Comments on “Stability Regions of Nonlinear Autonomous Dynamical Systems”

Abstract

The proofs of the groundbreaking theorems of [Chiang, Hirsch, and Wu. Stability Regions of Nonlinear Autonomous Dynamical Systems, IEEE Transactions on Automatic Control, 1988] rely on a lemma which states that if the stable manifold of a first hyperbolic closed orbit intersects transversely the unstable manifold of a second (possibly the same) hyperbolic closed orbit, then the dimension of the unstable manifold of the first is strictly less than the dimension of the unstable manifold of the second. However, we provide an example meeting the conditions of the lemma where the dimensions of the unstable manifolds are equal, thereby disproving the lemma. In particular, we present a hyperbolic closed orbit of a smooth vector field over three-dimensional Euclidean space whose stable and unstable manifolds have nonempty, transverse intersection.