Geodesic
On the instability of linear nonautonomous delay systems
Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 497-514
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The unstable properties of the linear nonautonomous delay system $x^{\prime }(t)=A(t)x(t)+B(t)x(t-r(t))$, with nonconstant delay $r(t)$, are studied. It is assumed that the linear system $y^{\prime }(t)=(A(t)+B(t))y(t)$ is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay $r(t)$ is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function $r(t)$ and the results depending on the asymptotic properties of the delay function.
The unstable properties of the linear nonautonomous delay system $x^{\prime }(t)=A(t)x(t)+B(t)x(t-r(t))$, with nonconstant delay $r(t)$, are studied. It is assumed that the linear system $y^{\prime }(t)=(A(t)+B(t))y(t)$ is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay $r(t)$ is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function $r(t)$ and the results depending on the asymptotic properties of the delay function.
Classification : 34D05, 34D09, 34D20, 34K06, 34K20
Keywords: Liapounov instability; $h$-instability; instability of delay equations; nonconstant delays 
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Naulin, Raúl. On the instability of linear nonautonomous delay systems. Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 497-514. http://geodesic.mathdoc.fr/item/CMJ_2003_53_3_a0/

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