*UCL Centre for Nonlinear Dynamics and its
Applications*

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One of the main motivations for the establishment of the Centre was the desire to apply the emerging ideas of nonlinear dynamics to practical problems in engineering. Today this area of research continues to form the core of the Centre's activities. Current projects in this area are described below, but it should be noted that new research topics are continually being developed.

The escape from a potential well is a universal problem in science and engineering. It arises in the buckling and post-buckling of elastic structures; in the study of the capsize of ships; in solid state physics, including the response of Josephson junctions; in electrical power engineering, where the loss of synchronization of generators can lead to the black-out of a power grid (see below); and in physical chemistry, for instance when modelling activation energies of molecular reactions. Under periodic excitation, escape under steady state conditions is governed by a complex web of regular and chaotic bifurcations; while escape under transient conditions is governed by the basins of attraction which can undergo elaborate metamorphoses. The onset of a fractal basin boundary can be predicted by Melnikov estimations of the first homoclinic tangency (see below), whilst the subsequent basin erosion process is governed by a later heteroclinic connection. Current effort is directed towards understanding these global bifurcation phenomena and developing methods for their prediction. The engineering relevance of the dramatic basin erosion that is observed under increasing excitation has been highlighted by the definition of various engineering integrity measures. This work has important implications for the wave-tank testing of ship hulls, where our new concept of a transient capsize diagram seems to offer a more rational approach to ship safety in waves.

Contact: J.M.T. Thompson.

Since chaotic systems exhibit extreme sensitivity to initial conditions it has for many years been generally thought that chaotic systems are neither predictable nor controllable. Thus chaos is often regarded as an annoying property to be avoided in the design of engineering systems. More recently, however, studies have shown that chaotic dynamics are not only controllable but can also be exploited to achieve useful goals. Utilisation of the properties of chaotic systems can achieve special advantages: a) small perturbations can lead to large effects; b) flexible switching between many different periodic orbits without changing the global configuration of the system. These benefits cannot be achieved in non-chaotic systems in which a large effect typically requires a large control. The great potential of the control of chaos has been demonstrated in applications in electronics, lasers, chemical reactions, communications and biological systems.

The aim of this research topic is to investigate basic concepts of chaotic control, develop new theoretical algorithms, and consider their implementation in a variety of systems. To date, three new methods have been introduced for controlling chaotic systems. The first approach is a parametric control scheme based on a linear approximation using a one step optimal process. The method does not require analysis of eigenvalues and eigenvectors and can be applied without using explicit knowledge of the system dynamics - an aspect which is particularly useful for experiments. The method is very simple especially when applied to Hamiltonian systems. To enhance the effectiveness for stabilising highly unstable orbits in a very noisy environment, the method is extended to a variational algorithm for flows based on multiple control surfaces. The second method is based on the contraction mapping theorem in the form of state variable feedbacks. The control strategy does not rely on linear approximation, thus control can be achieved regardless of whether the system state is close to the desired target point or not. To achieve a stabilisation of orbits with high accuracy, a third method with a self-locating function has been developed which can automatically detect the location of a desired orbit without having any explicit knowledge of the system dynamics or the exact location of the orbit. The robustness of each control scheme is also discussed including the effects of noise and overall stability of the control algorithms.

Contact: S.R. Bishop.

Sigma-delta modulation is an analogue to digital conversion technique used in many telecommunication, radar, sonar and audio applications. It employs high over-sampling rates to achieve a good signal to noise ratio by exchanging temporal resolution for amplitude resolution. Due to the negative feedback structure employed, this results in circuits which are especially insensitive to device imperfections and mismatches. However, sigma-delta modulators can exhibit complex dynamic behaviour, which has a direct bearing on their performance. In collaboration with Coventry University and the University of Manchester Institute of Science and Technology we are therefore undertaking a study of the dynamics of such circuits. First order modulators can be readily analysed using well-known results for circle maps, but unfortunately real-world circuits are usually second order or higher. Very little is understood about this case, and very few mathematical techniques exist to analyse it. The project is therefore proceeding with a mixture of experimental work and numerical simulation in an attempt to obtain better insight into the qualitative properties of higher order modulators.

