Global linearization of asymptotically stable systems without hyperbolicity

TL;DR

This paper extends the Hartman-Grobman theorem to asymptotically stable systems without hyperbolicity, providing conditions for topological conjugacy and differentiability in various dimensions.

Contribution

It generalizes linearization results to nonhyperbolic, asymptotically stable equilibria, including the entire basin of attraction and connections to the Poincaré conjecture.

Findings

Linearization on entire basin of attraction for complete vector fields

Existence of $C^{k}$-diffeomorphism in non-5-dimensional cases

Equivalence of $C^{k}$ regularity in 5D to the smooth Poincaré conjecture

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.