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Instabilities in threshold-diffusion equations with delay

Laing, C. R.; Coombes, S.


C. R. Laing


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. © 2008 Elsevier B.V. All rights reserved.


Laing, C. R., & Coombes, S. (2009). Instabilities in threshold-diffusion equations with delay. Physica D: Nonlinear Phenomena, 238(3), 264-272.

Journal Article Type Article
Acceptance Date Oct 15, 2008
Online Publication Date Nov 19, 2008
Publication Date Jan 1, 2009
Deposit Date Nov 19, 2008
Publicly Available Date Nov 19, 2008
Journal Physica D: Nonlinear Phenomena
Print ISSN 0167-2789
Electronic ISSN 0167-2789
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 238
Issue 3
Pages 264-272
Keywords delay, periodic orbit, Floquet exponent, travelling wave, global connection, Evans function
Public URL
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