Contact: S.R. Bishop, J. Stark.

Dynamical systems which involve impacts frequently arise in engineering. The resonant phenomenon of rattling, caused by repeated impacts at one or two motion limiting stops, often needs to be suppressed. In mechanical systems such rattling can lead to noise and wear. Even more seriously, in some offshore situations such as those motivating this work, excessive loads at the constraints can lead to complete and expensive failure.

The impact process introduces a strong nonlinearity into the dynamics, and hence impacting oscillators can exhibit the typical types of complicated behaviour associated with nonlinear dynamical systems. One of the main results of our research is the identification of bifurcational behaviour associated with part of an orbit of the dynamical system just touching a stop under the change of some system parameter. After this type of bifurcation the system can suddenly begin undergoing much more severe impacts, or can undergo chaotic motion. Experimental work has been carried out which clearly demonstrates the qualitative behaviour caused when these types of bifurcation occur. It is important to be aware of the possibility of such sudden jumps to what could be "dangerous" states in the design of structures which may undergo repeated impacts, and thus to avoid them. The dynamical behaviour of more general constrained dynamical systems is now also under investigation. As with impacting systems, the constrained nature of these systems ensures that their responses can be highly nonlinear. The effects of constraint-induced nonlinearities on the overall dynamical responses of this class of system is being studied both theoretically and experimentally.

Contact: S.R. Bishop.

Flashover is a phenomenon describing the transition by which a small localized fire with a relatively low burning rate suddenly undergoes a rapid increase in both size and intensity. In the evolution of a fire in a confined space the occurrence of flashover signals a highly dangerous transition in the nature and intensity of the fire and its potential effect on building and occupants. It is thus of paramount importance that an understanding of this critical change in the character of the fire be attained.

A recent collaborative research project with the Unit of Fire Safety Engineering at the University of Edinburgh applied the techniques of nonlinear dynamics to this problem. It examined a variety of current fire growth models and identified their most significant nonlinear features. This has led to the development of archetypal models incorporating such features. These models were subjected to in-depth analysis to determine mechanisms leading to instability. New computational techniques were devised to display fire growth within a building, permitting increasingly sophisticated interactive and automated computer experiments to be carried out.

Contact: S.R. Bishop.

Circulation within the ocean dramatically affects, and often determines the climate of many regions of the globe. For example, the Gulf Stream, which manifests itself as a warm surface current between the Gulf of Mexico and the North Atlantic, has a significant influence upon the climate of Northern Europe. Surface currents of this type are the result of circulation deep in the ocean and originate from differences in temperature between equatorial and polar regions, effectively creating a single large convective cell. However, the situation is complicated by the effect of evaporation at the equator and precipitation in polar regions, which lead to a gradient in ocean salinity (salt content). Ignoring the thermal circulation, this alone would produce a pressure gradient and hence a circulation, which would in fact be in the opposite direction to that produced by thermal forcing. When combined, the opposing thermal and saline driving forces are referred to as thermohaline circulation, which may produce more complex behaviour than that produced by either effect alone. Thermohaline circulation has been investigated using simple box models, in which a cross-section of the ocean is divided into a finite number of boxes which possess the average properties of those regions of the ocean. This system of boxes may be described by a set of simple nonlinear differential equations (based upon differences in temperature and salinity), which govern the transport of fluid between the regions. Such models demonstrate certain instabilities depending upon the choice of parameters used to describe the diffusion of fluid between the boxes. A collaborative project with the UK Meteorological Office has investigated the dynamics of such systems, and the instabilities which occur, by applying techniques developed in the study of other nonlinear dynamical systems.

Contact: S.R. Bishop.

