Comments on "Stability Regions of Nonlinear Autonomous Dynamical Systems"

The proofs of the groundbreaking theorems of [1] rely on a lemma, which states that if the unstable manifold of a first hyperbolic closed orbit intersects transversely the stable manifold of a second (possibly the same) hyperbolic closed orbit, then the dimension of the unstable manifold of the firs...

Full description

Saved in:
Bibliographic Details
Published in:IEEE transactions on automatic control 2021-12, Vol.66 (12), p.6194-6196
Main Authors: Fisher, Michael W., Hiskens, Ian A.
Format: Article
Language:English
Subjects:
Dynamic stability
Dynamical systems
Eigenvalues and eigenfunctions
Fields (mathematics)
Hyperbolic dynamical systems
Manifolds
nonlinear system stability
Nonlinear systems
Orbital stability
region of attraction boundary
Space vehicles
stability boundary
stable/unstable manifolds
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The proofs of the groundbreaking theorems of [1] rely on a lemma, which states that if the unstable manifold of a first hyperbolic closed orbit intersects transversely the stable manifold of a second (possibly the same) hyperbolic closed orbit, then the dimension of the unstable manifold of the first is strictly greater 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 C^\infty vector field over \mathbb {R}^3 whose stable and unstable manifolds have nonempty, transverse intersection.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2021.3061674