Attraction property of local center-unstable manifolds for differential equations with state-dependent delay

摘要

In the present paper we consider local center-unstable manifolds at a stationary point for a class of functional differential equations of the form $dot{x}(t)=f(x_{t})$ under assumptions that are designed for application to differential equations with state-dependent delay. Here, we show an attraction property of these manifolds. More precisely, we prove that, after fixing some local center-unstable manifold $W_{cu}$ of $dot{x}(t)=f(x_{t})$ at some stationary point $arphi$, each solution of $dot{x}(t)=f(x_{t})$ which exists and remains sufficiently close to $arphi$ for all $tgeq 0$ and which does not belong to $W_{cu}$ converges exponentially for $toinfty$ to a solution on $W_{cu}$.