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

Junya  Nishiguchi

doi:10.3934/dcds.2016048

Article Contents

2016, Volume 36, Issue 10: 5657-5679. Doi: 10.3934/dcds.2016048

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

Junya Nishiguchi^1,

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

Received: August 31, 2015

Revised: January 31, 2016

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 34K20, 34K25; Secondary: 93C23.

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