Bernd Krauskopf

Sensitivity of the Generic Transport Model upset dynamics to time delay

Bifurcation analysis has previously been applied to the NASA Generic Transport Model (GTM) to provide insight into open-loop upset dynamics and also the impact on and sensitivity of such behaviour to closing the loop with a flight controller. However, these studies have not considered time delay in the system: this arises in all feedback controllers and has specific relevance when remotely piloting a vehicle such as the NASA AirSTAR GTM with ground-based controllers. Developments in the AirSTAR programme, in which a sub-scale generic airliner model will be tested for loss-of-control conditions over long ranges, raise the prospect of increased adverse effects of time delay relative to previous testing. This paper utilises bifurcation analysis, supplemented with time histories, on the GTM numerical model with a LQR-PI controller to evaluate the sensitivity of the closed-loop system stability to time delay. In this paper, the impact of time delays in both a fixed-gain and a gain-scheduled version of the controller is presented in terms of stability of nominal and off-nominal solutions.

Nonlinear Dynamics of Interacting Populations

A brief outline of the ideas and methods of mathematical modelling of populations growth dynamics of isolated population predator-prey interaction competition and symbiosis local systems of three populations dissipative structures in predator-prey systems.

Mixed-Mode Oscillations with Multiple Time Scales

Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

Numerical Continuation Methods for Dynamical Systems

Path following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedels 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects and showcase examples of how numerical bifurcation analysis can be used in concrete applications. Topics that are treated include: interactive continuation tools, higher-dimensional continuation, the computation of invariant manifolds, and continuation techniques for slow-fast systems, for symmetric Hamiltonian systems, for spatially extended systems and for systems with delay. Three chapters review physical applications: the dynamics of a SQUID, global bifurcations in laser systems, and dynamics and bifurcations in electronic circuits.

A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

A unifying view of bifurcations in a semiconductor laser subject to optical injection

We are concerned with the dynamics and bifurcations of a single-mode semiconductor laser with optical injection, modeled by three-dimensional rate equations. Key bifurcations, namely saddle-node, Hopf, period-doubling, saddle-node of limit cycle and toms bifurcations, are followed over a wide range of injection strengths and detunings for different fixed values of the linewidth enhancement factor o~. In this way we present, to our best knowledge, the most far-reaching overview yet of the dynamics of injected semiconductor lasers. Our results compare very well with experimental studies and tie together information in the literature on different aspects of the behavior of optically injected lasers.

Resonant homoclinic flip bifurcations

This paper studies three-parameter unfoldings of resonant orbit flip and inclination flip homoclinic orbits. First, all known results on codimension-two unfoldings of homoclinic flip bifurcations are presented. Then we show that the orbit flip and inclination flip both feature the creation and destruction of a cusp horseshoe. Furthermore, we show near which resonant flip bifurcations a homoclinic-doubling cascade occurs. This allows us to glue the respective codimension-two unfoldings of homoclinic flip bifurcations together on a sphere around the central singularity. The so obtained three-parameter unfoldings are still conjectural in part but constitute the simplest, consistent glueings.

Two-dimensional global manifolds of vector fields

We describe an efficient algorithm for computing two-dimensional stable and unstable manifolds of three-dimensional vector fields. Larger and larger pieces of a manifold are grown until a sufficiently long piece is obtained. This allows one to study manifolds geometrically and obtain important features of dynamical behavior. For illustration, we compute the stable manifold of the origin spiralling into the Lorenz attractor, and an unstable manifold in zeta(3)-model converging to an attracting limit cycle. (c) 1999 American Institute of Physics.

Excitability and self-pulsations near homoclinic bifurcations in semiconductor laser systems

Many laser systems show self-pulsations with a large amplitude that are born suddenly in a homoclinic bifurcation. Just before the onset of these self-pulsations the laser is excitable where the excitability threshold is formed by the stable manifold of a saddle point. We show that there exists a special configuration, a codimension-two bifurcation called a non-central saddle-node homoclinic orbit, that acts as an organising centre of excitability in lasers. It is the key to understanding excitability in laser systems as diverse as lasers with saturable absorbers, lasers with optical injection and lasers with optical feedback.

Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity

We investigate a delay differential equation that models a pendulum stabilized in the upright position by a delayed linear horizontal control force. Linear stability analysis reveals that the region of stability of the origin (the upright position of the pendulum) is bounded for positive delay. We find that a codimension-three triple-zero eigenvalue bifurcation acts as an organizing centre of the dynamics. It is studied by computing and then analysing a reduced three-dimensional vector field on the centre manifold. The validity of this analysis is checked in the full delay model with the continuation software DDE-BIFTOOL. Among other things, we find stable small-amplitude solutions outside the region of linear stability of the pendulum, which can be interpreted as a relaxed form of successful control.