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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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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(y_{n}-y_{0}) ² r +
1/n, where 1/n(y_{n}-y_{0}) 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
y_{n}-y_{0}-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 y_{n}-y_{0}-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 y_{n}-y_{0}-rn is
unbounded either above or below, then there is a residual set of
orbits for which y_{n}-y_{0}-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.

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.* *

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

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.

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

**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

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.

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.*
*

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.

**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.

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.

**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.

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.

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.

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.

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(t_{n-1},t_{n}) over schedules
(t_{n})_{n³0} as t_{N} 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 t_{0}, to characterise the properties of such
schedules and to analyse their dependence on t_{0} and on the
parameters of the model.

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

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.

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.

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.

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

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

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

by M.E. Davies and K. M. Campbell.

**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.

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*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`