A phase-locked loop is an electronic circuit designed to demodulate frequency-modulated signals. Although widely used in a variety of communications systems ranging from satellites to portable telephones there are many aspects of its general behaviour which are not well understood. In particular current linear models of phase-locked loops do not explain much of the behaviour seen in actual working systems. This research project is therefore studying the full non-linear equations describing a phase-locked loop's dynamics, employing both analytical and computer simulation techniques .

The equation describing the phase-locked loop can be simplified to an equation of a pendulum with constant torque and forcing applied to it. This equation is also used to model other systems like the Josephson junction. The possible steady states of the equation and the robustness of these steady states have been analysed. These are related to the parameter regime in which the phase-locked loop is operating. As the values of the operating parameters change, the observed steady states change in magnitude or bifurcate into other types of behaviour. It is hoped that by understanding the possible behaviour and the parameter regimes of the interesting behaviour, the design of phase-locked loop circuits will have clearer guidelines.

Contact: J.M.T. Thompson.

Perhaps the single most important lesson to be drawn from the study of non-linear dynamical systems over the last few decades is that even simple deterministic dynamical systems can give rise to complex behaviour which is statistically indistinguishable from that produced by a completely random process. One obvious consequence of this is that it may be possible to describe apparently complex signals using simple non-linear models. This has led to the development of a variety of novel techniques for the manipulation of such "chaotic" time series. In appropriate circumstances, such algorithms are capable of achieving levels of performance which are far superior to those obtained using classical linear signal processing techniques. The Centre has wide interests in this area, with projects focusing on both applications, and the development of new theory and algorithms.

Virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems are based on the Takens Embedding Theorem. This typically allows us to reconstruct an unknown dynamical system which gave rise to a given observed scalar time series simply by constructing a new state space out of successive values of the time series. This provides the theoretical foundation for many popular techniques, including those for the measurement of fractal dimensions and Liapunov exponents, for the prediction of future behaviour, for noise reduction and signal separation, and most recently for control and targeting.

Recent work in the Centre has been concerned with a number of generalizations to Takens Theorem. In particular, existing versions assume that the underlying system is autonomous. Unfortunately this is not the case for many real systems; in the laboratory we often force an experimental system in order for it to exhibit interesting behaviour, whilst in the case of naturally occurring systems it is very rare for us to be able to isolate the system to ensure that there are no external influences. We have recently proved two versions of Takens Theorem which are relevant to forced systems: one is applicable to the case where the forcing is unknown, and the other to the situation where we are able to determine the state of the forcing system independently (usually because we are responsible for the forcing ourselves). In collaboration with the University of Manchester Institute of Science and Technology, we are extending these results to encompass certain types of random forcing, and investigating new algorithms for the analysis of nonlinear stochastic time series based on these ideas. Finally a related project is attempting to develop a framework for the analysis and processing of irregularly-sampled time series.

Contact: J.P.M. Heald, J. Stark.

One of the main consequences of Takens
Theorem is that time series generated by a nonlinear dynamical
system are predictable, and this has given rise to a wide range of
prediction algorithms. However, hitherto these have all suffered from
the disadvantage that they use batch processing. In other words they
build a model of the dynamics of the time series from a
pre-determined block of observations *x*_{1}, ... ,
*x _{n}*, but then have no means of updating the model
from further measurements

A different problem with most current prediction schemes is that they make no attempt to estimate the error inherent in any given prediction. In practical applications this is a serious drawback, since an estimate of the quality of a prediction is often as important as the actual predicted value. Based on ideas of Bayesian estimation we have developed an algorithm for estimating state-dependent noise levels in a time series and hence calculating error bars for predicted values. This has been found to work well for a single state space dimension, and efforts are currently being made to extend it to higher dimensions.

Contact: J.P.M. Heald, J. Stark.

A recursive filter, broadly speaking, is one whose output is fed back into the filter and allowed to affect its future operation. Within the context of conventional digital filtering such filters range from simple infinite impulse response (IIR) filters, used for a variety of signal processing tasks, including echo cancellation, signal equalisation and speech encoding, to recursive least squares (RLS) filters such as the well known Kalman filter.

