Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global linearization of asymptotically stable systems without hyperbolicity

Matthew D. Kvalheim, Eduardo D. Sontag

TL;DR

This work addresses extending the Hartman-Grobman linearization to nonhyperbolic but asymptotically stable equilibria, without requiring hyperbolicity and under minimal regularity assumptions. It constructs a topological (and, when possible, $C^k$) conjugacy $h$ between the nonlinear flow and a linear flow with a Hurwitz matrix $A$, achieving global linearization on the basin of attraction when the vector field is complete. The main contributions are a local $C^k$-regularity result for $n\neq 5$, a global linearization result, and a surprising equivalence between the $C^k$ statement at $n=5$ and the $4$-dimensional smooth Poincaré conjecture, revealing deep connections between dynamical systems and topology. These results provide a rigorous foundation for linear representations of nonlinear dynamics and have potential implications for Koopman-based analyses and data-driven linearization methods.

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 Poincaré conjecture.

Global linearization of asymptotically stable systems without hyperbolicity

TL;DR

This work addresses extending the Hartman-Grobman linearization to nonhyperbolic but asymptotically stable equilibria, without requiring hyperbolicity and under minimal regularity assumptions. It constructs a topological (and, when possible, ) conjugacy between the nonlinear flow and a linear flow with a Hurwitz matrix , achieving global linearization on the basin of attraction when the vector field is complete. The main contributions are a local -regularity result for , a global linearization result, and a surprising equivalence between the statement at and the -dimensional smooth Poincaré conjecture, revealing deep connections between dynamical systems and topology. These results provide a rigorous foundation for linear representations of nonlinear dynamics and have potential implications for Koopman-based analyses and data-driven linearization methods.

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 -diffeomorphism on the complement of the equilibrium if the vector field is and the underlying space is not -dimensional. We also show that the statement in the -dimensional case is equivalent to the -dimensional smooth Poincaré conjecture.

Paper Structure

This paper contains 3 sections, 4 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Proofs

Key Result

Theorem 1

Let $x_*$ be an asymptotically stable equilibrium for a uniquely integrable continuous vector field $f$ on an $n$-dimensional $C^\infty$ manifold $M$. There is an open neighborhood $U\subset M$ of $x_*$ such that, for any Hurwitz matrix $A\in \mathbb{R}^{n\times n}$, there is a topological embedding Moreover, if $n\neq 5$ and $f\in C^k$ with $k\in \mathbb{N}_{\geq 1}\cup \{\infty\}$, then there ex

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • proof : Proof of Theorem \ref{['th:local-lin']}
  • proof : Proof of Theorem \ref{['th:global-lin']}
  • proof : Proof of Proposition \ref{['prop:poincare']}
  • ...and 1 more