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Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback. (English) Zbl 1137.34038

This paper initiates a very interesting analysis of the dynamics generated by the nonlinear delayed differential equation

${x}^{\text{'}}\left(t\right)=-\mu x\left(t\right)+f\left(x\left(t-\tau \right)\right)\phantom{\rule{2.em}{0ex}}\left(*\right)$

with unimodal feedback. The paper provides a systematic study of the behaviour of the positive solutions to equation $\left(*\right)$ in the most interesting case when the unique positive equilibrium is larger than the critical point of the feedback $f$. The existence of a global attractor is shown which is a subset of some invariant closed interval $\left[\alpha ,\beta \right]$, which attracts every nontrivial nonnegative trajectory. An important step is to determine if $\left[\alpha ,\beta \right]$ belongs to the domain where ${f}^{\text{'}}$ is negative, which enables the applying of the powerful theory of delayed differential equations with monotone feedback. This situation occurs under further restrictions that are fulfilled for a wide range of parameters. Then the existence of heteroclinic orbits from the trivial equilibrium (unstable) to a periodic orbit oscillating around the positive equilibrium is established. But in some cases and for large delays it cannot be expected that all the nonnegative solutions enter and remain in the domain where the feedback is monotone, giving rise to complex dynamics. Both situations are illustrated with numerical simulations of Mackey-Glass and Nicholson’s blowflies equations, in different scenarios.

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 34K60 Qualitative investigation and simulation of models 34K28 Numerical approximation of solutions of functional-differential equations