Eugen Stumpf

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

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 66, 3 - 44 (2011)

 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.

Keywords:   Center-unstable manifold, functional differential equation, state-dependent delay

Notes:   This is a part of the author's doctoral dissertation \cite{stu15}, written under the supervision of Reiner Lauterbach (University of Hamburg) and Hans-Otto Walther (University of Giessen) at the University of Hamburg. The author wishes to express his thanks to both advisors for their outstanding support.

karin.martin@uni-rostock.de
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Titel des Artikels korrigiert am 07.06.2012