Title: A globally stable attractor that is locally unstable everywhere,
Abstract: We construct two examples of invariant manifolds that despite being locally
unstable at every point in the transverse direction are globally stable. Using
numerical simulations we show that these invariant manifolds temporarily repel
nearby trajectories but act as global attractors. We formulate an explanation
for such global stability in terms of the `rate of rotation' of the stable and
unstable eigenvectors spanning the normal subspace associated with each point
of the invariant manifold. We discuss the role of this rate of rotation on the
transitions between the stable and unstable regimes.