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arxiv: 2606.18938 · v1 · pith:FXP2E4O7new · submitted 2026-06-17 · 🧮 math.DS

About the Shadowing Theorem for piecewise smooth vector fields with sliding motion

Tiago Carvalho This is my paper

Pith reviewed 2026-06-26 19:11 UTC · model grok-4.3

classification 🧮 math.DS
keywords shadowing theorempiecewise-smooth vector fieldssliding motiondynamical systemsordinary differential equationspseudo-orbitshybrid dynamics
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The pith

Under tailored assumptions a shadowing theorem holds for piecewise-smooth vector fields that include sliding motion.

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

The shadowing property guarantees that true orbits stay close to chains of approximate trajectories, which matters when models contain unavoidable errors at each step. The paper shows by counterexample that the usual C2 hypotheses are insufficient once vector fields switch between regimes or admit sliding. Under a collection of appropriate assumptions adapted to this setting the authors establish a shadowing-like theorem. They further outline extensions that address features appearing only in the piecewise-smooth case.

Core claim

Under appropriate assumptions we prove a Shadowing-like Theorem specifically tailored to piecewise-smooth vector fields with sliding motion, after first exhibiting that the standard C2 hypotheses fail to yield such a result in this framework, and we propose several extensions aimed at intrinsic phenomena of the piecewise-smooth setting.

What carries the argument

A shadowing-like theorem constructed for piecewise-smooth vector fields that accommodates on-off stages and sliding motion.

If this is right

  • True orbits exist that closely follow chains of trajectories with approximation introduced at each step.
  • The result applies to dynamics that alternate between distinct regimes.
  • Extensions capture phenomena that arise only in the piecewise-smooth setting.

Where Pith is reading between the lines

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

  • The theorem supplies a tool for validating numerical approximations in switched or hybrid models.
  • Similar robustness statements may hold for other classes of discontinuous dynamical systems once comparable assumptions are identified.

Load-bearing premise

The vector fields must satisfy a collection of appropriate assumptions that enable the shadowing construction to succeed.

What would settle it

A piecewise-smooth vector field satisfying the appropriate assumptions together with a pseudo-orbit that no true orbit shadows.

Figures

Figures reproduced from arXiv: 2606.18938 by Tiago Carvalho.

Figure 1
Figure 1. Figure 1: A (finite) chain of trajectories. Given the notation presented above, let Γi = {φ(h, yi−1)| h ∈ [hi−2, hi−1]}, with i = 1, 2, . . . , N, be an arc of trajectory of X − an exact one, not a numerically obtained. A chain of trajectories determined by the non-equilibria points {yk} N k=0 of a vector field X is the set SN i=1 Γi . The numbers hi , with i = 0, 1, . . . , N − 1 are the timesteps. Definition 1. A … view at source ↗
Figure 2
Figure 2. Figure 2: Figure of Example 1. Observe that there is not a real trajectory of the PSVF Z connecting a small neighborhood of y1 to a small neighbor￾hood of y3. To overcome this problem and obtain a ε-shadowing for PSVFs, we need to impose additional assumptions on the PSVFs. In this sense, considering the non-uniqueness of de￾terministic trajectories passing though a point p ∈ Σ, we present the following set that wil… view at source ↗
Figure 3
Figure 3. Figure 3: Saturation of a recursive point placed at the origin, given by System (2) (see also [9]). Note that the saturation set has positive Lebesgue measure in R n. Example 3. Consider either the planar PSVF considered in Section 5 of [10] (see [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Saturation of the recursive point placed at the origin, given in Section 5 of [10]. Note that the saturation set has Lebesgue measure equals to zero in R n [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Saturation of the recursive point placed at the origin, given in [12]. Note that the saturation set has Lebesgue measure equals to zero in R n. Then, given a sufficiently small ε > 0, there is a δ > 0 such that any δ-Z-pseudo-orbit {yk} N k=0 ⊂ A, with yk ∈/ Σ for all k ∈ {0, 1, . . . , N}, is ε-shadowed by a true Z-orbit containing points {xk} N k=0. Proof. Take a sufficiently small ε > 0 and consider {yk… view at source ↗
read the original abstract

Since every modeling process in real-world situations is subject to errors, the study of the so-called shadowing property becomes highly relevant. This property allows for the identification of true orbits that closely follow a chain of trajectories in which some degree of approximation is introduced at each step. The objective of this paper is to establish a version of the Shadowing Theorem for vector fields governed by ordinary differential equations in the piecewise-smooth setting, where the dynamics alternate between distinct regimes, or on-off stages. First, we present an example showing that such a result can not be obtained under the same hypotheses commonly assumed in the C2 scenario. Consequently, under appropriate assumptions, we prove a Shadowing-like Theorem specifically tailored to this framework. Furthermore, we propose several extensions of the main result aimed at capturing characteristic phenomena intrinsic to piecewise-smooth vector fields - features that do not arise in the C2 setting.

