Nonlinear dynamics of delay systems: an overview

Abstract

Time delays play an important role in many fields such as engineering, physics or biology. Delays occur due to finite velocities of signal propagation or processing delays leading to memory effects and, in general, infinite-dimensional systems. Time delay systems can be described by delay differential equations and often include non-negligible nonlinear effects. This overview article introduces the theme issue 'Nonlinear dynamics of delay systems', which contains new fundamental results in this interdisciplinary field as well as recent developments in applications. Fundamentally, new results were obtained especially for systems with time-varying delay and state-dependent delay and for delay system with noise, which do often appear in real systems in engineering and nature. The applications range from climate modelling over network dynamics and laser systems with feedback to human balancing and machine tool chatter.This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Delay-induced stabilization of coupled oscillators

“…In contrast, studies in the field of nonlinear science have reported that transmission delays can stabilize systems [1][2][3]. For example, transmission delays between neurons are essential for inducing stable synchronization patterns in neural networks [4].…”

Time delay effects in the control of synchronous electricity grids

“…In general, systems with delay, also called time delay systems, can be described by delay differential equations (DDEs). It is known that delays can have both stabilizing as well as de-stabilizing effects 15,16 . In DDEs the stability of a fixed point can switch from stable to unstable and back again multiple times under variation of the delay 17,18 .…”

On norm-based estimations for domains of attraction in nonlinear time-delay systems

“…However, the focus was mainly on stability criteria for linear systems [9,30,60,65,84], be it Lyapunov-Krasovskii functionals [28,35,74], Lyapunov-Razumikhin functions [35], comparison principles [17,33,64], inputoutput approaches [8], or eigenvalue calculations [7,26,41,43]. By contrast, in technical applications, we encounter nonlinear systems [22,23,68,72,85]. Indeed, for a nonlinear system the Principle of Linearized Stability [21] allows to deduce stability or instability of equilibria, but the question arises: what is the practical relevance of knowledge about asymptotic stability if there is no knowledge about the domain of attraction?…”