Multi-Delay Differential Equations: A Taylor Expansion Approach

TL;DR

This paper introduces a Taylor expansion method to approximate critical delays in multi-delay differential equations, enabling stability analysis where exact formulas are unknown, especially for systems with two or three delays.

Contribution

It presents a novel approximation approach for multi-delay systems using Taylor expansions, providing formulas for critical delays and analyzing their accuracy.

Findings

Approximate formulas for critical delays in multi-delay systems.

Effective for systems with two or three delays.

Performance depends on system parameters and number of delays.

Abstract

It is already well-understood that many delay differential equations with only a single constant delay exhibit a change in stability according to the value of the delay in relation to a critical delay value. Finding a formula for the critical delay is important to understanding the dynamics of delayed systems and is often simple to obtain when the system only has a single constant delay. However, if we consider a system with multiple constant delays, there is no known way to obtain such a formula that determines for what values of the delays a change in stability occurs. In this paper, we present some single-delay approximations to a multi-delay system obtained via a Taylor expansion as well as formulas for their critical delays which are used to approximate where the change in stability occurs in the multi-delay system. We determine when our approximations perform well and we give extra analytical and numerical attention to the two-delay and three-delay settings.