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Section: New Results

Asymptotic behavior of critical imaginary roots for retarded differential equations

Participants : Islam Boussaada, Jie Chen [City University of Hong Kong] , Liana Felix [Universidad Autonoma de San Luis Potosi] , Keqin Gu [Southern Illinois University] , Fernando Mendez-Barrios [Universidad Autonoma de San Luis Potosi] , Dina Irofti, Silviu-Iulian Niculescu, Alejandro Martinez.

The behavior of characteristic roots of time-delay systems, when the delay is subject to small variations is investigated in [26]. We performed an analysis by means of the Weierstrass polynomial which are employed to study the stability behavior of the characteristic roots with respect to small variations on parameters. Analytic description and splitting properties of the Puiseux series expansions of critical roots are characterized by allowing a full description covering all the cases that can be encountered.

In the paper [21] the migration of double imaginary roots of the systems characteristic equation when two parameters are subjected to small deviations is geometrically investigated. Under the least degeneracy assumptions, the local stability crossing curve is shown to have a cusp at the point that corresponds to the double root, which divides the neighborhood of this point into two sectors (called S-sector and a G-sector). We have shown that 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. However, 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 a quantity that depends on the characteristic function and its derivatives up to the third order.