Instabilities in threshold-diffusion equations with delay

Laing, C. R.; Coombes, S.

Abstract

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.

Citation

Laing, C. R., & Coombes, S. (2009). Instabilities in threshold-diffusion equations with delay. Physica D: Nonlinear Phenomena, 238(3), 264-272. https://doi.org/10.1016/j.physd.2008.10.014

• 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: 1872-8022
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 238
• Issue: 3
• Pages: 264-272
• DOI: https://doi.org/10.1016/j.physd.2008.10.014
• Keywords: delay, periodic orbit, Floquet exponent, travelling wave, global connection, Evans function
• Public URL: https://nottingham-repository.worktribe.com/output/1026273
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0167278908003692?via%3Dihub