Asymptotic behaviour of a two-dimensional differential system with delay under the conditions of instability

Eighty fifth anniversary of birthday and scientific legacy of Professor Miloš Ráb

“…The asymptotic properties of the two-dimensional system with constant delay in the stable case were studied by J. Kalas and L. Baráová in [28], the asymptotic behavior of solutions under the conditions of instability was investigated by J. Kalas in [29], [30]. The equations with a finite system of constant delays were inspected by J. Rebenda in [49], [50].…”

“…The equations with a finite system of constant delays were inspected by J. Rebenda in [49], [50]. The asymptotic properties of solutions of the two-dimensional system with nonconstant delays were investigated by J. Kalas and J. Rebenda in [31], [32], [34], [51]. The asymptotic behavior of the solutions of systems with a finite number of nonconstant delays was examined by J. Kalas and J. Rebenda in [33] and by J. Rebenda and Z. Šmarda in [52]- [54].…”

Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

“…Related results for ordinary differential equations without delay can be found in [12]. The asymptotic behaviour of the solutions of (1.1') under the conditions of instability was studied in [7,8].…”

Asymptotic properties of an unstable two-dimensional differential system with delay

“…Stability and asymptotic properties of the solutions for the stable case of (0) are investigated in [2]. The unstable case of (0) was studied in [1]. In [2], it was shown that it is useful to investigate (0) also under different conditions, namely the conditions, when the shortened equation x (t) = A(t)x(t) is closer to a "focus" than to a "node" at origin.…”

“…The key tool will be a Razumikhin-type version of Ważewski topological method. Similarly to [1], we shall concentrate considerable attention to the problem of existence of bounded solutions or solutions tending to the origin as t → ∞. Related results for ordinary differential equations without delay can be found in [7] and [3].…”