# Recent thesis

### Nonlinear and Complex Systems Group, University of Loughborough

#### Supervisor: Paul Bressloff

This PhD thesis is now available online: http://www.lboro.ac.uk/departments/ma/research/ncsg/papers/index.html . Peter is now a Research Fellow in Theoretical Neuroscience at the Babraham Institute in Cambridge.

Source: Steve Coombes.

### Centre for Applied Dynamics and Optimization, Department of Mathematics and Statistics, University of Western Australia.

#### Supervisors: Kevin Judd (CADO, Department of Mathematics and Statistics, University of Western Australia) and Steve Stick (Department of Respiratory Medicine, Princess Margeret Hospital for Children)

Abstract:

Using inductance plethysmography it is possible to obtain a non-invasive measurement of the chest and abdominal cross-sectional area. These measurements are representative'' of the instantaneous lung volume. This thesis describes an analysis of the breathing patterns of human infants during quiet sleep using techniques of nonlinear dynamical systems theory. The purpose of this study is to determine if these techniques may be used to extend our understanding of the human respiratory system and its development during the first few months of life. Ultimately, we wish to use these techniques to detect and diagnose abnormalities and illness (such as apnea and sudden infant death syndrome) from recordings of respiratory effort during natural sleep.

Previous applications of dynamical systems theory to biological systems have been primarily concerned with the estimation of dynamic invariants: correlation dimension, Lyapunov exponents, entropy and algorithmic complexity. However, estimating these numbers has not proven useful in general. The study described in this thesis focuses on building models from time-series recordings and using these models to deduce properties of the underlying dynamical system. We apply a correlation dimension estimation algorithm in conjunction with well known surrogate data techniques and conclude that the respiratory system is not linear. To elucidate the nature of the nonlinearity within this complex system we apply a new type of radial basis modelling algorithm (cylindrical basis modelling) and generate new nonlinear surrogate data.

New nonlinear radial (cylindrical) basis modelling techniques have been developed by the author to accurately model this data. This thesis presents new results concerning the use of correlation integral based statistics for surrogate data hypothesis testing. This extends the scope of surrogate data techniques to include hypotheses concerned with broad classes of nonlinear systems. We conclude that the human respiratory system behaves as a periodic oscillator with two or three degrees of freedom. This system is shown to exhibit cyclic amplitude modulation (CAM) during quiet sleep.

By examining the eigenvalues of fixed points exhibited by our models, and the qualitative features of the asymptotic behaviour of these models we find further evidence to support this hypothesis. An analysis of Poincare sections and the stability of the periodic orbits of these models demonstrates that CAM is present in models of almost all data sets. Models which do not exhibit CAM often exhibit chaotic first return maps. Some models are shown to exhibit period doubling bifurcations in the first return map.

To quantify the period and strength of CAM we suggest a new statistic based on an information theoretic reduction of linear models. The models we utilise offer substantial simplification of autoregressive models and provide superior results. We show that the period of CAM present before a sigh and the period of subsequent periodic breathing are the same. This suggests that CAM is ubiquitous but only evident during periodic breathing. Physiologically, CAM may be linked to an autoresucitation mechanism. We observe a significantly increased incidence of CAM in infants at risk of sudden infant death syndrome and a higher incidence of CAM during apneaic episodes of bronchopulmonary dysplasic infants.

Source: Michael Small.

## Nonlinear model evaluation: $\iota$-shadowing, probabilistic prediction and weather forecasting

### Oxford Centre of Industrial and Applied Mathematics, Mathematical Institute, University of Oxford.

#### Supervisor: Dr Lenny Smith

Abstract

Physical processes are often modelled using nonlinear dynamical systems. If such models are relevant then they should be capable of demonstrating behaviour observed in the physical process. In this thesis a new measure of model optimality is introduced: the distribution of $\iota$ shadowing times defines the durations over which there exists a model trajectory consistent with the observations. By recognising the uncertainty present in {\em every} observation, including the initial condition, $\iota$ shadowing distinguishes model sensitivity from model error; a perfect model will always be accepted as optimal. The traditional root mean square measure may confuse sensitivity and error, and rank an imperfect model over a perfect one.

In a perfect model scenario a good variational assimilation technique will yield an $\iota$ shadowing trajectory but this is not the case given an imperfect model; the inability of the model to $\iota$ shadow provides information on model error, facilitating the definition of an alternative assimilation technique and enabling model improvement.

