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

Fingerprint

Delayed Feedback
Feedback
Characteristic equation
Switches
Stability Switch
Delay-differential Systems
Variable Delay
Unstable
Eigenvalue
Zero
Model
Demonstrate

Keywords

  • stability boundaries
  • delay ratio

Cite this

@article{1b97b4d12c884ad09c8180ef62a95b6e,
title = "Period to delay ratios near stability boundaries for systems with delayed feedback",
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.",
keywords = "stability boundaries, delay ratio",
author = "A.E. Jones and R.M. Nisbet and William Gurney and S.P. Blythe",
year = "1988",
month = "10",
doi = "10.1016/0022-247X(88)90160-6",
language = "English",
volume = "135",
pages = "354--368",
journal = "Journal of Mathematical Analysis and Applications",
issn = "0022-247X",
number = "1",

}

Period to delay ratios near stability boundaries for systems with delayed feedback. / Jones, A.E.; Nisbet, R.M.; Gurney, William; Blythe, S.P.

In: Journal of Mathematical Analysis and Applications, Vol. 135, No. 1, 10.1988, p. 354-368.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Period to delay ratios near stability boundaries for systems with delayed feedback

AU - Jones, A.E.

AU - Nisbet, R.M.

AU - Gurney, William

AU - Blythe, S.P.

PY - 1988/10

Y1 - 1988/10

N2 - 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.

AB - 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.

KW - stability boundaries

KW - delay ratio

U2 - 10.1016/0022-247X(88)90160-6

DO - 10.1016/0022-247X(88)90160-6

M3 - Article

VL - 135

SP - 354

EP - 368

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -