UK Nonlinear News, August 1999

Recent thesis

Dynamical Systems models of patchiness in estuaries

James R Stirling

Mathematical Sciences, Loughborough University

Supervisors: Andy Osbaldestin, Cecil F Scott and Ron Smith


Three models for the flow in estuaries are developed The first is a 2+1 dimensional time-periodic model of a flow in the vertical cross section. The second model adds an un-coupled velocity field U in the along-estuary direction. The final model is a 3+1 dimensional fully coupled, time-dependent flow. The transport of material in each model is studied using lobe diagrams. Such diagrams allow us to separate the flow into different regions and to calculate the transport of material between adjacent regions. Particular attention is given to curves which form partial barriers to transport (cantori, stable and unstable manifolds of hyperbolic stationary points) or complete barriers to transport (KAM curves and invariant circles).

These models are used to develop an understanding of both the mixing within the flow and the formation and leakages from patches of higher concentration within a cloud of pollution released into the estuary. Also the time taken for particles to exit the bounds of the estuary is investigated. As a result we get an understanding of which regions of the flow flush pollution out of the estuary in the least time and whether that flushing is seaward or landward. This understanding is applied to the problem of the optimal discharge of effluent into an estuary.

James can be contacted at

Source: Professor Ronald Smith.

Nonlinear Dynamics and Chaos in Electronic Oscillators

University College Dublin, Ireland

Gian Mario Maggio

Supervisor: Dr. Michael Peter Kennedy


Sinusoidal oscillators lie at the heart of modern communication systems. The drive towards increasingly higher operating frequencies forces circuit designers to use single-transistor L-C oscillators structures where feasible and to implement as much of the oscillator as possible in a single integrated circuit.

Oscillator design methods currently employed in the electronics industry for oscillators of this type are based primarily upon linear analysis. While linear methods have proven invaluable over many years in generating first-order approximations, they are severely limited in both qualitative and quantitative predictions.

In this work, we apply nonlinear analysis techniques in order to predict qualitatively the large-signal performance of an oscillator. In particular, the Colpitts oscillator is introduced as a paradigm for developing analysis and design techniques for sinusoidal oscillators. First, we discuss the birth of oscillation in terms of Hopf bifurcation theory. Then, characterisation of the signal generated is discussed. The limiting role played by period-doubling bifurcations with respect to nearly-sinusoidal oscillation is highlighted and we propose several methods for predicting the appearance of subharmonic behaviour.

By making use of bifurcation analysis and continuation techniques, we show how a complete picture of the dynamical behaviour of the circuit in terms of its parameters can be obtained. These results are compared with those obtained by simulation, and the limitations associated with simulation techniques are highlighted.

For the Colpitts oscillator we identify the main routes to chaos and we illustrate the local and global coexistence phenomena. Also, the organising role played by homoclinic bifurcations in the parameter space is stressed. Design rules are developed for applications requiring either sinusoidal oscillation or chaotic behaviour. Finally, the theoretical predictions are confirmed experimentally and the extension of this work to other oscillator configurations is discussed.

Source: Gian Mario Maggio (

Spatio-temporal patterns in differential-flow reactors

Razvan Alin Satnoianu

Department of Applied Mathematics, University of Leeds

Supervisors: Professors John Merkin and Steve Scott (Chemistry)


This thesis is concerned with the mathematical modelling and subsequent analysis of a new instability mechanism for pattern formation in physical systems, the so-called DIFI (Differential-Flow Induced Instability) and in particular its connection with convective instabilities in flow systems. The fundamental assumption behind DIFI is that, in a given system, certain species have a lower flow velocity than the others thus giving a wider spatial mobility degree for spontaneous pattern formation. This underlines that DIFI should be a more realistic pattern formation scenario than that originally proposed by Turing (1952) and as such finds a variety of applications in chemistry (DIFICI-Differential-Flow Induced Chemical Instability in Chemistry), biology (modelling of calcium waves under the action of an electric field and pattern formation on growing domains), ecology (plankton population modelling), physiology (heart and kidney modelling), chemical engineering and combustion (packed-bed reactor), laser instabilities, etc.

The aim of the present thesis is to build a theoretical understanding for the complex dynamical behaviour related to the DIFI mechanism when applied to model chemical systems (the DIFICI problem). As a preliminary, in the first chapter a simple general analysis is presented outlining the basic ingredients for DIFICI to occur in general reaction-diffusion-convection systems comprising two species.

