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# Recent Papers

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#### Modelling T Cell Activation

by Cliburn Chan

This thesis aims to provide a fuller understanding of how T cells can reliably recognise and respond appropriately to very small densities of foreign antigen on antigen presenting cells. The problem for the T cell is that it has to detect a small signal (foreign peptide-MHC molecules) in the presence of a large amount of noise (self peptide-MHC molecules). Since stochastic processes (e.g., ligand dissociation, recruitment of signalling components) play a large role in the recognition event, it is difficult to see how the T cell can achieve the experimentally documented levels of sensitivity and specificity. In the thesis, mathematical and computational models based on experimental cell signalling data show that feedback and cooperativity for both individual receptors and receptor populations are critical to achieve the observed sensitivity and specificity.

First, the standard model for TCR activation (McKeithan's kinetic proofreading model) is analysed and found to have several biological and theoretical problems, which limit its attractiveness for explaining the specificity and sensitivity of T cell activation. Based on this analysis, a new model that incorporates the essential elements of proofreading (i.e., delay followed by activation) and is more consistent with known TCR signalling biology is constructed. This new model predicts a role for the immune synapse and self ligands in amplifying and sustaining T cell signalling, as well as a possible role for multiple ITAMs to decrease the variance of the activation threshold.

The next model moves from the level of individual TCR to study interactions between a population of receptors. A Monte Carlo simulation of a lattice of TCR interacting with ligands is constructed, which integrates the most important models for T cell specificity (kinetic proofreading) and sensitivity (serial ligation), and incorporates recent evidence for cross-talk between neighbouring receptors. This simulation reveals that the specificity of T cell ligand discrimination can be significantly enhanced with cooperativity. Finally, the model suggests a resolution to the paradox of positive and negative selection on a similar set of ligands, and uses this to explain the surprising repertoire of transgenic mice that express the same peptide on all MHC II molecules.

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#### Delay Embeddings for Forced Systems: II. Stochastic Forcing

by J. Stark, D.S. Broomhead, M.E. Davies and J. Huke

Takens' Embedding Theorem forms the basis of virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems. It typically allows us to recon-struct 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 pro-vides the theoretical foundation for many popular techniques, including those for the meas-urement 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. Current versions of Takens' Theorem assume that the underlying system is autonomous (and noise free). Unfortunately this is not the case for many real systems. In a previous paper, one of us showed how to extend Takens' Theorem to deterministically forced systems. Here, we use similar techniques to prove a number of delay embedding theorems for arbitrarily and stochas-tically forced systems. As a special case, we obtain embedding results for Iterated Functions Systems, and we also briefly consider noisy observations.

Submitted to J. Nonlinear Science

Revised 15.7.02

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Strange Nonchaotic Attractors in Quasiperiodically Forced Systems

by R. Sturman

thesis submitted for the degree of Doctor of Philosophy to the University of London

Rotation Numbers for Quasi-Periodically Forced Monotone Circle Maps

by J. Stark, U Feudel, P. Glendinning and A. Pikovsky

RRotation numbers have played a central role in the study of (unforced) monotone circle maps. In such a case it is possible to obtain a priori bounds of the form r - 1/n ² 1/n(yn-y0) ² r + 1/n, where 1/n(yn-y0) is an estimate of the rotation number obtained from an orbit of length n with initial condition y0, and r is the true rotation number. This allows rotation numbers to be computed reliably and efficiently. Although Herman has proved that quasi-periodically forced circle maps also possess a well defined rotation number, independent of initial condition, the analogous bound does not appear to hold. In particular, two of the authors have recently given numerical evidence that there exist quasi-periodically forced circle maps for which yn-y0-rn is not bounded. This renders the estimation of rotation numbers for quasi-periodically forced circle maps much more problematical. In this paper, we derive a new characterization of the rotation number for quasi-periodically forced circle maps based upon integrating iterates of an arbitrary smooth curve. This satisfies analogous bounds to above and permits us to develop improved numerical techniques for computing the rotation number. Additionally, we consider the boundedness of yn-y0-rn. We show that if this quantity is bounded (both above and below) for one orbit, then it is bounded for all orbits. Conversely, if for any orbit yn-y0-rn is unbounded either above or below, then there is a residual set of orbits for which yn-y0-rn is unbounded both above and below. In proving these results we also present a min-max characterization of the rotation number. We evaluate the performance of an algorithm based on this, and on the whole find it to be inferior to the integral based method.

