Comments on Stability Regions of Nonlinear Autonomous Dynamical Systems

• 成果类型: Editorial Material
• 署名作者: Fisher, Michael W.; Hiskens, Ian A.
• 署名单位: University of Michigan System; University of Michigan
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2021.3061674
• 发表日期: 2021
• 页码: 6194-6196
• 关键词: Hyperbolic dynamical systems; nonlinear system stability; region of attraction boundary; stable/unstable manifolds; stability boundary

摘要

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-infinity vector field over R-3 whose stable and unstable manifolds have nonempty, transverse intersection.