Asynchronous discrete dynamical systems

Petr Stehlik; Stefan Siegmund

arxiv: 1907.01799 · v1 · pith:GXU65Y3Nnew · submitted 2019-07-03 · 🧮 math.DS

Asynchronous discrete dynamical systems

Stefan Siegmund , Petr Stehlik This is my paper

Pith reviewed 2026-05-25 10:04 UTC · model grok-4.3

classification 🧮 math.DS

keywords asynchronous discrete systemscoupled equationssample and holdstabilityperiodic time scalesdynamical properties

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The pith

Asynchronicity in sample-and-hold coupled discrete equations can significantly alter their stability and dynamical properties.

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

The paper examines two coupled discrete-time equations that update on different periodic time scales. It models their interaction with sample-and-hold coupling, where each equation's state is frozen between its own updates. The authors build an interpolating two-dimensional complex-valued system on the union of the time scales and an extrapolating four-dimensional system on their intersection. Stability results together with explicit examples and counterexamples demonstrate that the timing mismatch produces stability behaviors distinct from the synchronous case.

Core claim

We construct an interpolating two-dimensional complex-valued system on the union of the two time scales and an extrapolating four-dimensional system on the intersection of the two time scales. We discuss stability by several results, examples and counterexamples in various frameworks to show that the asynchronicity can have a significant impact on the dynamical properties.

What carries the argument

Sample-and-hold coupling between two asynchronous periodic discrete equations, analyzed through interpolating complex-valued and extrapolating real-valued auxiliary systems on combined and shared time scales.

If this is right

Stability of the coupled system can depend on the ratio between the two update periods.
There exist cases where the asynchronous system is stable while its synchronous version is unstable.
There exist cases where the asynchronous system is unstable while its synchronous version is stable.
The constructed interpolating and extrapolating systems provide a framework for stability analysis that accounts for the timing mismatch.

Where Pith is reading between the lines

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

The embedding technique into higher-dimensional systems on unions and intersections of time scales could be tested on networks with three or more asynchronously coupled equations.
If the impact of asynchronicity persists under small perturbations of the update times, it may suggest robustness properties of timing-based control in discrete networks.

Load-bearing premise

The coupling mechanism is exactly sample-and-hold, with each equation's state held constant until the next read by the other equation.

What would settle it

A specific pair of equations and update periods where every stability property of the asynchronous system matches that of the corresponding synchronous system, with no counterexamples appearing.

Figures

Figures reproduced from arXiv: 1907.01799 by Petr Stehlik, Stefan Siegmund.

**Figure 1.** Figure 1: Time scales T3 and T5 of a (3, 5)-asynchronous discrete dynamical system (2.1). The extended function zR≥0 realizes the principle of sample and hold in systems theory [15, Section 1.4] and can be used in the analysis of discrete-time equations [18]. For every τ ∈ Tν the value zTν (τ ) is sampled and held constant for one period ν, i.e., zR≥0 (t) = zTν (τ ) (t ∈ [τ, τ + ν)). Let µ > 0. The restriction zTµ :… view at source ↗

**Figure 2.** Figure 2: Time scales related to dynamically equivalent (2,3)- and (6,1)-asynchronous discrete dynamical [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

read the original abstract

We study two coupled discrete-time equations with different (asynchronous) periodic time scales. The coupling is of the type sample and hold, i.e., the state of each equation is sampled at its update times and held until it is read as an input at the next update time for the other equation. We construct an interpolating two-dimensional complex-valued system on the union of the two time scales and an extrapolating four-dimensional system on the intersection of the two time scales. We discuss stability by several results, examples and counterexamples in various frameworks to show that the asynchronicity can have a significant impact on the dynamical properties.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit constructions for modeling sample-and-hold coupling between two discrete systems on mismatched periodic time scales and uses examples to show that the mismatch can change stability.

