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Stability Threshold for Scalar Linear Periodic Delay Differential Equations

  Published:2016-09-14
 Printed: Dec 2016
  • Kyeongah Nah,
    Bolyai Institute, University of Szeged, Szeged H-6720, Aradi vértanúk tere 1., Hungary
  • Gergely Röst,
    Bolyai Institute, University of Szeged, Szeged H-6720, Aradi vértanúk tere 1., Hungary
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Abstract

We prove that for the linear scalar delay differential equation $$ \dot{x}(t) = - a(t)x(t) + b(t)x(t-1) $$ with non-negative periodic coefficients of period $P\gt 0$, the stability threshold for the trivial solution is $r:=\int_{0}^{P} \left(b(t)-a(t) \right)\mathrm{d}t=0,$ assuming that $b(t+1)-a(t)$ 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: delay differential equation, stability, periodic system delay differential equation, stability, periodic system
MSC Classifications: 34K20, 34K06 show english descriptions Stability theory
Linear functional-differential equations
34K20 - Stability theory
34K06 - Linear functional-differential equations
 

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