This example is a bit confusing, as, properly speaking, the stable, unstable and neutral manifolds do not divide up the coordinate space; they divide up the phase space. In this case, the phase space has the structure of a tangent manifold: for every point in space (a 3D position), there is the collection of "tangent vectors": all possible velocities a particle might have. Some position-velocity pairs are driven towards the center manifold, others are flung away from it. Those that are in the center manifold are susceptible to small perturbations that generally push them about randomly, and often push them out of the center manifold. That is, small perturbations tend to destabilize points in the center manifold: the center manifold behaves like a saddle point, or rather, an extended collection of saddle points. There are dramatic counterexamples to this idea of instability at the center manifold; see Lagrangian coherent structure for detailed examples.