Geodesic
Stability Threshold for Scalar Linear Periodic Delay Differential Equations
Canadian mathematical bulletin, Tome 59 (2016) no. 4, pp. 849-857

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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.
DOI : 10.4153/CMB-2016-043-0
Mots-clés : 34K20, 34K06, delay differential equation, stability, periodic system 
Nah, Kyeongah; Röst, Gergely. Stability Threshold for Scalar Linear Periodic Delay Differential Equations. Canadian mathematical bulletin, Tome 59 (2016) no. 4, pp. 849-857. doi: 10.4153/CMB-2016-043-0
@article{10_4153_CMB_2016_043_0,
     author = {Nah, Kyeongah and R\"ost, Gergely},
     title = {Stability {Threshold} for {Scalar} {Linear} {Periodic} {Delay} {Differential} {Equations}},
     journal = {Canadian mathematical bulletin},
     pages = {849--857},
     year = {2016},
     volume = {59},
     number = {4},
     doi = {10.4153/CMB-2016-043-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-043-0/}
}
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