Dynamical Systems, 17, 2002, 1-28.

Revised 24.5.01

A Nonlinear Dynamics Perspective of Moment Closure for Stochastic Processes

by J. Stark, P. Iannelli and S. Baigent

Nonlinear Analysis: Series A Theory and Methods, 47, 2001, 753-764.

#### Convergent Dynamics of Two Cells Coupled by a Nonlinear Gap Junction

by S. Baigent, J. Stark and A. Warner

Abstract: A theoretical model for two Xenopus cells linked by a gap junction is analysed (see Baigent et al below for development of this model). The model takes the form of a four-dimensional singular perturbation problem describing the evolution of the cell membrane potentials and the gating states of the gap junction. Depending upon the strength of the gap junction and the electrogenic pumping, there is either one stable steady state or two stable and one unstable steady states. Convergence of the dynamics to a steady state is proved by projecting the system onto a 2 dimensional inertial manifold and applying Dulac's test together with the Poincare-Bendixson theorem.

Nonlinear Analysis: Series A Theory and Methods 47, 2001, 257-268.

#### A Quasi-diagonal Approach to the Estimation of Lyapunov Spectra for Spatio-Temporal Systems from Multivariate Time Series

by R. Carretero-González, S. Ørstavik and J. Stark

Abstract: We describe methods of estimating the entire Lyapunov spectrum of a spatially extended system from multivariate time-series observations. Provided that the coupling in the system is short range, the Jacobian has a banded structure and can be estimated using spatially localised reconstructions in low embedding dimensions. This circumvents the "curse of dimensionality" that prevents the accurate reconstruction of high-dimensional dynamics from observed time series. The technique is illustrated using coupled map lattices as prototype models for spatio-temporal chaos and is found to work even when the coupling is not strictly local but only exponentially decaying..

Phys. Rev. E., 62, 2000, 6429-6439.

Revised 28.7.00

#### Cytokine-Modulated Regulation of Helper T Cell Populations

by A. Yates, C. Bergmann, J.L. van Hemmen, J. Stark and R. Callard

Abstract: Helper T (Th) cells are a crucial component of the adaptive immune system and are of fundamental importance in orchestrating the appropriate response to pathogenic challenge. They fall into two broad categories defined by the cytokines each produces. Th1 cells make interferon-$\gamma$ and are required for effective immunity to intracellular bacteria, viruses and protozoa whereas Th2 make IL-4 and are required for optimal antibody production to T dependent antigens. A great deal of experimental data on the regulation of Th1 and Th2 differentiation have been obtained but many essential features of this complex system are still not understood. Here we present a mathematical model of Th1/Th2 differentiation and cross regulation. We model Fas-mediated activation-induced cell death (AICD) as this process has been identified as an important mechanism for limiting clonal expansion and resolving T cell responses. We conclude that Th2 susceptibility to AICD is important for stabilising the two polarized arms of the T helper response, and that cell-cell killing, not suicide, is the dominant mechanism for Fas-mediated death of Th1 effectors. We find that the combination of the anti-proliferative effect of the cytokine TGF-b and the inhibiting influence of IL-10 on T cell activation are crucial controls for Th2 populations. We see that the strengths of the activation signals for each T helper cell subset, which are dependent on the antigen dose, co-stimulatory signals and the cytokine environment, critically determine the dominant helper subset. Switches from Th1- to Th2-dominance may be important in chronic infection and we show that this phenomenon can arise from differential Fas/FasL expression on T helper subsets, and asymmetries in the nature of the cross-suppressive cytokine interactions. Our model suggests that in some senses a predominantly type 2 reaction may well be the default' pathway for an antigen-specific immune response, due to these asymmetries.

