DELAY-DEPENDENT STABILITY SWITCHES IN A DELAY DIFFERENTIAL SYSTEM

In this study, stability properties of a linear delay differential system x '(t) = -ax(t - tau) - by(t), y '(t) = -cx(t) - dy(t - tau) are considered, where a, b, c, and d are real numbers and tau > 0. Some explicit necessary and sufficient conditions are presented for the zero solution of the system to be asymptotically stable. The results demonstrate that delay-dependent stability switches in the system can occur not only when bc < 0 but also when a > 0, b > 0, c > 0, and d > 0. Some examples are provided to illustrate the delay-dependent stability switches. The proof technique is based on careful analysis of the existence and the transversality of characteristic roots on the imaginary axis.