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On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. (English) Zbl 1068.34072
The authors consider a function $h(\lambda,\mu)$ that is analytic in $\lambda\in\bbfC$ and continuous in $(\lambda,\mu)\in\bbfC\times B$, where $B\subset \Bbb R^n$ is open and connected. They prove a theorem on the zeros of $h$ located in the right (complex) half plane. This result is applied to the characteristic equation $$\lambda=-b[e^{-\lambda\tau_1}+e^{-\lambda\tau_2}]-a$$ characterizing the stability behavior of the linear differential delay equation $$\frac{dx}{dz}=-ax(t)-b[x(t-\tau_1)+x(t-\tau_2)].$$ By this way, the authors study stability and bifurcation of a scalar equation with two delays modeling compound optical resonators.

34K20Stability theory of functional-differential equations
34K60Qualitative investigation and simulation of models
30D20General theory of entire functions
30C15Zeros of polynomials, etc. (one complex variable)