read the letter

The main point is that asynchronicity under sample-and-hold can shift dynamical properties, and they demonstrate it by building an interpolating 2D complex system on the union of the time scales plus an extrapolating 4D system on the intersection. These lifts let them analyze the coupled behavior in a single framework instead of dealing with two separate clocks directly. The constructions are concrete and tied to the periodic assumption, which keeps the argument consistent. They then run stability checks with examples and counterexamples in different settings to illustrate the impact, rather than claiming a broad theorem. That approach is straightforward and avoids overreach. The periodic time-scale restriction and the specific coupling type are stated up front, so there is no hidden generality problem. The evidence is example-driven, which is fine for showing that the effect can occur but leaves open how often it matters in practice. No circular reasoning appears in the setup. This work fits readers who already work on discrete systems or timing-mismatched models in control or numerics and want a way to embed the asynchrony into a higher-dimensional map. It is narrow enough that most people outside that niche will not need it, but the constructions are reproducible enough to be useful if the topic matches. The paper is coherent on its own terms and shows clear engagement with the modeling choice, so it deserves a serious referee even if revisions will be needed on the scope of the stability claims.

Referee Report

2 major / 2 minor

Summary. The manuscript examines two coupled discrete-time dynamical systems evolving on distinct periodic time scales under sample-and-hold coupling. It constructs an interpolating two-dimensional complex-valued system defined on the union of the time scales and an extrapolating four-dimensional system on their intersection, then analyzes stability properties through a combination of theorems, explicit examples, and counterexamples to demonstrate that asynchronicity can alter dynamical behavior.

Significance. The explicit constructions of the 2D interpolating and 4D extrapolating systems, together with targeted stability examples and counterexamples, provide concrete evidence that asynchronicity can change stability properties in coupled systems. This is a useful contribution for understanding asynchronous discrete dynamics under the stated coupling, with potential relevance to control and networked systems.

major comments (2)

[Constructions and stability analysis (main body, following abstract)] The central claim is existential and rests on the explicit constructions for periodic time scales; however, the stability results appear to rely on the periodicity assumption being built into the interpolating and extrapolating systems. It would strengthen the work to clarify in the main theorems whether the impact of asynchronicity persists for non-periodic time scales or is limited to the periodic case considered.
[4D extrapolating system construction] The sample-and-hold coupling is defined in the abstract; the 4D extrapolating system on the intersection should be checked for whether the extrapolation step introduces additional assumptions on the held states that could affect the counterexamples showing instability.

minor comments (2)

Notation for the union and intersection of time scales could be introduced earlier and used consistently to improve readability of the constructions.
A brief comparison table of the stability outcomes for the synchronous versus asynchronous cases would help readers quickly assess the impact shown by the examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. We address the major comments point by point below.

read point-by-point responses

Referee: [Constructions and stability analysis (main body, following abstract)] The central claim is existential and rests on the explicit constructions for periodic time scales; however, the stability results appear to rely on the periodicity assumption being built into the interpolating and extrapolating systems. It would strengthen the work to clarify in the main theorems whether the impact of asynchronicity persists for non-periodic time scales or is limited to the periodic case considered.

Authors: The manuscript is devoted exclusively to periodic time scales, as stated in the title, abstract, and the problem formulation. The interpolating 2D complex-valued system and the extrapolating 4D system are constructed by exploiting periodicity to define the dynamics on the union and intersection, respectively. All stability results, examples, and counterexamples are derived under this periodicity assumption. We agree that the scope should be stated more explicitly. We will add a clarifying sentence in the introduction and immediately preceding the main theorems noting that the demonstrated effects of asynchronicity are established for periodic time scales and that the non-periodic case lies outside the present work. revision: yes
Referee: [4D extrapolating system construction] The sample-and-hold coupling is defined in the abstract; the 4D extrapolating system on the intersection should be checked for whether the extrapolation step introduces additional assumptions on the held states that could affect the counterexamples showing instability.

