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On the global attractor of delay differential equations with unimodal feedback. (English) Zbl 1188.34102

For delay equations of the type

$\stackrel{˙}{x}\left(t\right)=-\mu x\left(t\right)+f\left(x\left(t-\tau \right)\right),$

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.

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 92D25 Population dynamics (general) 34K19 Invariant manifolds (functional-differential equations) 34K20 Stability theory of functional-differential equations