### 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 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 |

### 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

Jones, A. E., Nisbet, R. M., Gurney, W., & Blythe, S. P. (1988). Period to delay ratios near stability boundaries for systems with delayed feedback.

*Journal of Mathematical Analysis and Applications*,*135*(1), 354-368. https://doi.org/10.1016/0022-247X(88)90160-6