(0\,r)$
 \, $Y$ an infinite-dimensional subset of the space $C^1([-r\,0]\,\mathbb{R
 })$\, so that the equation $$x'(t)=-\alpha\\,x(t-d(x_t))$$ has a solution 
 whose {\it short segments} $x_t|_{[-1\,0]}$\, $t\ge0$\, are dense in the s
 pace $C^1([-1\,0]\,\mathbb{R})$. This implies a new kind of complicated be
 haviour of the flowline $[0\,\infty)\ni t\mapsto x_t\in C^1_r$. Reference:
  H. O. Walther\, {\em A delay differential equation with a solution whose 
 shortened segments are dense}.\\ J. Dynamics Dif. Eqs.\, to appear.\n\n \n
DTSTART:20180928T200000Z
DTEND:20180928T210000Z
LOCATION:Room 1104\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue
  Sherbrooke Ouest
SUMMARY:Hans-Otto Walther\, Universität Giessen
URL:/mathstat/channels/event/hans-otto-walther-univers
 itat-giessen-289935
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