# American Institute of Mathematical Sciences

October  2016, 36(10): 5657-5679. doi: 10.3934/dcds.2016048

## On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay

 1 Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

Received  September 2015 Revised  February 2016 Published  July 2016

A transcendental equation $\lambda + \alpha - \beta\mathrm{e}^{-\lambda\tau} = 0$ with complex coefficients is investigated. This equation can be obtained from the characteristic equation of a linear differential equation with a single constant delay. It is known that the set of roots of this equation can be expressed by the Lambert W function. We analyze the condition on parameters for which all the roots have negative real parts by using the graph-like'' expression of the W function. We apply the obtained results to the stabilization of an unstable equilibrium solution by the delayed feedback control and the stability condition of the synchronous state in oscillator networks.
Citation: Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048
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##### References:
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