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arxiv: 2505.21401 · v5 · submitted 2025-05-27 · 🧮 math.DS · cs.SY· eess.SY· math.OC

A generalized global Hartman-Grobman theorem for asymptotically stable semiflows

Wouter Jongeneel This is my paper

Pith reviewed 2026-05-19 12:55 UTC · model grok-4.3

classification 🧮 math.DS cs.SYeess.SYmath.OC
keywords Hartman-Grobman theoremsemiflowsasymptotic stabilityLyapunov functionsdiscontinuous vector fieldstopological equivalenceglobal linearization
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The pith

The generalized global Hartman-Grobman theorem extends to asymptotically stable semiflows generated by possibly discontinuous vector fields.

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

The paper extends a recent generalization of the global Hartman-Grobman theorem to handle semiflows that are asymptotically stable. It shows that topological properties of associated Lyapunov functions suffice to establish a homeomorphism between the semiflow and its linearization at the equilibrium. This removes the need for hyperbolicity and allows for discontinuities in the generating vector field. A sympathetic reader would care because many practical systems involve non-smooth dynamics where standard linearization fails.

Core claim

By leveraging topological properties of Lyapunov functions, the theorem establishes that asymptotically stable semiflows are topologically equivalent to their linearizations near equilibria, even when the vector fields are discontinuous.

What carries the argument

Lyapunov functions whose topological properties enable the construction of a conjugacy between the semiflow and the linearized flow.

Load-bearing premise

The topological properties of Lyapunov functions for asymptotically stable semiflows can be used in the same way as for continuous vector fields.

What would settle it

A counterexample consisting of an asymptotically stable semiflow generated by a discontinuous vector field where no such global homeomorphism to the linearization exists.

Figures

Figures reproduced from arXiv: 2505.21401 by Wouter Jongeneel.

Figure 1
Figure 1. Figure 1: Some orbits corresponding to the vector field (3), [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of Theorem II.2: any asymptotically sta [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The maps x 7→ τ ′ (x) and x 7→ r · e τ ′ (x) from Section III-A. Recall that we have set ε such that V −1 (ε) = Lε = S n−1 such that τ ′ (x) = ln(∥x∥ 2 2 ) for ∥x∥2 < 1 and τ ′ (x) = ∥x∥2 − 1 for ∥x∥2 ≥ 1. Of course, if φ happens to be a flow (i.e., t ∈ R), Theorem II.2 is also true, but slightly conservative as there is no need to construct B(0, r) and the like cf. [7, Thm. 2]. We also emphasize that we m… view at source ↗
Figure 4
Figure 4. Figure 4: Resulting semiflow from Section III-A, e.g., we visualize t 7→ ∥φe t 1 (y)∥2 for r = 1 and starting from ∥y∥2 = 2, that is, although the true system is finite-time stable, we first have exponential (asymptotic) decay for t ≤ ln(2), then we follow (6) (from B(0, r)) and observe a decay of the form t 7→ (1 − (t − ln(2))2 until we hit 0. for t ≤ √ θ = p ∥y∥2/r. Thus, finite-time stability is preserved, as it … view at source ↗
Figure 5
Figure 5. Figure 5: Resulting semiflow from Section III-B, e.g., we visualize t 7→ ∥φe t e (x)∥2 starting from ∥x∥2 = 2, that is, although the true system is merely asymptotically stable, we follow t 7→ ∥x∥2 −t (i.e., typical finite-time behaviour) until we hit the unit ball. Afterwards we have an asymptotic decay of the form t 7→ e −1/2(t−1)∥x∥2. theory, e.g., see [24], [25]. Moreover, these results are closely related to re… view at source ↗
read the original abstract

Recently, Kvalheim and Sontag provided a generalized global Hartman-Grobman theorem for equilibria under asymptotically stable continuous vector fields. By leveraging topological properties of Lyapunov functions, their theorem works without assuming hyperbolicity. We extend their theorem to a class of possibly discontinuous vector fields, in particular, to vector fields generating asymptotically stable semiflows.

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

2 major / 2 minor

Summary. The manuscript extends the generalized global Hartman-Grobman theorem of Kvalheim and Sontag from continuous asymptotically stable vector fields to a class of possibly discontinuous vector fields that generate asymptotically stable semiflows. It claims that topological properties of Lyapunov functions can be leveraged to obtain a homeomorphism conjugating the semiflow to its linearization at an equilibrium, without hyperbolicity assumptions.

