Insights into the multiplicity-induced-dominancy for scalar delay-differential equations with two delays

It has been observed in recent works that, for several classes of linear time-invariant time-delay systems retarded or neutral type with a single delay, if root its characteristic equation attains maximal multiplicity, then this is the rightmost spectral value, and hence it determines exponential behavior system, property usually referred to as multiplicity-induced-dominancy (MID). In paper, we investigate MID one simplest cases two delays, scalar delay-differential first order delayed terms zero. We discuss standard approach based on argument principle establishing single-delay some limitations case our simple system before proposing technique crossing imaginary roots that allows conclude holds setting.