| Contents |
| We consider a class of functional differential equations of the form
\begin{equation}\label{eq: FDE}
\dot{x}(t)=f(x_{t})
\end{equation}
with $f$ defined on an open subset of $C^{1}([-h,0],\mathbb{R}^{n})$, $h>0$.
Under certain conditions, which are typically satisfied in cases where Eq. \eqref{eq: FDE} represents an autonomous differential equation with state-dependent delay, the associated Cauchy problems define a smooth semiflow on a submanifold of $C^{1}([-h,0],\mathbb{R}^{n})$. In particular, it is known that at a stationary point of the semiflow there exist so-called local center-unstable manifolds. Here, we discuss an attraction property of these manifolds. More precisely, we prove that, after fixing some local center-unstable manifold $W_{cu}$ of Eq. \eqref{eq: FDE} at a stationary point $\phi$, each solutions which exists and remains close enough to $\phi$ for all $t\geq 0$ converges exponentially for $t\to\infty$ to a solution on the local center-unstable manifold $W_{cu}$. |
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