J. Theor. Biol., 206, 2000, 539-560.

#### Estimation of Intensive Quantities in Spatio-Temporal Systems from Time-Series

Abstract: We study multi-variate time-series generated by coupled map lattices exhibiting spatio-temporal chaos and investigate to what extent we are able to estimate various intensive measures of the underlying system without explicit knowledge of the system dynamics. Using the rescaling and interleaving properties of the Lyapunov spectrum of systems in a spatio-temporally chaotic regime and paying careful attention to errors introduced by sub-system boundary effects, we develop algorithms that are capable of estimating the Lyapunov spectrum from time series. We analyse the performance of these and find that the choice of basis used to fit the dynamics is crucial: when the local dynamics at a lattice site is well approximated by this basis we are able to accurately determine the full Lyapunov spectrum. However, as the local dynamics moves away from the space spanned by this basis the performance of our algorithm deteriorates.

Physica D, 147, 2000, 204-220.

Revised 1.8.00

#### Delay Reconstruction: Dynamics v Statistics

by J. Stark

Abstract: Traditionally delay reconstruction is seen as lying in the realm of dynamics, or even differential topology. It is thus perceived to be a largely automatic procedure that reconstructs an existing dynamical system. In this chapter we argue that it is as imprecise as all other parts of time series analysis and should be subject to as much statistical scrutiny as procedures such as modelling, prediction or noise reduction.

In Nonlinear Dynamics and Statistics, ed A.I. Mees, Birkhauser, 2001, ISBN: 0817641637.

Scaling of Intermittent Behaviour of a Strange Nonchaotic Attractor

by R. Sturman

We describe a type of intermittency present in a strange nonchaotic attractor of a quasiperiodically forced system. This has a similar scaling behaviour to the intermittency found in an attractor-merging crisis of chaotic attractors. By studying rational approximations to the irrational forcing we present a reasoning behind this scaling, which also provides insight into the mechanism which creates the strange nonchaotic attractor.

Physics Letters A, 259, 355-365.

#### Observing Complexity, Seeing Simplicity

by J. Stark

Abstract: This century has seen the formulation of a number of novel mathematical and computational frameworks for the study, characterization and control of various classes of complex phenomena. Most of these involve some non-trivial dynamics. In order to be of genuine use in the real world, it is essential that such theoretical developments are related to observed data. This essay is concerned with the question of how this might be achieved. In particular, it investigates how much information about a complex unknown system one can hope to recover from observations. The vast majority of theoretical analysis assumes that we have an accurate model of a system and that we know the variables that uniquely determine its state. In principle the application of such a theory to real problems requires the simultaneous measurement of all these variables. This is rarely feasible in practice, where often we will not even know what the important variables are. All that we may be able to achieve is to make a sequence of repeated measurements of one or more observables. The relationship between such observations and the state of the system is often uncertain. It is therefore unclear how much information about the behaviour of the system we can deduce from such measurements. It turns out that for a certain class of mathematically idealized systems it is in principle possible to reconstruct the whole system from a sequence of measurements of just a single observable. As a consequence, we may be able to build remarkably simple models of apparently complex looking behaviour. We shall outline the theoretical framework behind this remarkable result, discuss its limitations and its generalizations to more realistic systems. Finally, we shall speculate that the complexity of theoretical models may sometimes outstrip our ability to detect them in real data.

Phil. Trans. Royal Society A, 358, (2000), 41-61.

#### Thermodynamic Limit from Small Lattices of Coupled Maps

by R. Carretero-González, S. Ørstavik, J.Huke, D.S.Broomhead and J. Stark

Abstract: We compare the behaviour of a small truncated coupled map lattice with random inputs at the boundaries with that of a large deterministic lattice essentially at the thermodynamic limit. We find exponential convergence for the probability density, predictability, power spectrum, and two-point correlation with increasing truncated lattice size. This suggests that spatio-temporal embedding techniques using local observations cannot detect the presence of spatial extent in such systems and hence they may equally well be modelled by a local low dimensional stochastically driven system.

Physics Review Letters, 83, (1999), 3633-3636.

#### Estimation of Noise Levels for Models of Chaotic Dynamical Systems

by J. Heald and J. Stark

Abstract: We investigate how far it is possible to identify and separate dynamical noise from measurement noise in observed nonlinear time series. Using Bayesian methods, we derive estimates for the two noise levels, and find that, given a good model of the dynamics, these can give accurate results even if the dynamical noise level is orders of magnitude smaller than the measurement noise level; whereas a simple calculation of RMS error badly understates the dynamical noise. We argue that this should allow better estimates of the underlying dynamical time-series, and so better predictions of its immediate future and fundamental dynamical properties.

