  This is a contribution to the program of dynamical approach to mean curvature
flow initiated by Colding and Minicozzi. In this paper, we prove two main
theorems. The first one is local in nature and the second one is global. In
this first result, we pursue the stream of ideas of \cite{CM3} and get a slight
refinement of their results. We apply the invariant manifold theory from
hyperbolic dynamics to study the dynamics close to a closed shrinker that is
not a sphere. In the second theorem, we show that if a hypersurface under the
rescaled mean curvature flow converges to a closed shrinker that is not a
sphere, then a generic perturbation on initial data would make the flow leave a
small neighborhood of the shrinker and never come back.
