Counting Roots of the Characteristic Equation for Linear Delay-Differential Systems

A formula is given that counts the number of roots in the positive half plane of the characteristic equation for general real, constant coefficient, linear delay-differential systems. The formula is used to establish necessary and sufficient conditions for asymptotic stability of the zero solution of linear delay-differential systems. The formula is potentially useful in verifying stability hypotheses that arise in bifurcation analysis of autonomous delay-differential systems. Application of the formula to Hopf bifurcation theory for delay-differential systems is discussed, and an example application to an equation with two delays is given.