A globally stable attractor that is locally unstable everywhere

TL;DR

This paper presents examples of invariant manifolds that are locally unstable everywhere yet act as global attractors, challenging traditional stability notions and highlighting the importance of eigenvector rotation rates.

Contribution

It introduces invariant manifolds with local instability but global stability, explained through the rate of rotation of eigenvectors, supported by numerical simulations.

Findings

Invariant manifolds can be globally stable despite local instability.

Numerical simulations demonstrate the manifolds' role as global attractors.

The rate of rotation of eigenvectors influences stability transitions.

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.