Global linearization of asymptotically stable systems without hyperbolicity

2025/02/11 by Matthew D. Kvalheim, Eduardo D. Sontag, Kvalheim, Matthew D. +1 · 3 citations
Mathematics · #34D20 #34D23 #37C75 #57R60 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Electrical engineering #FOS: Mathematics #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · pdf · doi:10.48550/arxiv.2502.07708

openalex publication_date 2025/02/11 · openalex created_date 2025/02/13 · openalex updated_date 2026/07/28

Abstract

We give a proof of an extension of the Hartman-Grobman theorem to nonhyperbolic but asymptotically stable equilibria of vector fields. Moreover, the linearizing topological conjugacy is (i) defined on the entire basin of attraction if the vector field is complete, and (ii) a C^{k≥ 1}-diffeomorphism on the complement of the equilibrium if the vector field is C^k and the underlying space is not 5-dimensional. We also show that the C^k statement in the 5-dimensional case is equivalent to the 4-dimensional smooth Poincaré conjecture.

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