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arxiv: 2305.18288 · v8 · submitted 2023-05-29 · 🧮 math.DS · cs.SY· eess.SY· math.OC

Linearizability of flows by embeddings

Matthew D. Kvalheim , Philip Arathoon This is my paper

Pith reviewed 2026-05-24 08:54 UTC · model grok-4.3

classification 🧮 math.DS cs.SYeess.SYmath.OC
keywords linearizabilityembeddingsdynamical systemsflowsHartman-Grobman theoremFloquet theoremattractorsinvariant manifolds
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The pith

Necessary and sufficient conditions characterize when flows on connected compact spaces or spaces with compact attractors admit linearizing embeddings into Euclidean space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the class of continuous-time dynamical systems that can be globally linearized through an embedding into a linear system on higher-dimensional Euclidean space. It restricts attention to systems whose state spaces are connected and either compact or contain at least one nonempty compact attractor. For these systems the authors obtain necessary and sufficient conditions under which C^k linearizing embeddings exist, for any k including infinity. The results yield checkable necessary conditions and extend the Hartman-Grobman and Floquet theorems to the stated topological settings while relating linearizability to symmetry, topology, and invariant manifolds.

Core claim

For continuous-time dynamical systems on connected state spaces that are compact or contain a nonempty compact attractor, necessary and sufficient conditions exist for the existence of linearizing C^k embeddings into linear systems on higher-dimensional Euclidean space.

What carries the argument

Linearizing C^k embeddings that conjugate the given flow to a linear flow on a higher-dimensional Euclidean space.

If this is right

  • Several checkable necessary conditions for global linearizability follow directly from the main criteria.
  • The Hartman-Grobman theorem extends to the connected compact or compact-attractor setting.
  • The Floquet normal form theorem extends beyond its classical settings under the same topological hypotheses.
  • Linearizability is placed in explicit relationship with symmetry, topology, and invariant manifold theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The conditions may permit verification of linearizability for concrete examples such as flows on spheres or tori without explicit construction of the embedding.
  • Invariant manifold techniques could supply constructive methods for producing the embeddings when the conditions hold.
  • Analogous characterizations might be sought for discrete-time maps or for non-compact spaces lacking compact attractors.

Load-bearing premise

The state space must be connected and either compact or contain at least one nonempty compact attractor.

What would settle it

A concrete dynamical system on a connected compact manifold that satisfies the stated necessary and sufficient conditions yet possesses no linearizing embedding of any class C^k, or conversely a system that admits such an embedding while violating the conditions.

Figures

Figures reproduced from arXiv: 2305.18288 by Matthew D. Kvalheim, Philip Arathoon.

Figure 1
Figure 1. Figure 1: Examples of flows that are linearizable by C 0 embed￾dings. This follows from Theorem 2 since each state space may be viewed as a compact subset of 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∞ embedding, as a direct construction shows. Remark 4. Theorem 2 explicitly restricts attention to … view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to solve the linearizability problem for continuous-time dynamical systems on connected state spaces that are either compact or contain at least one nonempty compact attractor. It derives necessary and sufficient conditions for the existence of linearizing C^k embeddings (k in N union {infinity}) into linear systems on higher-dimensional Euclidean space, and obtains corollaries consisting of checkable necessary conditions together with extensions of the Hartman-Grobman and Floquet theorems beyond their classical local settings.

Significance. If the derivations hold, the work supplies a complete characterization of globally linearizable flows within the stated topological class. This would constitute a substantive advance by linking linearizability to symmetry, topology, and invariant-manifold theory while extending two classical local results to a global setting under explicit compactness or attractor hypotheses.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'several checkable necessary conditions' is stated without enumeration; listing the principal ones (or their section references) already in the abstract would improve immediate readability.
  2. [Introduction] The manuscript should include a short table or explicit list in the introduction that cross-references each corollary to the corresponding necessary-and-sufficient condition from which it follows.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. We are pleased that the potential significance of the necessary and sufficient conditions for C^k linearizing embeddings, along with the corollaries extending Hartman-Grobman and Floquet theorems, is recognized.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives necessary and sufficient conditions for C^k linearizing embeddings on connected compact or attractor-containing state spaces. The abstract and claim structure present these as direct consequences of the topological setting and embedding problem, with no fitted parameters, self-definitional reductions, or load-bearing self-citations reducing the central result to its inputs. The derivation chain is self-contained within standard dynamical systems and manifold theory; no enumerated circularity pattern applies.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Pure mathematical characterization relying on standard assumptions from topology and dynamical systems; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A generalized global Hartman-Grobman theorem for asymptotically stable semiflows

    math.DS 2025-05 unverdicted novelty 7.0

    Extends the Kvalheim-Sontag generalized global Hartman-Grobman theorem to asymptotically stable semiflows generated by possibly discontinuous vector fields without requiring hyperbolicity.

  2. Global Linearization of Parameterized Nonlinear Systems with Stable Equilibrium Point Using the Koopman Operator

    math.DS 2026-04 unverdicted novelty 6.0

    Koopman operators provide a global linearization of parameterized nonlinear systems with stable equilibria into finite-dimensional linear systems that depend continuously on the parameter.

Reference graph

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