Physics Review Letters, 84 (2000), 2366-2369.

#### Semi-Uniform Ergodic Theorems and Applications to Forced Systems

by R. Sturman and J. Stark

Abstract: In nonlinear dynamics an important distinction exists between uniform bounds on growth rates, as in the definition of hyperbolic sets, and non-uniform bounds as in the theory of Liapunov exponents. In rare cases, for instance in uniquely ergodic systems, it is possible to derive uniform estimates from non-uniform hypotheses. This allowed one of us to show in a prior paper that a strange non-chaotic attractor for a quasiperiodically forced system could not be the graph of a continuous (but not smooth) function. This had been a conjecture for some time. In this paper we generalize the uniform convergence of time averages for uniquely ergodic systems to arbitrary systems. In particular, we show how conditions on growth rates with respect to all the invariant measures of a system can be used to derive one sided uniform convergence in both the Birkhoff and the Sub-Additive Ergodic Theorems. We apply the latter to show that any invariant set for a quasiperiodically forced system must support an invariant measure with a non-negative maximal normal Liapunov exponent; in other words it must contain some "non-attracting" orbits. This was already known for the few examples of strange non-chaotic attractors that have rigorously been proved to exist. Finally, we generalize our semi-uniform ergodic theorems to arbitrary skew product systems and discuss the application of such extensions to the existence of attracting invariant graphs.

Nonlinearity, 13, (2000), 113-143.

#### Nonlinear Noise Reduction: Bayesian Perspectives

by J. Heald and J. Stark

Abstract: Bayes' theorem is a mathematical identity in probability theory, which defines a consistent and coherent structure for assigning probabilities to models and model-elements given data. In nonlinear time-series analysis, such quantities of interest may include any or all of the following: the underlying dynamics (including the appropriate model order or regularisation parameters); the noise levels (with separate levels for measurement and dynamical noise); any state dependence of the noise (eg multiplicative noise); and identification of the measurement noise in the observed time series, with a view to its reduction or removal. Starting from a very simple case of noise reduction when the dynamics and the noise levels are known, we discuss how Bayesian methods can be applied and then widened to address each of the above.

1998 International Symposium on Nonlinear Theory and its Applications, 3, Presses polytechniques et universitaires romandes, (1998), 1293-1296.

#### Scaling and Interleaving of Sub-system Lyapunov Exponents for Spatio-temporal Systems

by R. Carretero-González, S. Ørstavik, J.Huke, D.S.Broomhead and J. Stark

Abstract: The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a sub-system in a very cost effective way. In this work we present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. It is inspired by the stability analysis of the homogeneous evolution in a one-dimensional coupled map lattice but appears to be equally valid in a much wider range of cases. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative sub-system volume) for one and two-dimensional lattices in spatio-temporal chaotic regimes. In doing so we notice that the Lyapunov spectra for consecutive sub-system sizes are interleaved and we discuss the possible ways in which this may arise. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension and Kolmogorov-Sinai entropy) finding better convergence as the sub-system size is increased than with conventional rescaling.

Chaos, 9, (1999), 466-482.

#### Reconstruction and Cross-Prediction in Coupled Map Lattices Using Spatio-Temporal Embedding Techniques

by S. Ørstavik and J. Stark

Abstract: We use a mix of temporal and spatial delay embedding techniques to carry out reconstruction and cross-prediction on time series generated by a coupled map lattice. We find that spatio-temporal delay reconstructions give better predictability than standard methods using either time delays only or spatial delays only. We also observe that in all these cases it is completely infeasible to rigorously embed the original spatio-temporal system since this would require impractically large embedding dimensions. Despite this, it proves possible to make good short term predictions in embedding dimensions as low as 4. We discuss a possible explanation of this apparent paradox and briefly describe a tentative theoretical framework for reconstructing high dimensional systems that this suggests.

Phys. Lett. A., 247, (1998), 145-160.

