Invariance properties in the root sensitivity of time-delay systems with double imaginary roots

If iω∈iR is an eigenvalue of a time-delay system for the delay τ0τ0 then iω is also an eigenvalue for the delays τk≔τ0+k2π/ωτk≔τ0+k2π/ω, for any k∈Zk∈Z. We investigate the sensitivity, periodicity and invariance properties of the root iω for the case that iω is a double eigenvalue for some τkτk. It turns out that under natural conditions (the condition that the root exhibits the completely regular splitting property   if the delay is perturbed), the presence of a double imaginary root iω for some delay τ0τ0 implies that iω is a simple root for the other delays τkτk, k≠0k≠0. Moreover, we show how to characterize the root locus around iω. The entire local root locus picture can be completely determined from the square root splitting of the double root. We separate the general picture into two cases depending on the sign of a single scalar constant; the imaginary part of the first coefficient in the square root expansion of the double eigenvalue.