### Abstract

Original language | English |
---|---|

Pages (from-to) | 354-368 |

Number of pages | 15 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 135 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 1988 |

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### Keywords

- stability boundaries
- delay ratio

### Cite this

*Journal of Mathematical Analysis and Applications*,

*135*(1), 354-368. https://doi.org/10.1016/0022-247X(88)90160-6

}

*Journal of Mathematical Analysis and Applications*, vol. 135, no. 1, pp. 354-368. https://doi.org/10.1016/0022-247X(88)90160-6

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

Research output: Contribution to journal › Article

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 -