Stability Analysis of Pantograph Delay Differential Equations

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.

• The paper develops a stability classification for scalar pantograph equations using exact power-series bounds, Lyapunov–Krasovskii and Chetaev functionals, and numerical simulations.
• Rigorous results identify sufficient instability and stability conditions, while simulations suggest stability for all q in (0,1) when a<0 and a<b<-a and reveal an empirical threshold b*(a) in delay-dependent regimes.
• Applying the framework to a proportional-delay Mackey–Glass model yields bifurcation thresholds, ultra-slow oscillations, sustained chaos, and coexisting chaotic attractors for selected parameter values.

Overview and motivation

This paper by Sachin Bhalekar studies the scalar pantograph (proportional-delay) equation

$\dot{x}(t)=ax(t)+bx(qt), \qquad 0<q<1,$

where the delay $\tau(t)=(1-q)t$ grows unboundedly with time. The work addresses a structural obstacle: unlike constant-delay equations, the exponential ansatz $x(t)=e^{\lambda t}$ fails here because the terms $e^{\lambda t}$ and $e^{\lambda qt}$ cannot be factored into a characteristic equation, and Laplace-transform techniques are not directly applicable. The analysis therefore combines three tools: the exact power-series solution

$$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$

Lyapunov–Krasovskii and Lyapunov–Chetaev functionals, and numerical simulation via a purpose-built predictor-corrector scheme. The stability framework is then applied to a proportional-delay analogue of the Mackey–Glass equation, where bifurcation thresholds and chaotic dynamics are identified.

Analytical stability criteria

The paper establishes four rigorous results for the linear pantograph equation.

Instability from the series solution. Two elementary sufficient conditions follow directly from bounding the product in the series representation. If $a>0$ and $b>0$, then $\prod_{j=0}^{n-1}(a+bq^j)>a^n$, so $x(t)>e^{at}$ and the solution is unbounded. Similarly, if $\tau(t)=(1-q)t$0, then $\tau(t)=(1-q)t$1, again giving instability. Both hold for all $\tau(t)=(1-q)t$2.

Asymptotic stability via Lyapunov–Krasovskii. Using the functional

$\tau(t)=(1-q)t$3

the derivative along trajectories is a quadratic form whose matrix is negative definite when optimized over $\tau(t)=(1-q)t$4 at $\tau(t)=(1-q)t$5. This yields the sufficient condition

$\tau(t)=(1-q)t$6

The paper explicitly notes this condition is sufficient but not necessary: numerics show asymptotic stability holds throughout the strictly larger region $\tau(t)=(1-q)t$7, $\tau(t)=(1-q)t$8 for all $\tau(t)=(1-q)t$9, making the Lyapunov–Krasovskii criterion conservative.

Instability via Lyapunov–Chetaev. With $x(t)=e^{\lambda t}$0, one has $x(t)=e^{\lambda t}$1 whenever $x(t)=e^{\lambda t}$2, so trajectories starting arbitrarily close to the origin move away; the system is unstable for all $x(t)=e^{\lambda t}$3 under this single scalar condition.

Numerical observations and the empirical threshold $x(t)=e^{\lambda t}$4

A trapezoidal predictor-corrector method with linear interpolation of $x(t)=e^{\lambda t}$5 between grid points validates the analytical criteria and extends the classification to regions not covered by theory:

• In the region $x(t)=e^{\lambda t}$6, $x(t)=e^{\lambda t}$7, $x(t)=e^{\lambda t}$8 (outside the Lyapunov–Krasovskii condition), solutions still converge to zero — damped oscillations for $x(t)=e^{\lambda t}$9, monotone convergence for $e^{\lambda t}$0 — though convergence is ultra-slow for small $e^{\lambda t}$1.
• For $e^{\lambda t}$2 and $e^{\lambda t}$3, simulations reveal a threshold $e^{\lambda t}$4: if $e^{\lambda t}$5, the system is stable for large $e^{\lambda t}$6 and unstable (with ultra-slow oscillatory divergence) for small $e^{\lambda t}$7; if $e^{\lambda t}$8, it is unstable for every $e^{\lambda t}$9. The computed pairs $e^{\lambda qt}$0 are well approximated by the quadratic fit

$e^{\lambda qt}$1

• If $e^{\lambda qt}$2 with $e^{\lambda qt}$3, divergence is immediate and monotone for any $e^{\lambda qt}$4.

