Shape of transition layers in a differential--delay equation

Wattis, Jonathan A.D.

Abstract

We use asymptotic techniques to describe the bifurcation from steady-state to a periodic solution in the singularly perturbed delayed logistic equation ?x?(t) = ?x(t)+ ? f(x(t ? 1)) with ? ? 1. The solution has the form of plateaus of approximately unit width separated by narrow transition layers. The calculation of the period two solution is complicated by the presence of delay terms in the equation for the transition layers, which induces a phase shift that has to be calculated as part of the solution. High order asymptotic calculations enable both the shift and the shape of the layers to be determined analytically, and hence the period of the solution. We show numerically that the form of transition layers in the four-cycles is similar to that of the two-cycle, but that a three-cycle exhibits different behaviour.

Citation

Wattis, J. A. (in press). Shape of transition layers in a differential--delay equation. IMA Journal of Applied Mathematics, https://doi.org/10.1093/imamat/hxx011

• Journal Article Type: Article
• Acceptance Date: Apr 13, 2017
• Online Publication Date: May 4, 2017
• Deposit Date: Apr 20, 2017
• Publicly Available Date: May 4, 2017
• Journal: IMA Journal of Applied Mathematics
• Print ISSN: 0272-4960
• Electronic ISSN: 1464-3634
• Publisher: Oxford University Press
• Peer Reviewed: Peer Reviewed
• DOI: https://doi.org/10.1093/imamat/hxx011
• Keywords: Asymptotic analysis, differential--delay equation, ; transition layers
• Public URL: https://nottingham-repository.worktribe.com/output/859200
• Publisher URL: https://academic.oup.com/imamat/article-lookup/doi/10.1093/imamat/hxx011
• Additional Information: This is a pre-copyedited, author-produced version of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record Jonathan A. D. Wattis; Shape of transition layers in a differential-delay equation. IMA J Appl Math 2017 hxx011. doi: 10.1093/imamat/hxx011 is available online at: https://academic.oup.com/imamat/article-lookup/doi/10.1093/imamat/hxx011