#### Inertial Manifolds for Dynamics of Cells Coupled by Gap Junctions

by P. Iannelli, S. Baigent and J. Stark

Abstract We study the asymptotic dynamics of a coupled system of ordinary differential equations, which arises from a model of a network of cells coupled by gap junctions (see below for model formulation and stability analysis). The variables in this model are the electrochemical potentials of a chemical species in each cell and the states of the various gap junctions. The dynamics of the chemical species is much faster than that of the gap junctions leading to a singular perturbation problem. We show that for biologically realistic parameters, the system has a globally attracting smooth invariant manifold which is the graph of a function from gap junction states to electrochemical potential. It is therefore the gap junction dynamics which controls the overall behaviour of the system. Rather than using a standard singular perturbation approach, which fails to give explicit estimates of the size of allowed perturbations, we employ inertial manifold techniques. These are usually applied to systems ofthe form du/dt =-Au+V(u) where u belongs to a Hilbert space and A is a positive linear operator satisfying a so called "gap condition". Since our system fails to meet this condition, we generalize these methods to a class of systems of the form du/dt = -A(u)u+V(u).

Dynamics and Stability of Systems, 13, (1998), 187-213.

#### Optimal Scheduling in a Periodic Environment

by R.S. MacKay, S. Slijepcevic and J. Stark

Abstract: We consider the problem of scheduling a sequence of identical actions in situations where the benefit obtained from an action depends partly on the time since the last action and partly on time periodic factors. A simple example would be a model of a bus shuttle service where the profit from running a bus could reasonably be expected to depend on the time of day and the time since the last bus. Mathematically, such problems can be formulated as maximising sum_{n=1}^N V(tn-1,tn) over schedules (tn)n³0 as tN goes to infinity for functions V(t,t') satisfying the periodicity condition V(t+T,t'+T) = V(t,t'). Such periodicity is incompatible with concavity of V which is the normal hypothesis under which such dynamic programming problems are normally studied. Under suitable conditions we show that this problem is related to so called Aubry-Mather theory in the dynamics of area-preserving maps. We extend this theory in order to prove the existence of optimising schedules for each initial condition t0, to characterise the properties of such schedules and to analyse their dependence on t0 and on the parameters of the model.

Nonlinearity, 13, (2000), 257-297.

#### Invariant Graphs for Forced Systems

by J. Stark

Many applications of nonlinear dynamics involve forced systems. We consider the case where for a fixed input the driven system is contracting; this is for instance the situation in certain classes of filters, and in the study of synchronization. When such contraction is uniform, it is well known that there exists a globally attracting invariant set which is the graph of a function G from the driving state space to the driven state space. If the contraction is sufficiently strong then G is smooth. We describe the theoretical framework for such results and discuss a number of applications to the filtering of time series, to syn chronization and to quasiperiodically forced systems. We go on to give recent generalizations to the case of non-uniform contraction (proofs in Regularity of Invariant Graphs for Forced Systems below); that is contraction measured by Liapunov exponent like quantities. We conclude with a number of new results for quasiperiodic forcing; a corollary of these is that a strange non-chaotic attractors for such a system cannot be the graph of a continuous function, nor roughly speaking can it have an open attracting neighbourhood; in other words its closure must contain some repelling orbits.

Physica D, 109 (1997), 163-179.

#### Takens Embedding Theorems for Forced and Stochastic Systems

by J. Stark, D.S. Broomhead, M.E. Davies and J. Huke

Nonlinear Analysis, 30, (1997), 5303-5314.

This is a summary of recent results on extending Takens Embedding Theorem to both deterministically and stochastically forced systems.

#### Modelling the Control of Ovulation and Polycystic Ovary Syndrome

by A. Chavez-Ross, S. Franks, H.D. Mason, K. Hardy and J. Stark

Abstract: The control of ovulation in mammalian species appears to be a highly robust process. The primary mechanism is believed to be competition amongst a group of developing follicles, mediated by a hormonal feedback loop involving in the first instance the pituitary. Successful follicles reach maturity and ovulate, the remainder atrophy and die. A model of this control process has been derived by Lacker and his group. Based on simple qualitative assumptions about the hormonal feedback loop, this is able to reflect many of the basic physiological features of ovulation in mammals. However, a fundamental hypothesis of Lacker¹s work is that all follicles are identical and respond to hormonal signals in precisely the same way. Not only is this improbable, but it also leads to several aspects of the model which are qualitatively unrealistic, most notable of these is its inability to accurately model the condition known as Polycystic Ovary Syndrome. This common malfunction of the ovulatory control mechanism accounts for up to three-quarters of cases of anovulatory infertility in humans and its understanding is therefore of considerable medical significance. In this paper we extend the analysis of Lacker¹s model to the case of non-identical follicles; this allows us to obtain behaviour much closer to that observed in PCOS patients and to draw some tentative conclusions about the mechanisms underlying this condition.