Authors: The 4D extrapolating system is obtained directly from the sample-and-hold mechanism: each component holds its sampled value constant between its own update instants, with no further assumptions imposed on the held states. The counterexamples demonstrating instability are constructed so that the asynchronicity (and the resulting mismatch in held values) is the source of the instability; the extrapolation step itself does not alter this mechanism. We will insert a short verification paragraph in the section defining the 4D system confirming that the counterexamples remain valid under the stated extrapolation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper advances an existential claim that asynchronicity can impact dynamics, addressed through explicit constructions of a 2D interpolating system on the union of time scales and a 4D extrapolating system on their intersection, under the stated sample-and-hold coupling. Stability analysis proceeds via standard frameworks, targeted examples, and counterexamples. No load-bearing steps reduce by definition, fitted parameters renamed as predictions, or self-citation chains; the periodic time-scale assumption is explicitly built into the constructions, rendering the argument internally consistent without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are indicated in the abstract; the work rests on standard mathematical frameworks for discrete dynamical systems and stability analysis.

axioms (1)

standard math Standard properties of discrete dynamical systems and stability theory in various frameworks
The paper invokes established stability concepts to discuss the impact of asynchronicity.

pith-pipeline@v0.9.0 · 5619 in / 1107 out tokens · 26909 ms · 2026-05-25T10:04:34.721098+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We construct an interpolating two-dimensional complex-valued system on the union of the two time scales and an extrapolating four-dimensional system on the intersection of the two time scales.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the asynchronicity can have a significant impact on the dynamical properties

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications , Birkh¨ auser, Boston, 2001

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S. Elaydi, S., S. Zhang, Stability and periodicity of diﬀerence equations with ﬁnite delay. Funkcial. Ekvac 37(3) (1994), 401–413

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P. Klemperer, Competition when Consumers have Switching Costs: An Overview with Applications to Industrial Organization, Macroeconomics, and International Trade , The Review of Economic Studies 62(1995), no. 4, 515–539

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Kelley, A

W. Kelley, A. Peterson, Diﬀerence Equations. An Introduction with Applications , Academic Press, London, 2001

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R. Lagunoﬀ, A. Matsui, Asynchronous choice in repeated coordination games. Econometrica 65 (1997), 1467–1477

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J. Libich, P. Stehl´ ık,Endogenous Monetary Commitment, Economic Letters. 112 (2011), 103–106

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J. Libich, P. Stehl´ ık,Incorporating Rigidity and Commitment in the Timing Structure of Macroeco- nomic Games, Economic Modelling. 27 (2010), 767–781

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Ogata, Discrete-time control systems, Prentice Hall Englewood Cliﬀs, NJ, 1995

K. Ogata, Discrete-time control systems, Prentice Hall Englewood Cliﬀs, NJ, 1995

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P¨ otzsche, S

C. P¨ otzsche, S. Siegmund, F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales , Discrete Contin. Dyn. Syst. 9 (2003), no. 5, 1223–1241

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W. Shou, C. T. Bergstrom, A. K. Chakraborty, F. K. Skinner. Theory, models and biology. eLife, 4(2015), e07158. http://doi.org/10.7554/eLife.07158

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Slav´ ık,Dynamic equations on time scales and generalized ordinary diﬀerential equations, J

A. Slav´ ık,Dynamic equations on time scales and generalized ordinary diﬀerential equations, J. Math. Anal. Appl. 385 (2012), 534–550

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J. Tobin. Money and Finance in the Macroeconomic Process, Journal of Money, Credit and Banking 14 (1982), no.2, 171–204

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Q. Yu, J. Fish, Temporal homogenization of viscoelastic and viscoplastic solids subjected to locally periodic loading, Computational Mechanics 29 (2002), no.3, 199–211. 16 i := 1 Tµ (t) j := 1 Tµ (σ(t)) k := 1 Tν (t) ℓ := 1 Tν (σ(t)) Pictogram Aijkℓ 1 1 1 1 ( 1+µα 0 µβ 0 1 0 0 0 νγ 0 1+νδ 0 0 0 1 0 ) =:A1111 1 1 1 0 ( 1+µα 0 µβ 0 1 0 0 0 0 0 1 0 0 0 1 0 )...

