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Stability Threshold for Scalar Linear Periodic Delay Differential Equations

Published online by Cambridge University Press:  20 November 2018

Kyeongah Nah  and
Gergely Röst
Kyeongah Nah
Affiliation:
Bolyai Institute, University of Szeged, Szeged H-6720, Aradi vértanúk tere 1., Hungary e-mail: knah@math.u-szeged.hu e-mail: rost@math.u-szeged.hu
Gergely Röst
Affiliation:
Bolyai Institute, University of Szeged, Szeged H-6720, Aradi vértanúk tere 1., Hungary e-mail: knah@math.u-szeged.hu e-mail: rost@math.u-szeged.hu
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Abstract

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We prove that for the linear scalar delay differential equation

$$\dot{x}\left( t \right)=-a\left( t \right)x\left( t \right)+b\left( t \right)x\left( t-1 \right)$$

with non-negative periodic coefficients of period $p\,>\,0$, the stability threshold for the trivial solution is $r\,:=\int_{0}^{p}{\left( b\left( t \right)-a\left( t \right) \right)}dt\,=\,0$, assuming that $b\left( t+1 \right)-a\left( t \right)$ does not change its sign. By constructing a class of explicit examples, we show the counter-intuitive result that, in general, $r\,=\,0$ is not a stability threshold.

Keywords

34K2034K06delay differential equationstabilityperiodic system
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 849 - 857
Copyright
Copyright © Canadian Mathematical Society 2016

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