@Preben Alsholm yes, Preben, that may well be all there is to it, but what the center manifold theory alerts us to is the possibility of a situation where the center manifold could be locally stable in the saddle-path sense (e.g. one stable trajectory out of an infinity of unstable ones) even when the Jacobian exhibits a positive eigenvalue.
If the dynamic system is a representation of a phenomenon observed in one of the physical sciences, chances are that such a stable path is of no practical relevance: The odds of an atom (or whatever) starting with just the exact initial values are essentially zero.
By contrast, in economics, for instance, the critical values (the stationary-state continuum, as economists usually call the indeterminate critical values) could represent a situation of maximum profit, in the context of a situation where rational, forward-looking, perfectly-informed maximizers would do everything in their power to get there, including selecting initial values that put them on a trajectory that gets there (if the choice of these initial values is up to them, the case for flow values but not generally for stock values). Thus in practice, it could, perhaps, be important to know of the existence of a stable trajectory on a center-manifold projection when the Jacobian exhibits a positive eigenvalue.
Well this is my limited understanding of the issues involved. My references are the 1981 book by Jack Carr (Applications of Centre Maniford Theory) and the 1990 (revised 2003) Springer book by Wiggins (Introduction to Applied Nonlinear Dynamical Systems and Chaos). Edit: Please do correct me if my interpretation is flawed.