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

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

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

Redouane Qesmi ^1, and Hans-Otto Walther ^2,

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

Abstract
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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.

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

Mathematics Subject Classification: Primary: 34K19; Secondary: 37L10.

Citation: Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009

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