Migration of imaginary roots of multiplicity three and four under small deviation of two delays in time-delay systems

Abstract

This paper studies the migration pattern of characteristic imaginary roots of multiplicity three and four in time-delay systems with two delays when the delay parameters undergo small deviations. Stability analysis for such problems is often based on Puiseux series, as multiple roots are not differentiable with respect to delay parameters. However, in this paper the approach is more traditional without using Puiseux series. In the case of triple roots, we show that the stability crossing curves are smooth; when a perturbation occurs in the delay parameter space, two roots move to one half-plane and one root to the other half-plane. The case of quadruple root is more complicated as the stability crossing curve has a cusp. Thus, in the neighbourhood of the critical point, the delay parameter space is divided in an S-sector and a G-sector. When the parameters move into the G-sector, two roots move to the right half-plane, and the other two roots move to the left half-plane. When the parameters move into the S-sector, then three of the roots move to one half-plane, and the remaining root moves to the other half-plane, depending on the conditions.