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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 $h\left(\lambda ,\mu \right)$ that is analytic in $\lambda \in ℂ$ and continuous in $\left(\lambda ,\mu \right)\in ℂ×B$, where $B\subset {ℝ}^{n}$ 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

$\lambda =-b\left[{e}^{-\lambda {\tau }_{1}}+{e}^{-\lambda {\tau }_{2}}\right]-a$

characterizing the stability behavior of the linear differential delay equation

$\frac{dx}{dz}=-ax\left(t\right)-b\left[x\left(t-{\tau }_{1}\right)+x\left(t-{\tau }_{2}\right)\right]·$

By this way, the authors study stability and bifurcation of a scalar equation with two delays modeling compound optical resonators.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K60 Qualitative investigation and simulation of models 30D20 General theory of entire functions 30C15 Zeros of polynomials, etc. (one complex variable)
##### Keywords:
zeros of quasi-polynomials; Liapunov’s first method