An algebraic approach to delay differential systems

2017

We introduce a framework for the description of a large class of delay-differential algebraic systems, in which we study three core problems: first we characterize abstractly the well-posedness of the initial-value problem, then we design a practical test for well-posedness based on a graph-theoretic representation of the system; finally, we provide a general stability criterion. We apply each of these results to a structure that commonly arises in the control of delay systems.

Mathematical Problems in Engineering, 2018

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019

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...

Nonlinear Physical Science, 2011

Periodic motions in DDE (Differential-Delay Equations) are typically created in Hopf bifurcations. In this chapter we examine this process from several points of view. Firstly we use Lindstedt's perturbation method to derive the Hopf Bifurcation Formula, which determines the stability of the periodic motion. Then we use the Two Variable Expansion Method (also known as Multiple Scales) to investigate the transient behavior involved in the approach to the periodic motion. Next we use Center Manifold Analysis to reduce the DDE from an an infinite dimensional evolution equation on a function space to a two dimensional ODE (Ordinary Differential Equation) on the center manifold, the latter being a surface tangent to the eigenspace associated with the Hopf bifurcation. Finally we provide an application to gene copying in which the delay is due to an observed time lag in the transcription process.

Automatica, 2004

2009

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2012

The analysis of nonlinear delay-differential systems (DDE's) subjected to external forcing is difficult due to the infinite dimensionality of the space in which they evolve. To simplify the analysis of such systems, the present work develops a non-homogeneous center manifold (CM) reduction scheme, which allows the derivation of a time-dependent order parameter equation in finite dimensions. This differential equation captures the major dynamical features of the delayed system. The forcing is assumed to be small compared to the amplitude of the autonomous system, in order to cause only small variations of the fixed points and of the autonomous CM. The time-dependent CM is shown to satisfy a non-homogeneous partial differential equation. We first briefly review CM theory for DDE's. Then we show, for the general scalar case, how an ansatz that separates the CM into one for the autonomous problem plus an additional time-dependent order-two correction leads to satisfying results. The paper then details the application to a transcritical bifurcation subjected to single or multiple periodic forcings. The validity limits of the reduction scheme are also highlighted. Finally, we characterize the specific case of additive stochastic driving of the transcritical bifurcation, where additive white noise shifts the mode of the probability density function of the state variable to larger amplitudes.

The analysis and numerical solution of initial value problems for linear delay differential-algebraic equations (DDAEs) is discussed. Characteristic properties of DDAEs are analyzed and the differences between causal and noncausal DDAEs are studied. The method of steps is analyzed and it is shown that it has to be modified for general DDAEs. The classification of ordinary delay differential equations (DDEs) is generalized to DDAEs, and a numerical solution procedure for general retarded and neutral DDAEs is constructed. The properties of the algorithm are studied and the theoretical results are illustrated with a numerical example.