# American Institute of Mathematical Sciences

July  2009, 23(3): 1009-1033. doi: 10.3934/dcds.2009.23.1009

## Center-stable manifolds for differential equations with state-dependent delays

 1 Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3, Canada 2 Mathematisches Institut, Universität Gieβen, Arndtstr. 2, 35392 Gieβen

Received  November 2007 Revised  June 2008 Published  November 2008

Consider the functional differential equation (FDE) $\dot{x}(t)=f(x_t)$ with $f$ defined on an open subset of the space $C^1=C^1([-h,0],\R^n)$. Under mild smoothness assumptions, which are designed for the application to differential equations with state-dependent delays, the FDE generates a semiflow on a submanifold of $C^1$ with continuously differentiable time-$t$-maps. We show that at a stationary point continuously differentiable local center-stable manifolds of the semiflow exist. The proof uses results of Chen, Hale and Tan and of Krisztin about invariant manifolds of time-$t$-maps and their invariance properties with respect to the semiflow.
Citation: Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009
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