Invariance and Noninvariance of Center Manifolds of Time-t Maps with Respect to the Semiflow
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This paper contains two examples related to the problem of whether the local center manifolds of the time-t maps of a semiflow at an equilibrium point are also invariant under the semiflow itself. An ordinary differential equation in ${\Bbb{R}}^2$ is given to show that, for almost all choices of the localization functions, the center manifold of the time-1 map at the origin is not locally invariant under the flow. The second example is an abstract functional differential equation. Although a variation-of-constants formula is not known to exist in the phase space, we prove that the classical approach works: The semiflow can be modified outside a neighborhood of the equilibrium point so that the center manifold of the time-t map of the modified semiflow will be locally invariant under the original semiflow.