Chapter 2 describes a simple prototype model for a differential-flow reactor and this is employed to study the initiation and propagation of reaction-diffusion-convection waves in a system based on a simple isothermal autocatalytic reaction step. Chapter 3 considers a more complex situation when the chemistry is based on cubic autocatalator kinetics taking the flow reactor to be with recycling. A variety of analytical and numerical methods uncover the structure of the solutions which arise. In chapter 4 the same chemical system is considered in an open flow reactor (without recycling). By taking the flow rate of the autocatalyst species as the main bifurcation parameter it is shown that a rich set of complex structures arise in the domain of the DIFICI-Hopf interaction. The whole parameter domain is completely mapped out through the use of the powerful methods of amplitude equations. Chapter 5 extends the previous analysis for the case of a periodically forced system. The complex dynamic behaviour seen is analysed in terms of the excitation frequency windows with a detailed bifurcation analysis. Chapter 6 is devoted to an analysis of the spatio-temporal chaotic behaviour arising in the DIFICI-Hopf domain alluded to in chapter 4. Finally, in chapter 7 the main conclusions are drawn and suggestions for further work are presented.

The full text is available to read and/or download (pdf format) from my web site:

I am currently working at Oxford University at the Mathematical Institute as a Research Assistant in a project involving a study of bacterial chemotaxis.

Source: Razvan Satnoianu (

POD Methods in Baroclinic Flows

Adam Vercingetorix Stephen

Mathematics Institute, Oxford University


This thesis investigates the hypothesis that baroclinic wave flows in the thermally-driven rotating annulus experiment may be modelled by low-dimensional sets of coupled, nonlinear ordinary differential equations. Data sets of baroclinic flows were obtained from both new experimental work and numerical simulation runs. An enhanced image analysis system was installed and tested, and new experiments were undertaken to obtain measurements of the velocity field in both steady wave and amplitude vacillation flows. The Proper Orthogonal Decomposition (POD) was applied to extract spatial patterns called Empirical Orthogonal Functions (EOFs) from these data sets.

This technique defines the optimal orthogonal basis for a linear modal decomposition of a signal in the sense that it compresses the variance of the signal into the leading order modes more efficiently than any other decomposition. A method to construct low-order models of baroclinic instability by projecting the two-layer quasi-geostrophic equations onto truncated series of EOFs is presented. The formulation of these low-order models was validated by reproducing the results of two standard spectral models. The complete specification of any particular model using this method requires the definition of a forcing scheme and calibration by estimating several dimensionless parameters. Two forcing schemes are presented and estimates of the parameter values are calculated from experimental measurements and numerical simulations.

The resulting models are analysed using a combination of solution continuation and numerical integration methods to seek the forcing/parameter combination which optimises agreement between model and observations.

Source: Dr Peter L Read

The Takens--Bogdanov Bifurcation with D4 Symmetry

Alun K. Thomas

School of Mathematical Sciences, University of Nottingham

Supervisors: P.C. Matthews and D.S. Riley


In this thesis, the equations for the Takens--Bogdanov bifurcation with D4 symmetry are studied.

After the normal form for the equations has been derived, the extent to which the equations may be solved analytically is covered. Three types of stationary solution are first identified, along with the conditions necessary for their linear stability. Analysis of periodic solutions forming at Hopf bifurcations from each of the three stationary solutions is then undertaken. It is shown that whilst some of these periodic solutions always emanate from subcritical Hopf bifurcations, other Hopf bifurcations can be supercritical. Stability criteria for the periodic solutions emanating from these supercritical Hopf bifurcations are calculated. Further analysis of these stationary and periodic solutions then reveals the form, and location in parameter space, of a number of heteroclinic and homoclinic connections.

A special case of the normal form, modelling a system of four coupled LRC oscillators in a square arrangement, is studied numerically. The existence of some of the analytically derived periodic solutions and their primary bifurcations is verified, as are the locations of some analytically derived global bifurcations. Secondary and subsequent bifurcations from these periodic solutions are also located, which leads to the identification of additional solution branches, and their bifurcations, that cannot be calculated analytically.

Source: Paul Matthews (

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