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