Instabilities in threshold-diffusion equations with delay

Laing, Carlo and Coombes, Stephen Instabilities in threshold-diffusion equations with delay. Physica D . ISSN 0167-2789 (In Press)

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The introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio-temporal chaos. In this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria, periodic orbits and travelling waves. Importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function. Using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback. Direct numerical simulations are used to confirm our bifurcation analysis, and to probe some of the rich behaviour possible for mixed feedback.

Item Type: Article
Keywords: delay, periodic orbit, Floquet exponent, travelling wave, global connection, Evans function
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
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Depositing User: Coombes, Prof Stephen
Date Deposited: 19 Nov 2008 09:46
Last Modified: 04 May 2020 20:34

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