Migration of double imaginary characteristic roots under small deviation of two delay parameters

Abstract

This paper studies the migration of double imaginary roots of the characteristic equation for systems with two delays when the delay parameters are subjected to small deviations. As the double roots are not differentiable with respect to the delay parameters, Puiseux series is often used in such a situation in the literature. In this article, we study the “least degenerate” case, and a more traditional analysis was used without Puiseux series. It was found that the local stability crossing curve has a cusp at the point in the parameter space that causes the double root, and it divides the neighborhood of this point into a G-sector and an S-sector. When the parameters move into the G-sector, one of the roots moves to the right half plane, and the other moves to the left half plane. When the parameters move into the S-sector, both roots move either to the left half plane or the right half plane depending on the sign of some value explicitly expressed in terms of derivatives of the characteristic function up to the third order.