Traditionally, the mathematical analysis of filters assumes that the input signal is given by a stochastic process. However there is increasing interest in using recursive filters in so called "chaotic" time series analysis. In such applications the input signal to the filter is then derived from a chaotic deterministic dynamical system. Obviously, a stochastic analysis is then inappropriate and new methods have to be developed to understand the filter's behaviour. Our approach has been largely through the theory of invariant manifolds for skew products, and this has led to both a new insight into existing results about IIR filters, and the development of a new framework for the analysis of the stability of RLS filters; the latter in particular has been a long standing problem within the signal processing community.

Contact: J. Stark.

In most applications, we are unlikely to be given a pure chaotic time series. Instead, typically we will be asked to manipulate a mixture of a chaotic time series and some other signal. The latter may represent noise, in which case we want to remove it, or it may be a signal that we wish to detect, in which case we want to extract it from our source time series and discard the chaotic part. An example of the latter might be a faint speech signal masked by deterministic "noise" coming from some kind of vibrating machinery, such as an air conditioner. In both cases, the mathematical problem that we face amounts to separating the original time series into its two components. A variety of schemes have been developed in the last few years to perform this task.

Our work has concentrated on improving the efficiency and stability of such algorithms. Thus for instance many noise reduction schemes suffer from instability in the vicinity of homoclinic tangencies. Such tangencies are common in real chaotic systems, severely limiting the usefulness of these techniques. We have therefore investigated more sophisticated approaches to noise reduction, culminating in a robust scheme based upon the Levenberg-Marquardt algorithm, which appears to be insensitive to the presence of homoclinic tangencies. Further improvements can be achieved if extra information, either about the noise or about the deterministic signal is available; for example the noise may be slowly varying. Generalizations of this are under investigation, including applications to under-sampled time series.

Similarly, all noise reduction algorithms involve the choice of a number of parameters; often these have to be set essentially arbitrarily. We are therefore attempting develop a Bayesian framework for noise reduction which will hopefully allow a more rational selection of these parameters, as well as an objective evaluation of the performance of the resulting algorithms.

Contact: J.P.M. Heald, J. Stark.

Virtually all of the attention in the area of chaotic time series has been on the case of a single scalar signal. Many important applications, however, involve the simultaneous measurement of observables at different spatial locations, leading to multivariate spatio-temporal time series. This raises a variety of interesting questions, of both a theoretical and a practical nature. We are just beginning to address some of these, in collaboration with the University of Manchester Institute of Science and Technology.

One particular application of these ideas is the analysis of road traffic congestion, in collaboration with the London Centre for Transport Studies at UCL. Many motorways are now being fitted with sensors which can collect large volumes of detailed information about traffic flow. Typically such sensors are placed at regular intervals along a motorway (say every 500m), and collect traffic information at regular intervals in time (say every minute), giving rise to a spatio-temporal time series. The analysis of such data is obviously important from the point of view of both planning future roads and managing traffic flow on existing roads, but is currently lagging far behind our ability to collect the data. We are therefore intending to investigate the relevance of techniques from nonlinear dynamics to the characterisation and prediction of such time series, with a particular emphasis on the detection of incipient congestion.

Contact: J. Stark.

Biology and medicine is an area rich with potential applications of nonlinear dynamics, many of which have hitherto had relatively little attention. The Centre has begun to develop this area relatively recently, but a number of projects, described below, have already become established, and several more are under consideration. To enhance collaboration in the field of mathematical biology at UCL, the Centre has recently started a regular seminar series on this topic.

