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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 \(\lambda\in\mathbb{C}\) and continuous in \((\lambda,\mu)\in\mathbb{C}\times B\), where \(B\subset \mathbb R^n\) 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.

34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
30D20 Entire functions of one complex variable, general theory
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)