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November  2009, 24(4): 1215-1224. doi: 10.3934/dcds.2009.24.1215

On the global attractor of delay differential equations with unimodal feedback

Eduardo Liz 1, and  Gergely Röst 2,

1. 

Departamento de Matemática Aplicada II, E.T.S.E. Telecomunicación, Universidade de Vigo, Campus Marcosende, 36310 Vigo, Spain
2. 

Analysis and Stochastics Research Group, Hungarian Academy of Sciences, Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1.

Received  February 2008 Revised  October 2008 Published  May 2009

We give bounds for the global attractor of the delay differential equation $ \dot 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 delay is sufficiently small, then all solutions 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.
Keywords: unimodal feedback, delay differential equation, global attractor, Schwarzian derivative., Nicholson's blowflies equation, Mackey-Glass equation.
Mathematics Subject Classification: 34K20, 34D4.
Citation: Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215
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