The resulting stability diagram partitions the parameter plane into delay-independent stable regions, delay-independent unstable regions, and an intermediate delay-dependent region. A recurring qualitative feature near boundaries is ultra-slow convergence or divergence, consistent with the effective delay $e^{\lambda qt}$5 growing without bound so that oscillation periods increase with time.

The $e^{\lambda qt}$6-analogue of the Mackey–Glass equation

The paper proposes the proportional-delay Mackey–Glass system

$e^{\lambda qt}$7

It has equilibria at $e^{\lambda qt}$8 and at $e^{\lambda qt}$9. Linearization about the origin reduces exactly to the pantograph equation with $$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$0, $$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$1; since $$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$2, the Chetaev theorem implies the zero equilibrium is unstable for all $$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$3.

Linearization about the nonzero equilibria gives $$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$4 and $$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$5, which yields explicit bifurcation thresholds in $$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$6:

Regime	Condition on $$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$7	Stability of $$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$8
Delay-independent stability	$$x(t)=x(0)\sum_{n=0}^{\infty}\frac{t^n}{n!}\prod_{j=0}^{n-1}\left(a+bq^j\right),$$9	Stable for all $a>0$0
Delay-dependent stability	$a>0$1	Stable only for sufficiently large $a>0$2
Instability	$a>0$3	Unstable for all $a>0$4

For the representative case $a>0$5, $a>0$6, these give $a>0$7 and $a>0$8. Simulations confirm the predictions: at $a>0$9 solutions converge to $b>0$0 depending on the sign of $b>0$1; at $b>0$2 the solution is stable for $b>0$3 but oscillates around the equilibrium for $b>0$4, with period increasing in time as expected from the growing delay. At $b>0$5 with $b>0$6, the system exhibits sustained chaotic oscillations, and notably the paper reports coexisting chaotic attractors (e.g., at $b>0$7, $b>0$8), indicating multistability in the proportional-delay regime.

Limitations and open questions

Two central results rest on numerical evidence rather than proof. First, the claim that the entire region $b>0$9, $\prod_{j=0}^{n-1}(a+bq^j)>a^n$0 is asymptotically stable for every $\prod_{j=0}^{n-1}(a+bq^j)>a^n$1 remains a conjecture supported by computation; the Lyapunov–Krasovskii argument covers only the subregion $\prod_{j=0}^{n-1}(a+bq^j)>a^n$2. Second, the threshold curve $\prod_{j=0}^{n-1}(a+bq^j)>a^n$3 is purely empirical, obtained from tabulated simulations and fitted quadratically; no analytic derivation or sharp bound is provided, and its accuracy away from sampled values of $\prod_{j=0}^{n-1}(a+bq^j)>a^n$4 is unverified. Additionally, the chaos observed in the nonlinear $\prod_{j=0}^{n-1}(a+bq^j)>a^n$5-analogue is demonstrated through phase portraits and time series rather than through quantitative measures such as Lyapunov exponents, and the coexistence of attractors is reported only for specific parameter values.

Conclusion

The paper develops a stability classification for the pantograph equation that circumvents the absence of a characteristic equation, combining exact series bounds, Lyapunov-type functionals, and tailored numerics into a three-region stability diagram featuring delay-independent stability, delay-dependent stability, and instability. Applying this classification to a proportional-delay Mackey–Glass equation yields closed-form bifurcation thresholds and reveals chaotic dynamics with coexisting attractors. The main open problems are a rigorous proof of stability in the full region $\prod_{j=0}^{n-1}(a+bq^j)>a^n$6, $\prod_{j=0}^{n-1}(a+bq^j)>a^n$7 for all $\prod_{j=0}^{n-1}(a+bq^j)>a^n$8, and an analytic characterization of the empirically observed threshold $\prod_{j=0}^{n-1}(a+bq^j)>a^n$9.