Global linearization without hyperbolicity

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http://arxiv.org/abs/2502.07708

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https://doi.org/10.48550/arXiv.2502.07708
 http://hdl.handle.net/11603/37947

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UMBC Mathematics and Statistics Department
UMBC Faculty Collection

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Author/Creator ORCID

https://orcid.org/0000-0002-2662-6760

Date

2025-02-13

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journal articles
preprints
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This work was written as part of one of the author's official duties as an Employee of the United States Government and is therefore a work of the United States Government. In accordance with 17 U.S.C. 105, no copyright protection is available for such works under U.S. Law.
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Subjects

Mathematics - Dynamical Systems
Computer Science - Systems and Control
Electrical Engineering and Systems Science - Systems and Control

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 $C^{k\geq 1}$ 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 Poincar\'{e} conjecture.