Period to delay ratios near stability boundaries for systems with delayed feedback

A.E. Jones, R.M. Nisbet, William Gurney, S.P. Blythe

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

We show how, with a suitable choice of a “free” parameter, period to delay ratios near stability boundaries may be found for delay-differential systems with a single delay, and with a characteristic equation of the form F(λ) + G(λ)e−λτ = 0. When F and G do not depend on the delay, τ itself is a natural choice for the free parameter, and the the period to delay ratio can be easily found for given values of the parameters of F and G. It is shown that if more than one stability switch occurs for such a system as τ is increased, then the period to delay ratio will become progressively smaller with each stable-unstable change. By considering a model with a variable delay, we demonstrate how to determine period to delay ratios when the characteristic equation is such that F and G themselves depend on τ, and show that for the model considered, the period must always lie between τ and 2τ. An Appendix considers the appearance of zero eigenvalues in such characteristic equations.
Original languageEnglish
Pages (from-to)354-368
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Volume135
Issue number1
DOIs
Publication statusPublished - Oct 1988

Keywords

  • stability boundaries
  • delay ratio

Fingerprint Dive into the research topics of 'Period to delay ratios near stability boundaries for systems with delayed feedback'. Together they form a unique fingerprint.

  • Cite this