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 τ0 then iω is also an eigenvalue for the delays, for any k ∈ ℤ. We investigate the sensitivity and other properties of the root iω for the case that iω is a double eigenvalue for some τk. It turns out that under natural conditions, the presence of a double imaginary root iω for some delay τ0 implies that iω is a simple root for the other delays τkk ≠ 0. Moreover, we show how to characterize the root locus around iω. The entire local root locus picture can be 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.