Animals of different species are both very similar and very different. The cell types found in mice, frogs, flies and worms are very similar, but the way in which they are organized to generate the whole animal is very different. The cells are spatially arranged into a pattern, whose form represents an animal. Since all animals derive from a single cell, the fertilized egg, the pattern characteristic of any individual must be generated during embryonic development - this process is called pattern formation. Understanding the mechanisms underlying the way in which pattern is generated during embryonic development is an important biological problem that remains to be resolved. One of the main approaches to this problem has been through the building of mathematical models of the developing embryo, with the aim of duplicating the qualitative behaviour observed in real organisms. Unfortunately, whilst current models can often reproduce some of the properties of real embryos, they in general do not incorporate realistic biological information, nor do they lead to experimentally verifiable predictions.

In an attempt to overcome these objections we have embarked on a collaborative research project with the Department of Anatomy and Developmental Biology at UCL to devise models of the early embryo which both make use of its known biological properties and give rise to testable predictions. These focus on cell-cell signalling via the exchange of small molecules through a specialized intercellular structure, the gap junction. The embryo is thus modelled as a simple circuit or network consisting of the cells and the junctions between them. In mathematical terms, this leads to systems of coupled ordinary differential equations, in contrast to the partial differential equations usually used to model pattern formation. We are currently investigating the dynamics of such networks, with the aim of understanding which kinds of patterns they can give rise to.

Contact: S. Baigent, J. Stark.

In collaboration with Kate Hardy and Steve Franks of the Imperial College School of Medicinewe are trying to understand the mechanisms that control mammalian ovulation and its failure. The ovary contains a large number of immature dormant follicles, each one of which contains an egg. Such follicles continuously leave the dormant state and begin to develop. The vast number of these atrophy and die before they reach maturity and usually only a single follicle ovulates in each reproductive cycle. It is important both medically and scientifically to understand how this follicle is selected, and why such selection fails in PCOS, which is the single largest cause of anovulatory infertility. We were able to take an existing model and extend it so that it exhibited behaviour characteristic of PCOS, and hence drew some tentative conclusions about the causes of PCOS which are consistent with current biological thinking. Unfortunately, the important feedback loop in the model is an abstract one, and hence cannot be related to biological parameters, and we are therefore currently working towards more biologically realistic versions.

Contact: J. Stark.

Between puberty and the menopause, a woman will normally ovulate a single egg during each menstrual cycle. Such eggs are stored in specialized structures called follicles, in her two ovaries. All the follicles and eggs that a woman ever possesses are laid down in a dormant state in her ovaries approximately half way through her foetal development. Throughout her life there is a steady stream of follicles that initiate growth. It takes approximately 6 months for them to reach a stage where they can ovulate, and during this period the vast majority (at least 99.9%) of follicles die. When the stock of follicles is exhausted, a woman enters the menopause. The mechanisms controlling initiation are largely unknown, and difficult to study experimentally. A understanding of these is both of fundamental scientific interest, and of great significance to the many women in which the process fails to function properly, leading to infertility and other adverse consequences (eg increased risk of diabetes, obesity and cardiovascular disease).

Some data is available on how the number of follicles in the ovary changes with time. This suggests a process analogous to radioactive decay though clearly the detailed mechanisms must be very different. In collaboration with Kate Hardy and Steve Franks of the Imperial College School of Medicinewe propose to construct and analyse models of follicle initiation. By comparing these models to observed experimental data we shall propose plausible mechanisms for initiation. Such models in turn will suggest new experiments, and by iterating this process we intend to gain a better understanding of follicle development.

Contact: J. Stark.

Cytokines are small soluble proteins that convey information between cells. To date more than one hundred have been identified and cloned and the literature is full of descriptions of their biological activities. Interactions between cytokines and their target cells are extremely complex, and attempts to understand how they regulate cellular growth and differentiation have depended mostly on the reductionist approach of identifying the many activities each has on a cell or cellular system of interest, and then trying to fit them together in some sort of coherent biological model. Although enormous progress has been made in cataloguing the many cytokine activities, the sheer complexity of the system has so far defeated any attempt to understand how they work together in a co-ordinated fashion to control cellular function.