J. Math. Biol., 36, (1997), 95-118.

#### Modelling the Effect of Gap Junction Nonlinearities in Systems of Coupled Cells

by S. Baigent,J. Stark and A. Warner

J. Theor. Biol. 186 (1997), 223-239.

Abstract: A general model for the transfer of ions and molecules between two cells via a gap junction is presented. This involves a dynamical system consisting of two parts: the dynamics of intracellular concentrations and electrical potentials, and the dynamics of the gating in the gap junction. The analysis focuses on a particular approximation in which the concentration changes are assumed negligible, so that the dynamics then describes the evolution of the potential differences across the cell membranes. If the gating dynamics are sufficiently slow in comparison to the cell dynamics, it is argued that, under certain conditions on the membrane pump mechanisms actively transporting matter across the membrane, there can be no periodic orbits. The other extreme, where the junctional gating dynamics is much slower than that of the cells, is also considered. Under certain assumptions about the gating mechanism it is again argued that periodic behaviour is not possible. It is shown that, in theory, coupled Xenopus cell pairs can exhibit hysteretic behaviour for certain cell and junctional parameter ranges, although data from recent electrophysiological experiments on Xenopus cell pairs however suggests that this hysteretic behaviour is outside the usual physiological regime. However, it is suggested that cells may operate under conditions in which hysteretic changes in the concentrations of ion and molecules could arise from alterations in active membrane pumping, if the gap junctions are sufficiently sensitive to concentration changes.

See above for further analysis of this model

#### Regularity of Invariant Graphs for Forced Systems

by J. Stark

Erg. Theory and Dyn. Sys., 19, (1999), 155-199.

Abstract:Many applications of nonlinear dynamics involve forced systems. We consider the case where for a fixed input the driven system is contracting; 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 a special case of the well known concept of an inertial manifold for more general systems. 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. The aim of this paper is to extend this result in two directions: firstly where we only have uniform contraction for a compact invariant set of input states, and secondly where the contraction rates are non-uniform (and hence defined by Liapunov exponents and analogous quantities). In both cases the invariant graph is only defined over closed subsets of the input space, and hence we need to define an appropriate notion of smoothness for such functions. This is done in terms of the Whitney Extension Theorem: a function is considered Whitney smooth if it satisfies the conditions of this theorem and hence can be extended to a smooth function of the whole input space.

Revised 18.2.98

#### Delay Embeddings of Forced Systems: I Deterministic Forcing

by J. Stark

J. Nonlinear Sci., 9, 255-332.

Abstract:Takens Embedding Theorem forms the basis of virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems. It 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. Current versions of Takens Theorem 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. In this paper we therefore prove two versions of Takens Theorem relevant to forced systems: one applicable to the case where the forcing is unknown, and the other to the situation where we are able to independently determine the state of the forcing system (usually because we are responsible for the forcing ourselves). In a subsequent paper we shall show how to extend these results to give an analogue of Takens Theorem for randomly forced systems, leading to a new framework for the analysis of time series arising from nonlinear stochastic systems.

Revised 16.3.98

#### Linear Recursive Filters and Nonlinear Dynamics

Abstract: In this paper we investigate the effects of filtering a chaotic time series with an IIR filter. Using the Kaplan and Yorke conjecture it has been argued that such filtering can result in an increase in information dimension. Here we show that the filter dynamics induces an extended dynamical system and that this system possesses a globally attracting invariant graph over the base system. Using this framework we obtain sufficient conditions on the filter dynamics that guarantee that the dimension remains unchanged.

Postscript (218 K)

UCL >> CNDA >> History > People > Research > Preprints > MSc > Seminars > Contacts
Created 19/9/95; Last modified 19/9/1995 jpmh

UCL Centre for Nonlinear Dynamics and its Applications,
University College London, Gower Street, London, WC1E 6BT, UK.

cnda@ucl.ac.uk`