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Zbl 1068.34072
Ruan, Shigui; Wei, Junjie
On the zeros of transcendental functions with applications to stability of delay differential equations with two delays.
(English)
[J] Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 10, No. 6, 863-874 (2003). ISSN 1201-3390; ISSN 1918-2538/e

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.
[Klaus R. Schneider (Berlin)]
MSC 2000:
*34K20 Stability theory of functional-differential equations
34K60 Applications of functional-differential equations
30D20 General theory of entire functions
30C15 Zeros of polynomials, etc. (one complex variable)

Keywords: zeros of quasi-polynomials; Liapunov's first method

Citations: JFM 42.0351.04

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