In collaboration with the Institute of Child Health we are beginning to develop and analyse models of the cytokine network. It is initially unrealistic to attempt to model the whole network, and hence we are initially focusing on relatively simple models of only a part of the system, namely the cytokine interactions that regulate human TH1 and TH2 T cell subset growth and differentiation and the information flow required for IgE antibody production. Most of the interacting cytokines involved in this sub-system have been identified and characterized, and in vitro culture techniques in which predictions made by non-linear dynamical modelling can be tested are well established. As experience with the model grows, it will gradually be refined and additional interactions will be incorporated. The tools of modern non-linear dynamics will be used to investigate a number of fundamental issues: the number and properties of the equilibria of the system, the possibility of oscillations or more complex "chaotic" dynamics, the robustness and stability of the system to various classes of perturbations, and the effect of structural changes such as the removal of cytokines and/or receptors from the system.

Contact: J. Stark.

Together with immunologist Andrew George at Imperial College School of Medicine, we aim to develop a fuller understanding of how engagement of the T cell receptor activates the T cell. This event involves a number of stochastic processes, such as ligand dissociation, and so it is difficult to understand how the cell can respond in a specific and sensitive manner to foreign antigen. Recently we have used an interdisciplinary approach of combining mathematical and computer modelling with basic biochemical data to address this problem. We have made a stochastic cellular automata model and have shown, for the first time in the context of eukaryotic cells, that cross talk between receptors can be used to enhance the specificity of ligand recognition with little loss of sensitivity. We aim to go on and use this approach to more fully explore the following aspects of T cell recognition

- Is the action of altered ligands (antagonists and partial agonists) explained by cross talk between receptors?
- Does the presence of MHC bearing the incorrect ligand (>99.9% of MHC) alter the sensitivity and specificity of the signalling event?
- What role does thymic selection have on the nature of signalling events?
- How can alloreactivity be accommodated in relationship to serial triggering and receptor cooperativity?

Contact: J. Stark, Cliburn Chan

This has developed out of the work on modelling gap junctions described above. See abstract for a description of completed research. We have now turned our attention towards stochastic models of parasite dynamics developed by Prof. V. Isham in the Department of Statistical Science. Many of the mathematical tools used for this work are closely related to those for the study of invariant graphs for skew product systems below.

Contact: S. Baigent, J. Stark.

Together with Kate Hardy and Prof. Lord Winston of the Imperial College School of Medicine we have recently begun to develop models of programmed cell death in pre-implantation human embryos. Such embryos exhibit surprisingly high levels of cell death, and high rates of developmental arrest during the first week in vitro. The relation between the two is unclear and difficult to determine by conventional experimental approaches, partly due to limited numbers of embryos. We have there-fore applied a mixture of experiment and mathematical modelling to show that observed levels of cell death can only be reconciled with the high levels of embryo arrest seen in the human if the developmental competence of embryos is already established at the zygote stage, and environmental factors merely modulate this. We also predicted, and subsequently verified experimentally, that cell death did not occur during the first few cell divisions.

Contact: J. Stark.

Many of the above applications require the development of new theoretical concepts. The Centre therefore maintains a vigorous research programme in several barnches of more abstract dynamical systems.

Many problems in engineering and the applied sciences are modelled by low dimensional forced oscillators. Examples include vibrating mechanical and structural (e.g. beams, bridges) engineering systems, marine systems (e.g. ships, oil platforms), electronic circuits (see below) and devices (e.g. Josephson junctions), as well as biological oscillators. Whilst the linear behaviour of such systems has long been well understood, the potential dynamics of nonlinear oscillators is far more complex and many open problems remain. The Centre has a long history of research in this area. Current effort has concentrated on the particular topics of Melnikov's method.

