A Remark on Resonance and Beat in a Homogeneous Linear Delay System

We consider a homogeneous system of linear delay differential equations (DDEs) with internal coupling. It is well known that such systems can exhibit oscillatory solutions via Hopf bifurcation when the system parameters, including the delay, satisfy certain critical conditions. In this paper, we show that this oscillatory behavior can lead to resonance-like amplification in one variable, even in the absence of any external forcing. The phenomenon arises when the natural frequency of the internal forcing, induced by delay and coupling, matches the system’s oscillatory mode. Furthermore, when distinct delays are introduced in each equation, a beat phenomenon naturally occurs due to the detuning between internal frequencies.