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

Jones, A.E. and Nisbet, R.M. and Gurney, William and Blythe, S.P. (1988) Period to delay ratios near stability boundaries for systems with delayed feedback. Journal of Mathematical Analysis and Applications, 135 (1). pp. 354-368. ISSN 0022-247X

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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.