Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linearizability of flows by embeddings

Matthew D. Kvalheim, Philip Arathoon

TL;DR

This work addresses when a nonlinear continuous-time dynamical system on a connected state space can be globally linearized by a $C^k$ embedding into a linear system on $\mathbb{R}^n$. It develops necessary and sufficient conditions for four regimes: smooth/continuous compact spaces and attractor basins, with $k\ge 0$ or $k=\infty$, revealing that linearizability exactly corresponds to the flow being a 1-parameter subgroup of a torus action on the invariant set and to the existence of asymptotic phase structures. The results unify and extend Hartman-Grobman and Floquet theory in a global, embedding-based framework and connect linearizability to symmetry, topology, and invariant-manifold concepts, while providing corollaries that give checkable conditions and Koopman-operator perspectives. These insights constrain the possibility of global linearizing embeddings and have implications for data-driven Koopman methods, shedding light on when such embeddings can be expected to exist or be computationally feasible.

Abstract

We consider the problem of determining the class of continuous-time dynamical systems that can be globally linearized in the sense of admitting an embedding into a linear system on a higher-dimensional Euclidean space. We solve this problem for dynamical systems on connected state spaces that are either compact or contain at least one nonempty compact attractor, obtaining necessary and sufficient conditions for the existence of linearizing $C^k$ embeddings for $k\in \mathbb{N}_{\geq 0}\cup \{\infty\}$. Corollaries include (i) several checkable necessary conditions for global linearizability and (ii) extensions of the Hartman-Grobman and Floquet normal form theorems beyond the classical settings. Our results open new perspectives on linearizability by establishing relationships to symmetry, topology, and invariant manifold theory.

Linearizability of flows by embeddings

TL;DR

This work addresses when a nonlinear continuous-time dynamical system on a connected state space can be globally linearized by a embedding into a linear system on . It develops necessary and sufficient conditions for four regimes: smooth/continuous compact spaces and attractor basins, with or , revealing that linearizability exactly corresponds to the flow being a 1-parameter subgroup of a torus action on the invariant set and to the existence of asymptotic phase structures. The results unify and extend Hartman-Grobman and Floquet theory in a global, embedding-based framework and connect linearizability to symmetry, topology, and invariant-manifold concepts, while providing corollaries that give checkable conditions and Koopman-operator perspectives. These insights constrain the possibility of global linearizing embeddings and have implications for data-driven Koopman methods, shedding light on when such embeddings can be expected to exist or be computationally feasible.

Abstract

We consider the problem of determining the class of continuous-time dynamical systems that can be globally linearized in the sense of admitting an embedding into a linear system on a higher-dimensional Euclidean space. We solve this problem for dynamical systems on connected state spaces that are either compact or contain at least one nonempty compact attractor, obtaining necessary and sufficient conditions for the existence of linearizing embeddings for . Corollaries include (i) several checkable necessary conditions for global linearizability and (ii) extensions of the Hartman-Grobman and Floquet normal form theorems beyond the classical settings. Our results open new perspectives on linearizability by establishing relationships to symmetry, topology, and invariant manifold theory.
Paper Structure (10 sections, 17 theorems, 28 equations, 1 figure)

This paper contains 10 sections, 17 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Illustration of results
  4. Results and examples
  5. The smooth compact case
  6. The continuous compact case
  7. The continuous attractor basin case
  8. The smooth attractor basin case
  9. Proofs
  10. Conclusion

Key Result

Theorem 1

Fix $k\in \mathbb{N}_{\geq 1}\cup \{\infty\}$. Let $\Phi$ be a $C^k$ flow on a compact $C^k$ manifold $X$. Then $(X,\Phi)$ is linearizable by a $C^k$ embedding if and only if $\Phi$ is a $1$-parameter subgroup of a $C^k$ torus action.

Figures (1)

  • Figure 1: Examples of flows that are linearizable by $C^0$ embeddings. This follows from Theorem \ref{['th:cont-compact']} since each state space may be viewed as a compact subset of $\mathbb{R}^3$, and each flow is a 1-parameter subgroup of a $C^0$ torus (circle) action with finitely many orbit types. The rightmost example is actually linearizable by a $C^\infty$ embedding, as a direct construction shows.

Theorems & Definitions (43)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Proposition 1
  • Corollary 1
  • ...and 33 more