Eur. Phys. J. Special Topics 191, 91-103 (2010)
https://doi.org/10.1140/epjst/e2010-01343-7

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Complex dynamics in delay-differential equations with large delay

¹ Weierstrass Institute for Applied Analysis and Stochastics, 10117 Berlin, Germany
² Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
³ Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany

Received: 22 November 2010
Revised: 23 December 2010
Published online: 18 February 2011

Abstract

We investigate the dynamical properties of delay differential equations with large delay. Starting from a mathematical discussion of the singular limit τ → ∞, we present a novel theoretical approach to the stability properties of stationary solutions in such systems. We introduce the notion of strong and weak instabilities and describe a method that allows us to calculate asymptotic approximations of the corresponding parts of the spectrum. The theoretical results are illustrated by several examples, including the control of unstable steady states of focus type by time delayed feedback control and the stability of external cavity modes in the Lang-Kobayashi system for semiconductor lasers with optical feedback.