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[1] [1]

Aubry and G

D. Aubry and G. Puel, Two-timescale homogenization method for the modeling of material fatigue , IOP Conference Series: Materials Science and Engineering 10 (2010), no.1, article no. 012113

work page 2010

[2] [2]

Bohner, A

M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications , Birkh¨ auser, Boston, 2001

work page 2001

[3] [3]

Elaydi, S., S

S. Elaydi, S., S. Zhang, Stability and periodicity of diﬀerence equations with ﬁnite delay. Funkcial. Ekvac 37(3) (1994), 401–413

work page 1994

[4] [4]

Hassibi, S.P

A. Hassibi, S.P. Boyd, J.P. How, Control of asynchronous dynamical systems with rate constraints on events, Proceedings of the 38th IEEE Conference on Decision and Control (1999), 1345–1351

work page 1999

[5] [5]

Heli¨ ovaara, R

K. Heli¨ ovaara, R. V¨ ais¨ anen, C. Simon,Evolutionary ecology of periodical insects, Trends in Ecology and Evolution 9(1994), no. 12, 475–480

work page 1994

[6] [6]

N. J. Higham, Functions of matrices: theory and computation . SIAM, 2008

work page 2008

[7] [7]

Hilger, Analysis on measure chains – a uniﬁed approach to continuous and discrete calculus , Results Math

S. Hilger, Analysis on measure chains – a uniﬁed approach to continuous and discrete calculus , Results Math. 18 (1990), 18–56

work page 1990

[8] [8]

P. Klemperer, Competition when Consumers have Switching Costs: An Overview with Applications to Industrial Organization, Macroeconomics, and International Trade , The Review of Economic Studies 62(1995), no. 4, 515–539

work page 1995

[9] [9]

Kelley, A

W. Kelley, A. Peterson, Diﬀerence Equations. An Introduction with Applications , Academic Press, London, 2001

work page 2001

[10] [10]

Lagunoﬀ, A

R. Lagunoﬀ, A. Matsui, Asynchronous choice in repeated coordination games. Econometrica 65 (1997), 1467–1477

work page 1997

[11] [11]

Libich, P

J. Libich, P. Stehl´ ık,Endogenous Monetary Commitment, Economic Letters. 112 (2011), 103–106

work page 2011

[12] [12]

Libich, P

J. Libich, P. Stehl´ ık,Incorporating Rigidity and Commitment in the Timing Structure of Macroeco- nomic Games, Economic Modelling. 27 (2010), 767–781

work page 2010

[13] [13]

L¨ utkepohl,Handbook of Matrices, John Wiley & Sons, 1997

H. L¨ utkepohl,Handbook of Matrices, John Wiley & Sons, 1997

work page 1997

[14] [14]

J. D. Murray, Mathematical Biology II., Springer, 2003

work page 2003

[15] [15]

Ogata, Discrete-time control systems, Prentice Hall Englewood Cliﬀs, NJ, 1995

K. Ogata, Discrete-time control systems, Prentice Hall Englewood Cliﬀs, NJ, 1995

work page 1995

[16] [16]

P¨ otzsche, S

C. P¨ otzsche, S. Siegmund, F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales , Discrete Contin. Dyn. Syst. 9 (2003), no. 5, 1223–1241

work page 2003

[17] [17]

W. Shou, C. T. Bergstrom, A. K. Chakraborty, F. K. Skinner. Theory, models and biology. eLife, 4(2015), e07158. http://doi.org/10.7554/eLife.07158

work page doi:10.7554/elife.07158 2015

[18] [18]

Slav´ ık,Dynamic equations on time scales and generalized ordinary diﬀerential equations, J

A. Slav´ ık,Dynamic equations on time scales and generalized ordinary diﬀerential equations, J. Math. Anal. Appl. 385 (2012), 534–550

work page 2012

[19] [19]

J. Tobin. Money and Finance in the Macroeconomic Process, Journal of Money, Credit and Banking 14 (1982), no.2, 171–204

work page 1982

[20] [20]

Q. Yu, J. Fish, Temporal homogenization of viscoelastic and viscoplastic solids subjected to locally periodic loading, Computational Mechanics 29 (2002), no.3, 199–211. 16 i := 1 Tµ (t) j := 1 Tµ (σ(t)) k := 1 Tν (t) ℓ := 1 Tν (σ(t)) Pictogram Aijkℓ 1 1 1 1 ( 1+µα 0 µβ 0 1 0 0 0 νγ 0 1+νδ 0 0 0 1 0 ) =:A1111 1 1 1 0 ( 1+µα 0 µβ 0 1 0 0 0 0 0 1 0 0 0 1 0 )...

work page 2002