Stability Analysis of Pantograph Delay Differential Equations

TL;DR

This paper analyzes the stability of pantograph delay differential equations with proportional delays, deriving criteria for stability regions, supported by simulations, and explores a related chaotic delay differential equation.

Contribution

It introduces analytic stability criteria for pantograph delay differential equations and examines a proportional-delay Mackey--Glass model, advancing understanding of their dynamical behavior.

Findings

Derived explicit stability boundaries for pantograph delay equations

Numerical simulations confirm the sharpness of stability criteria

Explored chaotic dynamics in a proportional-delay Mackey--Glass model

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.