The existence and C¹-smoothness of local center-unstable manifolds for differential equations with state-dependent delay

MSC:	34K19	Invariant manifolds

Abstract:   The purpose of this work is to construct $C^{1}$-smooth local center-unstable manifolds at a stationary point for a class of functional differential equations of the form $\dot{x}(t)=f\left(x_{t}\right)$. Here the function $f$ under consideration is defined on an open subset of the space $C^{1}([-h,0], \mathbb{R}^{n})$, $h>0$, and satisfies some mild smoothness conditions which are often fulfilled when $f$ represents the right-hand side of a differential equation with state-dependent delay.