Hans-Otto Walther, Universität Giessen

Titre/Title : A delay differential equation with a solution whose shortened segments are dense

Resume/Abstract :
Simple-looking autonomous delay differential equations $$x'(t)=f(x(t-r))$$ with a real function $f$ and single time lag $r>0$ can generate complicated (chaotic) solution behaviour, depending on the shape of $f$. The same could be shown for equations 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 $x$ between $t-r$ and $t$ defined as the function $x_t:[-r,0]\to\mathbb{R})$ given by $x_t(s)=x(t+s)$. So the delay alone may be responsible for complicated solution behaviour. In both cases the complicated behaviour which could be established occurs in a thin dust-like invariant subset of the infinite-dimensional Banach space or manifold of functions $[-r,0]\to\mathbb{R}$ on which the delay equation defines a nice semiflow. The lecture presents 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 space $C^1([-1,0],\mathbb{R})$. This implies a new kind of complicated behaviour 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.