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- Title
Shape of transition layers in a differential-delay equation.
- Authors
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 ∈ ... (t) = - x(t) + λf (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 behaviours.
- Subjects
BIFURCATION theory; PERTURBATION theory; LOGISTIC functions (Mathematics); LOGISTIC distribution (Probability); DIFFERENTIAL equations; ASYMPTOTIC theory of algebraic ideals
- Publication
IMA Journal of Applied Mathematics, 2017, Vol 82, Issue 3, p681
- ISSN
0272-4960
- Publication type
Article
- DOI
10.1093/imamat/hxx011