Melnikov's method is a perturbation technique widely used to estimate the parameter location at which a homoclinic tangency occurs. This is an indicator of complex motion and, loosely speaking, implies the existence of "chaos" in the system. However, it is not always appreciated that this particular behaviour is often a comparatively insignificant feature of the dynamics. Although it exists as a solution of the system equations, it can be extremely difficult to excite a physical system into the associated "chaotic" motions. Other regimes of complex motion occur which are much more readily encountered but for which there exist no comparable mathematical techniques to estimate their locations. This has motivated recent work in the Centre on the application of topological methods such as lobe dynamics and symbolic dynamics to the study of homoclinic tangles. This in turn has demonstrated the importance of higher order tangencies (referred to as Birkhoff signature changes) which are associated with these more "accessible" occurrences of complex response and chaotic instability. Using a simple energy interpretation of Melnikov's method, it has been shown how this could be extended to detect these higher order, but more important, phenomena.

Contact: J.M.T. Thompson.

The parametrically excited pendulum can exhibit a variety of dynamical behaviour including stable equilibria, periodic oscillations, continuous rotations, and motions which include rotations and oscillations: called tumbling solutions. Chaotic behaviour which is either oscillatory, rotating or tumbling, can also be viewed. Research over recent years has used numerical simulations and theoretical treatments to identify possible motions and provide an understanding of the bifurcational structure. A topological analysis using knot and braid theory has revealed much about the underlying dynamics.

Contact: S.R. Bishop.

Using a wide variety of computational and geometrical techniques we are exploring the bifurcational behaviour of a number of carefully chosen archetypal models of dynamical systems drawn from engineering and the applied sciences. Of particular interest at the moment are the indeterminate jumps to resonance that have been identified as robust and very typical events in a variety of softening oscillators under both direct and parametric excitation. They arise, for example, when the saddle of a saddle-node fold bifurcation is located on a fractal basin boundary, and give rise to a totally unpredictable jump to two or more disparate solutions, one of which might signal the failure of an engineering system.

Contact:J.M.T. Thompson.

Many dynamical systems of both practical and theoretical importance consist of one system driven by another. The simplest, best known, and most widely studied case of this is that of periodic forcing, but a variety of other examples, involving more complex forms of forcing, can be found throughout engineering and the applied sciences. The standard approach to studying the dynamics of driven systems is to extend the state space of the driven system to include the driving system; systems of this kind are often called skew products. Where such systems are designed to perform some useful task, the forced system will usually be stable in some sense or another, at least in the absence of any forcing; this is for instance the situation in certain classes of filters, and in the study of synchronization. When this contraction is uniform, it can easily be shown that there exists a globally attracting invariant set which is the graph of a function from the driving state space to the driven state space; this is analogous to the well known concept of an inertial manifold. If the driving state space is a manifold and the contraction is sufficiently strong this invariant set is a normally hyperbolic manifold, and hence smooth. Unfortunately, in many applications uniform estimates of contraction rates are not available. The aim of this project is therefore to generalize such results to non-uniform contraction rates. The invariant graph is then only defined over a set of full measure of input conditions and hence we need to use an appropriate notion of its smoothness. This turns out to be that given by the Whitney Extension Theorem. Using standard ideas from Pesin theory, we have been able to show that the invariant graph is indeed smooth in this sense, with the smoothness depending on average contraction rates. We are currently investigating the structural stability and other properties of such invariant graphs.

Contact: J. Stark.

A different class of forced systems arises if we restrict the type of forcing we wish to consider, as opposed to restricting the class of forced system, as above. After periodic, the next simplest class of forcing to consider is that of two forcing frequencies whose periods are not rationally related. Such systems are called quasiperiodic, and appear to exhibit interesting new phenomena not found in periodically forced systems. In collaboration with Queen Mary and Westfield College and the University of Potsdam, Germany, we are trying to understand several aspects of such systems, partly using the tools of invariant manifolds.

Contact: J. Stark.

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*Original by J.Stark 1993; Converted by rtftohtml 16/3/95 jpmh;
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*UCL Centre for Nonlinear Dynamics and its Applications,
University College London, Gower Street, London, WC1E 6BT,
UK.*

`cnda@ucl.ac.uk`