On the Global Attractor of Delay Differential Equations with Unimodal Feedback

TL;DR

This paper establishes bounds for the global attractor of certain delay differential equations with unimodal feedback, providing insights into their long-term behavior regardless of delay size.

Contribution

It introduces new bounds for the global attractor of delay differential equations with unimodal feedback and negative Schwarzian derivative, extending previous results to larger delays.

Findings

Solutions enter the monotone decreasing domain under certain conditions.

The sharpest interval containing the global attractor is determined for any delay.

Numerical examples illustrate the theoretical results with Nicholson's and Mackey-Glass equations.

Abstract

We give bounds for the global attractor of the delay differential equation $x'(t) =-\mu x(t)+f(x(t-\tau))$, where $f$ is unimodal and has negative Schwarzian derivative. If $f$ and $\mu$ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where f is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In this situation we determine the sharpest interval that contains the global attractor for any delay. In the absence of that condition, improving earlier results, we show that if the d5A5Aelay is sufficiently small, then all solution enter the domain where $f'$ is negative. Our theorems then are illustrated by numerical examples using Nicholson's blowflies equation and the Mackey-Glass equation.