Asymptotic properties of a two-dimensional differential system with a bounded nonconstant delay under the conditions of instability

Abstract. A useful method for the investigation of the asymptotic behaviour of the solutions of two-dimensional systems of ordinary differential equations is the method of complexification. This method was used e. g. in papers [6] and [5], where asymptotic properties of a real two-dimensional system x ′ = A(t)x + h(t, x) were studied. In the present contribution we shall examine the asymptotic nature of the solutions of a real two-dimensional system of retarded differential equations x′(t) = A(t)x(t) + B(t)x(t−r)+h(t, x(t), x(t−r)), where r> 0 is a constant delay, A, B and h being matrix functions and a vector function, respectively. The method of complexification transforms this system to one equation with complex-valued coefficients. Stability and the asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional and by virtue of the Ważewski topological principle. The contribution has a character of an overview article, nevertheless several results are given in a somewhat modified and more general form than those given in [4] and [3]