A globally stable attractor that is locally unstable everywhere

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.