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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

x ˙(t)=-μx(t)+f(x(t-τ)),

the method of describing invariant sets for the semiflow by invariant intervals for the map g:=μ -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 τ 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.

34K25Asymptotic theory of functional-differential equations
92D25Population dynamics (general)
34K19Invariant manifolds (functional-differential equations)
34K20Stability theory of functional-differential equations