Smoothness Properties of Semiflows for Differential Equations with State-Dependent Delays

Differential equations with state-dependent delay can often be written as $\dot x(t)=f(x_t)$ with a continuously differentiable map $f$ from an open subset of the space $C^1=C^1([-h,0],\mathbb R^n)$, $h>0$, into $\mathbb R^n$. In a previous paper we proved that under two mild additional conditions the set $X=\{\phi\in U:\dot\phi(0)=f(\phi)\}$ is a continuously differentiable $n$-codimensional submanifold of $C^1$, on which the solutions define a continuous semiflow $F$ with continuously differentiable solution operators $F_t=F(t,\,\cdot\,)$, $t\geqslant 0$. Here we show that under slightly stronger conditions the semiflow $F$ is continuously differentiable on the subset of its domain given by $t>h$. This yields, among others, Poincaré return maps on transversals to periodic orbits. All hypotheses hold for an example which is based on Newton's law and models automatic position control by echo.