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Asymptotic properties of an unstable two-dimensional differential system with delay. (English) Zbl 1114.34058

Summary: The asymptotic behaviour of the solutions is studied for the real unstable two-dimensional system
\[ x'(t)={\mathbf A}(t)x(t)+{\mathbf B}(t)x(t-r)+h(t,x(t),x(t-r)), \]
where \(r>0\) is a constant delay. It is supposed that \(\mathbf A\), \(\mathbf {B}\) and \(h\) are matrix functions and a vector function, respectively. The method of investigation is based on the transformation of the real system into an equation with complex-valued coefficients. Asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional and by means of Ważewski’s topological principle.

MSC:

34K25 Asymptotic theory of functional-differential equations
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