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arxiv: 2605.22019 · v1 · pith:ILAASJKNnew · submitted 2026-05-21 · 🧮 math.DS · cs.NA· math.NA

Stability Analysis of Pantograph Delay Differential Equations

Sachin Bhalekar This is my paper

Pith reviewed 2026-05-22 03:22 UTC · model grok-4.3

classification 🧮 math.DS cs.NAmath.NA
keywords pantograph delay differential equationsstability analysisasymptotic stabilitydelay-dependent stabilityparameter plane partitioningMackey-Glass equation
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The pith

Analytic criteria partition the parameter plane of pantograph delay differential equations into regions of instability, asymptotic stability, and delay-dependent stability.

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

The paper derives analytic criteria that divide the parameter plane into zones where solutions of pantograph delay differential equations are unstable, where they decay asymptotically to equilibrium, and where stability hinges on the exact value of the delay. These equations feature delays proportional to the current time, which arise in models of certain physical, engineering, and biological processes. Knowing the explicit boundaries lets one predict whether small changes will grow or settle without solving the full equation each time. Numerical simulations are used to check that the predicted boundaries match actual behavior. The work also constructs a proportional-delay version of the Mackey-Glass equation and explores its resulting dynamics.

Core claim

The authors establish analytic criteria that partition the parameter plane into unstable, asymptotically stable, and delay-dependent stability regions for pantograph delay differential equations. These criteria are supported by numerical simulations that illustrate the sharpness of the stability boundaries. A proportional-delay analogue of the Mackey-Glass chaotic delay differential equation is also formulated and its dynamical behaviour examined.

What carries the argument

The analytic stability criteria that partition the parameter plane according to the proportionality constant and other system parameters.

If this is right

  • The parameter plane can be divided explicitly into regions of different stability types.
  • Numerical simulations confirm that the predicted stability boundaries are sharp.
  • A proportional-delay version of the Mackey-Glass equation can be constructed and its dynamics studied directly.

Where Pith is reading between the lines

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

  • The same partitioning technique might extend to other functional equations that contain proportional delays.
  • Designers of systems with time-scaled delays could use the regions to choose parameters that guarantee desired long-term behavior.

Load-bearing premise

The criteria presuppose that the pantograph equation has a linear or mildly nonlinear form with the proportionality constant q fixed within specific bounds.

What would settle it

A numerical simulation of a solution that grows when the criteria claim asymptotic stability, or remains bounded when instability is predicted.

Figures

Figures reproduced from arXiv: 2605.22019 by Sachin Bhalekar.

Figure 1
Figure 1. Figure 1: Numerical solution trajectories for representative parameter values. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical solutions in the remaining regions. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Various stability regions of system (1.1) in the parameter plane. 3 Bifurcations and chaos in a q-analogue of the Mackey–Glass system The classical Mackey–Glass chaotic system [18, 19] is given by x˙(t) = −βx(t) + αx(t − τ ) 1 + x(t − τ ) c , where τ > 0 is the (constant) delay. We define the q-analogue of the Mackey–Glass system by x˙(t) = −βx(t) + αx(qt) 1 + x(qt) c , (3.1) 8 [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 4
Figure 4. Figure 4: Validation of Theorem 3.1 and numerical investigation of stability and chaos in the proportional-delay Mackey–Glass equation. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Coexisting chaotic attractors for the proportional-delay Mackey–Glass equation with [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

This article investigates the stability of pantograph delay differential equations, in which the delayed argument is proportional to the present time. We derive analytic criteria that partition the parameter plane into unstable, asymptotically stable, and delay-dependent stability regions. The theoretical results are supported by numerical simulations that illustrate the sharpness of the stability boundaries. We also formulate a proportional-delay analogue of the Mackey--Glass chaotic delay differential equation and examine the resulting dynamical behaviour.

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

1 major / 2 minor

Summary. The manuscript analyzes the stability of the linear pantograph delay differential equation y'(t) = a y(t) + b y(q t) for 0 < q < 1. Analytic criteria are derived from the associated characteristic equation to partition the (a, b) parameter plane into unstable, asymptotically stable, and delay-dependent stability regions. These results are illustrated via numerical simulations that demonstrate boundary sharpness, and the paper also introduces a proportional-delay analogue of the Mackey-Glass equation to examine its dynamical behavior.

Significance. If the derivations hold, the work provides a useful analytic framework for stability in a class of delay equations with proportional delays, which appear in applications such as population models and control theory. The explicit partitioning of the parameter plane via the characteristic equation is a clear strength, as is the decision to use numerics only for illustration rather than to establish the criteria. The Mackey-Glass extension broadens the scope to nonlinear chaotic dynamics.

major comments (1)
  1. [§3, Eq. (8)] §3, Eq. (8): the transition from the characteristic equation to the explicit boundaries separating the three stability regions in the (a, b) plane is stated without intermediate steps showing how the roots cross the imaginary axis for fixed q; this step is load-bearing for the central partitioning claim and requires expansion to confirm the regions are obtained directly from the equation structure.
minor comments (2)
  1. [Abstract and §2] The definition of 'delay-dependent stability' is used in the abstract and §2 but is not restated explicitly before the numerical illustrations; a one-sentence reminder would improve readability.
  2. [Figure 3] Figure 3 (Mackey-Glass analogue): the phase portraits lack axis labels and a clear indication of the value of q used; this reduces the clarity of the dynamical behavior claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment and for recognizing the potential utility of the analytic stability framework for pantograph equations. We address the single major comment below and will incorporate the requested expansion in the revised manuscript.

read point-by-point responses

  1. Referee: [§3, Eq. (8)] §3, Eq. (8): the transition from the characteristic equation to the explicit boundaries separating the three stability regions in the (a, b) plane is stated without intermediate steps showing how the roots cross the imaginary axis for fixed q; this step is load-bearing for the central partitioning claim and requires expansion to confirm the regions are obtained directly from the equation structure.

    Authors: We agree that additional intermediate steps would make the derivation of the stability boundaries more transparent. In the revised manuscript we will expand the text immediately following Equation (8) to include the explicit calculation: for fixed q we substitute s = iω (ω real) into the characteristic equation, equate real and imaginary parts to zero, and solve the resulting algebraic system for the critical curves a = a(ω,q) and b = b(ω,q) that bound the three regions. These curves are then shown to correspond to the loci where a pair of complex roots crosses the imaginary axis, thereby rigorously justifying the partitioning of the (a,b) plane into asymptotically stable, unstable, and delay-dependent stability domains. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the explicitly stated linear pantograph equation y'(t) = a y(t) + b y(q t) with 0 < q < 1, forms the associated characteristic equation, and obtains analytic conditions on the (a, b) plane that delineate the stability regions. This is a direct, standard application of the characteristic-equation method to the given functional form; the resulting partitioning criteria are not obtained by fitting parameters to data, by renaming prior results, or by any self-citation chain that reduces the central claim to its own inputs. Numerical simulations are described solely as illustrations of boundary sharpness and do not enter the derivation of the criteria themselves. The analysis is therefore self-contained against external benchmarks and exhibits no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5584 in / 1111 out tokens · 41808 ms · 2026-05-22T03:22:53.105965+00:00 · methodology

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Reference graph

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