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.