Analyzing stability of a delay differential equation involving two delays

TL;DR

This paper investigates the stability of a generalized delay differential equation with two delays, deriving conditions for stability, identifying chaotic oscillations in unstable regions, and proposing a numerical solution scheme.

Contribution

It introduces stability criteria for a two-delay fractional differential equation and provides a numerical method for solving such equations, with analysis of chaotic behavior.

Findings

Chaotic oscillations occur only in unstable regions.

Derived critical delay values for stability.

Proposed a numerical scheme for solving the equations.

Abstract

Analysis of the systems involving delay is a popular topic among applied scientists. In the present work, we analyze the generalized equation $D^{\alpha} x(t) = g\left(x(t-\tau_1), x(t-\tau_2)\right)$ involving two delays viz. $\tau_1\geq 0$ and $\tau_2\geq 0$. We use the the stability conditions to propose the critical values of delays. Using examples, we show that the chaotic oscillations are observed in the unstable region only. We also propose a numerical scheme to solve such equations.