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Zbl 1188.34102
Liz, Eduardo; Röst, Gergely
On the global attractor of delay differential equations with unimodal feedback. (English)
[J] Discrete Contin. Dyn. Syst. 24, No. 4, 1215-1224 (2009). ISSN 1078-0947; ISSN 1553-5231/e

For delay equations of the type $$\dot x(t) = -\mu x(t) + f(x(t-\tau)),$$ the method of describing invariant sets for the semiflow by invariant intervals for the map $g := \mu^{-1} f$ (it goes back to a paper by Ivanov and Sharkovsky) is skillfully employed. New results which work under less restrictive assumptions on smallness of the delay $ \tau$ are proved. In examples like Nicholson's blowflies equation or the Mackey-Glass equation, it is possible to prove that the attractor is contained in a monotonicity interval of $f$, so that the results on delayed monotone feedback apply.
[Bernhard Lani-Wayda (Giessen)]

MSC 2000:: *34K25 Asymptotic theory of functional-differential equations
92D25 Population dynamics
34K19 Invariant manifolds
34K20 Stability theory of functional-differential equations

Keywords: delay differential equation; unimodal feedback; invariant intervals; global attractor; Nicholson's blowflies equation; Mackey-Glass equation; Schwarzian derivative

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