Significance. If the extension is valid, the result would meaningfully enlarge the applicability of non-hyperbolic Hartman-Grobman theorems to systems with discontinuities, including certain hybrid or Filippov-type models. The reliance on Lyapunov-function topology rather than linearization is a conceptual strength that could yield falsifiable predictions for concrete discontinuous examples once the proof details are clarified.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1 and its proof: the assertion that Lyapunov functions for the discontinuous semiflow retain the same topological properties (properness, sublevel-set structure, and inducement of a homeomorphism) as in the continuous Kvalheim-Sontag setting is load-bearing for the central claim, yet the manuscript provides no explicit verification or additional hypotheses on the discontinuity set that would guarantee preservation of these properties when orbits lose uniqueness or the flow map loses continuity.
  2. [§4] §4 (Proof of the main result): the argument appears to invoke the continuous-case construction directly after citing the existence of a Lyapunov function, without addressing how discontinuities affect the continuity of the conjugacy map or the properness of the Lyapunov function along orbits; a concrete check or counter-example ruling out failure modes would be required to support the extension.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should explicitly state the precise regularity assumptions placed on the discontinuity set (e.g., measure-zero or Filippov-regular) so that readers can immediately assess the scope of the extension.
  2. [§2 and §3] Notation for the semiflow (e.g., the symbol used for the discontinuous vector field versus the generated semiflow) is introduced inconsistently between the statement of Theorem 3.1 and the subsequent lemmas; a uniform notation table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where additional clarification would strengthen the manuscript. We agree that the extension to discontinuous vector fields requires explicit verification that the key topological properties survive, and we will revise accordingly to make the arguments self-contained.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1 and its proof: the assertion that Lyapunov functions for the discontinuous semiflow retain the same topological properties (properness, sublevel-set structure, and inducement of a homeomorphism) as in the continuous Kvalheim-Sontag setting is load-bearing for the central claim, yet the manuscript provides no explicit verification or additional hypotheses on the discontinuity set that would guarantee preservation of these properties when orbits lose uniqueness or the flow map loses continuity.

    Authors: We acknowledge that the manuscript would benefit from an explicit verification step. The topological properties follow from the definition of asymptotic stability for the semiflow itself (continuous in time, with compact sublevel sets that are positively invariant), rather than from continuity of the generating vector field. To address the concern directly, we will insert a short lemma in §3 that derives properness, sublevel-set compactness, and the homeomorphism-inducing property solely from the semiflow axioms and the stability assumption. This lemma will note that non-uniqueness of orbits is already accommodated by working with the semiflow map rather than the vector field, so no additional hypotheses on the discontinuity set are required beyond those already stated for the semiflow to be well-defined and asymptotically stable. revision: yes

  2. Referee: [§4] §4 (Proof of the main result): the argument appears to invoke the continuous-case construction directly after citing the existence of a Lyapunov function, without addressing how discontinuities affect the continuity of the conjugacy map or the properness of the Lyapunov function along orbits; a concrete check or counter-example ruling out failure modes would be required to support the extension.

    Authors: We agree that the proof in §4 should be expanded to spell out the effect of discontinuities. The conjugacy is built from the Lyapunov function using the semiflow orbits; because the semiflow is continuous in the time variable and the stability assumption guarantees that orbits approach the equilibrium while remaining in compact sublevel sets, the resulting map is a homeomorphism. We will revise the proof to include a dedicated paragraph that (i) confirms continuity of the conjugacy by composing the continuous time parametrization with the level-set structure and (ii) verifies properness along orbits directly from the semiflow definition. We will also add a brief remark ruling out the most common failure modes (e.g., jumps across discontinuity surfaces) by observing that any such behavior is already constrained by the global asymptotic stability hypothesis. revision: yes

Circularity Check

0 steps flagged

No circularity: extension relies on external cited theorem

full rationale

The paper extends the Kvalheim-Sontag generalized global Hartman-Grobman theorem (an external citation) to asymptotically stable semiflows generated by possibly discontinuous vector fields, leveraging topological properties of Lyapunov functions in a manner analogous to the continuous case. No load-bearing step reduces by construction to a self-citation, fitted input renamed as prediction, or self-definitional loop; the central claim introduces independent content regarding discontinuities while assuming the cited result's applicability. This is a standard non-circular extension with external support.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of asymptotically stable semiflows and on Lyapunov functions possessing suitable topological properties that survive the passage to the discontinuous setting.

axioms (1)
  • domain assumption Asymptotically stable semiflows admit Lyapunov functions with topological properties sufficient to mimic the continuous-case argument.
    Invoked in the abstract when stating the extension leverages the same topological properties.

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