Topological horseshoes and delay differential equations

Abstract

We show that if an ordinary differentialequation $x'=f(x)$, where $x\in \mathbb R^n$ and $f \in\mathcal C^1$, has a topological horseshoe, then thecorresponding delay equation $x'(t)=f(x(t-h))$ for small $h >0$also has a topological horseshoe, i.e. symbolic dynamics and aninfinite number of periodic orbits. A method of computation of $h$is given in terms of topological properties of solutions ofdifferential inclusion $x'(t) \in f(x(t)) +\bar B(0,\delta)$.