Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T20:39:07.214Z Has data issue: false hasContentIssue false

Limiting profiles for periodic solutions of scalar delay differential equations

Published online by Cambridge University Press:  12 July 2007

Y. Chen
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3

Abstract

Let f(·, λ) : R→R be given so that f(0, λ) = 0 and f(x, λ) = (1 + λ)x + ax2 + bx3 + o(x3) as x → 0. We characterize those small values of ε > 0 and λ ∈ R for which there are periodic solutions of periods approximately 1/k with k ∈ N of the delay equations When a = 0, these periodic solutions approach square waves if b < 0 or pulses if b > 0 as ε → 0. These results are similar to those obtained by Chow et al. and Hale and Huang, where the case of f(x, λ) = −(1 + λ)x + ax2 + bx3 + o(x3) as x → 0 is considered. However, when a ≠ 0, all these periodic solutions approach pulses as ε → 0; an interesting phenomenon that cannot happen in the case considered by Chow et al. and Hale and Huang.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)