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On the instability of linear nonautonomous delay systems. (English) Zbl 1080.34543
Summary: The unstable properties of the linear nonautonomous delay system \(x'(t) = A (t) x (t) + B (t) x (t - r (t))\), with nonconstant delay \(r(t)\), are studied. It is assumed that the linear system \(y'(t) = (A (t) + B (t)) y (t)\) 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.

MSC:
34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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