While the $\iota$ shadowing time of a model defines a limit of predictability, it does not validate the model as a predictor. Ensemble forecasting provides the preferred approach for evaluating the uncertainty in predictions, yet questions remain as to how best to construct ensembles. The formation of ensembles is contrasted in perfect and imperfect model scenarios in systems ranging from the analytically tractable to the Earth's atmosphere, thereby addressing the question of whether the apparent simplicity often observed in very high-dimensional weather models fails `even in or only in' low-dimensional chaotic systems. Simple tests of the consistency between constrained ensembles and their methods of formulation are proposed and illustrated. Specifically, the commonly held belief that initial uncertainties in the state of the atmosphere of realistic amplitude behave linearly for two days is tested in operational numerical weather prediction models and found wanting: nonlinear effects are often important on time scales of 24 hours. Through the kind consideration of the European Centre for Medium-range Weather Forecasting, the modifications suggested by this are tested in an operational model.

Source: Isla Gilmour ( gilmour@maths.ox.ac.uk).

## On the Continuous and Discrete Third Painlevé Equations

### Institute of Mathematics & Statistics, University of Kent

#### Supervisor: Professor Peter A Clarkson

Abstract

The main topic of this thesis is the third Painlevé equation and its discrete analogue. The Painlevé equations were derived around the turn of the century and have been well studied. Interest was rekindled in the 1970s with the discovery that symmetry reductions of soliton equations solvable by the inverse scattering technique, often yielded Painlevé equations. In contrast, discrete Painlevé equations have only recently been discovered.

The first part, and by far the larger part, of this thesis studies the third Painlevé equation and the discrete third Painlevé equation. One-parameter families are studied using the algorithmic method developed by Albrecht, Mansfield and Milne (1996). It is proved that the known one-parameter conditions given by Gromak (1977) are a full set. Hierarchies of rational solutions of the discrete third Painlevé equation, for particular values of the parameters, are constructed by applying Schlesinger transformations to simple seed solutions. A direct analogy to the hierarchies of rational solutions of the third Painlevé equation given by Milne (1995) is established. An analogy is also established between the Bessel function solutions of the discrete third Painlevé equation given by Kajiwara, Ohta and Satsuma (1995) and the known hierarchies of Bessel function solutions of the third Painlevé equation (Milne (1995)). The concluding piece of work in this part investigates a method of generating discrete Painlevé equations by using the auto-Bäcklund transformations of a continuous Painlevé equation. As an example, discrete Painlevé equations are generated from the auto-Bäcklund transformations of the third Painlevé equation. These calculations stress the importance of certain conditions and leave many open questions.

Classical symmetry reductions of the viscous Cahn Hilliard system are studied in the second part of this thesis. It is highlighted how the calculation depends on the arbitrary parameters and functions present in the system.

Source: Professor Peter A Clarkson.

## Vibro-impact Dynamics of Engineering Systems

### Centre for Nonlinear Dynamics and Its Applications, University College London

#### Supervisor: Professor Steven Bishop

Abstract

Many problems in engineering involve systems with vibrating components which can impact with a rigid boundary. The aim of this research is to improve our understanding of the dynamics associated with such vibro-impact systems, thus enabling more accurate mathematical modelling of associated engineering problems.

Initially we consider a single degree of freedom impact oscillator system. Such systems have been widely studied and are discussed here in terms of fundamental impacting motions and nonlinear mappings. The bifurcational behaviour of the system is studied and compared to results from an experimental system. In particular we consider the codimension two bifurcations which occur and demonstrate how these events can be used to maintain periodic motion of the system during parameter variation.

We then consider developing a structured approach for parameter estimation using experimentally recorded data. An energy analysis for single degree of freedom systems is presented. This is used to develop a method for estimating the coefficient of restitution and damping parameters using a time series obtained from a physical system.

Using a purpose built experimental impact load cell we estimate of impact forces for an impacting beam system. We discuss the analysis of impulse spike data, and show how system dynamics can be reconstructed using interspike intervals.

We then consider the dynamics of multi-degree of freedom impact oscillators, including sticking and chatter motions, using a two degree of freedom lumped mass model as an example. The energy analysis developed for the single degree of freedom system is extended to multi-degree of freedom systems and the effect of energy transfer between modes and reduction in coefficient of restitution is discussed.

Finally we consider how such work may be applied to continuous systems such as the cantilever beam using a Galerkin approach. We compare these results qualitatively with experimental data from flexible vibro-impacting beams.

Source: David Wagg

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