On the global attractor of delay differential equations with unimodal feedback
By Eduardo Liz, Gergely Rost. Discrete & Continuous Dynamical Systems - A, 2008
52 citations recorded by the source corpus.
We give bounds for the global attractor of the delay differential equation $ \dot x(t)=-\mu x(t)+f(x(t-\tau))$, where $f$ is unimodal and has negative Schwarzian derivative. If $f$ and $\mu$ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where $f$ is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In this situation we determine the sharpest interval that contains the global attractor for any delay. In the absence of that condition, improving earlier results, we show that if the delay is sufficiently small, then all solutions enter the domain where $f'$ is negative. Our theorems then are illustrated by numerical examples using Nicholson's blowflies equation and the Mackey-Glass equation.
Subfields: Computer Networks and Communications, Control and Systems Engineering, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/c1fc072460cf541fe25c78e0fb8a2adcf5922186
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Citations in the corpus
- On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities (2014, unknown): “, [1,3–5,9–13,15–17,20,23,24,27] and the references therein.”
- Map dynamics versus dynamics of associated delay reaction–diffusion equations with a Neumann condition (2010, background): “The more general form (1.1) has also been extensively and intensively studied (e.g. Kuang 1993; Walther 1995; Cooke et al. 1999; Györi & Trofimchuk 1999; Krisztin et al. 1999; Faria 2006; Röst & Wu 2007; Krisztin 2008; Liz & Röst 2009, in press; and references therein).”
- Nicholson's blowflies differential equations revisited: Main results and open problems (2010, background): “For autonomous equation (4) the solution estimates and attracting sets were studied in [33,34].”