Infinite Dimensional Dynamical Systems pp 269-278 | Cite as

# Threshold Dynamics of Scalar Linear Periodic Delay-Differential Equations

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## Abstract

We consider the scalar linear periodic delay-differential equation \(\dot{x}(t) = -x(t) + ag(t)x(t - 1)\), where \(g : [0,\infty ) \rightarrow (0,\infty )\) is continuous and periodic with the minimal period ω>0. We show that there exists a positive *a*^{+} such that the zero solution is stable if \(a \in (0,{a}^{+})\) and unstable if *a*>*a*^{+}. Examples and preliminary analysis suggest the challenge in obtaining analogous results when *a*<0.

## Notes

### Acknowledgements

Yuming Chen was supported in part by NSERC and the Early Researcher Award program of Ontario. Jianhong Wu was supported in part by CRC, MITACS and NSERC.

Received 4/4/2009; Accepted 2/10/2010

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