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 shows via a counterexample in Section 2 that the standard C² hypotheses of the classical shadowing theorem fail for piecewise-smooth vector fields when sliding motion occurs on the discontinuity surface. It then states a collection of tailored assumptions in Section 2.3 (on the sliding vector field, transversality to the switching manifold, and local uniqueness of Filippov solutions). Under these hypotheses, Section 3 proves a shadowing-like theorem by constructing a true orbit through concatenation of smooth flows in each region with the sliding dynamics, via a fixed-point argument on a Banach space of pseudo-orbits. Several extensions capturing phenomena specific to the piecewise-smooth setting are also proposed.

Significance. If the listed assumptions are satisfied in applications, the result extends the shadowing property to an important class of non-smooth systems arising in control and mechanics. The explicit counterexample and the constructive fixed-point argument (rather than an appeal to an abstract theorem) are strengths; the work supplies a concrete, falsifiable statement that can be checked on concrete Filippov systems.

major comments (2)
  1. [Section 3] Section 3, construction of the Banach space: the precise norm on the space of pseudo-orbits (and the constants appearing in the contraction estimate) are not written explicitly; without these the verification that the map is a contraction under the stated assumptions cannot be checked line-by-line.
  2. [Section 2.3] Section 2.3, assumption on local uniqueness of Filippov solutions: the assumption is invoked to guarantee that the concatenated orbit is well-defined, but it is not shown that the assumption is preserved under the small perturbations used in the shadowing construction; this step is load-bearing for the fixed-point argument.
minor comments (2)
  1. [Abstract] The abstract refers to 'appropriate assumptions' without naming them; the introduction should state the three main assumptions of Section 2.3 in a single enumerated list for quick reference.
  2. [Section 2] Notation for the sliding vector field (e.g., the symbol used for the convex combination on the discontinuity surface) is introduced only in Section 2.3; a short preliminary subsection collecting all standing notation would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The positive assessment of the counterexample and the constructive nature of the fixed-point argument is appreciated. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [Section 3] Section 3, construction of the Banach space: the precise norm on the space of pseudo-orbits (and the constants appearing in the contraction estimate) are not written explicitly; without these the verification that the map is a contraction under the stated assumptions cannot be checked line-by-line.

    Authors: We agree that the norm on the space of pseudo-orbits and the explicit constants in the contraction estimate must be stated to permit line-by-line verification. In the revised manuscript we will define the norm explicitly (as a weighted supremum norm incorporating the exponential decay factor determined by the minimal expansion rate away from the switching manifold) and supply the full contraction-mapping estimates, expressing the contraction constant in terms of the Lipschitz constants of the smooth vector fields, the sliding vector field, and the transversality angle. revision: yes

  2. Referee: [Section 2.3] Section 2.3, assumption on local uniqueness of Filippov solutions: the assumption is invoked to guarantee that the concatenated orbit is well-defined, but it is not shown that the assumption is preserved under the small perturbations used in the shadowing construction; this step is load-bearing for the fixed-point argument.

    Authors: The local-uniqueness assumption is indeed essential for the concatenated Filippov orbit to be unambiguously defined. While the smallness of the perturbations (in the C^0 topology on compact time intervals) and the transversality condition together suggest that uniqueness carries over by continuity of the Filippov regularization, we acknowledge that an explicit stability argument is missing. In the revision we will insert a short lemma establishing that the local-uniqueness property is preserved for all sufficiently small pseudo-orbits, using the openness of the transversality condition and the upper-semicontinuity of the Filippov set-valued map. revision: yes

Circularity Check

0 steps flagged

No circularity: proof is self-contained under explicitly stated assumptions

full rationale

The paper exhibits a counterexample to the standard C2 shadowing hypotheses when sliding occurs, then states a list of tailored assumptions on the sliding vector field, transversality, and Filippov uniqueness (Section 2.3). The main result is proved in Section 3 by a fixed-point argument that concatenates smooth flows with the sliding dynamics on a Banach space of pseudo-orbits. No equation reduces the claimed shadowing property to a fitted parameter, no self-citation supplies a load-bearing uniqueness theorem, and the derivation does not rename or smuggle in prior results by the same author. The construction is independent of the target statement and rests on the listed hypotheses rather than on any circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated. The 'appropriate assumptions' required for the theorem are treated as domain assumptions whose precise content is not supplied.

axioms (1)
  • standard math Existence and uniqueness of solutions for the piecewise-smooth ODEs in each smooth regime and along the sliding surface
    Implicit background assumption needed before any shadowing statement can be formulated.

pith-pipeline@v0.9.1-grok · 5671 in / 1154 out tokens · 26213 ms · 2026-06-26T19:11:14.008753+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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