Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback

Röst, Gergely and Wu, Jianhong (2007) Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463 (2086). pp. 2655-2669. ISSN 1364-5021 (print), 1471-2946 (online)

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Abstract

The dynamics generated by the delay differential equation \dot{x}(t) = -mu x(t) + f(x(t - tau)) with unimodal feedback is studied. The existence of the global attractor is shown and bounds of the attractor are given. We find attractive invariant intervals and give sufficient conditions that guarantee that all solutions enter the domain where f' is negative with respect to a positive equilibrium, so the results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In particular, the existence of heteroclinic orbits from the trivial equilibrium to a periodic orbit oscillating around the positive equilibrium is established. Numerical examples using Nicholson's blowfies equation and the Mackey Glass equation are provided to illustrate the main results.

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