UK Nonlinear News, May 2002

Recent thesis


A bundle view of parameter-dependent boundary value problems

Francis R. Austin

Department of Mathematics and Statistics, University of Surrey

Supervisor: Professor T. Bridges

Abstract

The thesis develops a range of tools for geometric and topological analysis of differential eigenvalue problems. The main part of the thesis considers holomorphic families of differential equations on a finite interval with prescribed parameter-dependent boundary conditions. The Gardner-Jones bundle, which was introduced for linearized reaction-diffusion equations, is generalised and applied to this abstract class of BVPs. It is proved that any characteristic determinant of the BVP on a Jordan curve can be characterised geometrically as the determinant of a transition function associated with the Gardner-Jones bundle. The topology of the bundle, represented by the Chern number, then yields precise information about the number of non-trivial solutions in a prescribed subset of parameter space. This bundle framework is applied to examples from hydrodynamic stability theory and the linearized complex Ginzburg-Landau equation. Other geometric aspects of eigenvalues problems which are developed in the thesis include the study of line bundles over multi-parameter families of Hermitian matrices, curvature structure of general k-parameter families of ODEs, and topology of parameter-dependent ODEs with periodic coefficients.

Source: Tom Bridges.


Modelling dolphin hydrodynamics: the numerical analysis and hydrodynamic stability of flow past compliant surfaces

Leanne Allen

Department of Mathematics and Statistics, University of Surrey

Supervisor: Professor T. Bridges

Abstract

The thesis has three main parts: (a) development of a numerical framework for numerically integrating differential equations on exterior algebra spaces, (b) study of the stability of three-dimensional rotating flows interacting with a compliant surface, and (c) study of the flow past a three-dimensional swept wing with a compliant surface as a model for flow past dolphin fins. The results show that passive compliance has a retarding effect on transition on swept wings which is qualitatively similar to the 2D case. On the other hand, addition of rotation in the fluid diminishes the effect of compliance on transition delay. The numerical framework has been shown to be useful in a wide range of other problems as well, including the stability of solitary waves.

Source: Tom Bridges.


Sensitive Dependence on Initial Conditions: Chaos and Stochastic Processes

Simone Giannerini

Supervisor Professor Howell Tong

Abstract

The main purpose of the thesis is to review how the concept of initial value sensitivity, motivated by chaos theory and well defined for a deterministic process, has been studied in a stochastic environment in order to bring (i) new insights and contributions to the statistical analysis of time series; (ii) a natural, generalised framework for dynamical system theory. In particular, we have focused our attention on the issues that arise from the definition, the estimation and the implementation of measures of initial value sensitivity both in the deterministic and in the stochastic settings.

The thesis is available upon request from the author.

Source:Simone Giannerini ( giannerini@stat.unibo.it).


Time scale interaction in low-order climate models

Lennaert van Veen

University of Utrecht

Supervisors: Professor F. Verhulst and Professor J.D. Opsteegh

Abstract

Over the last decades, the study of climate variability has attracted ample attention. The observation of structural climatic change has lead to questions about the causes and mechanisms involved. The task to understand interactions in the complex climate system is particularly difficult because of the lack of observational data, spanning a period of time typical for natural climate variability.

One way around this problem is to represent the earth's climate in a computer model, as a set of prognostic equations. However, if the model under consideration is to faithfully represent the real climate system, it has to be large in terms of the number of degrees of freedom. In meteorology, the study of simplified models of low dimension has lead to many a fundamental insight. The study of extremely simplified climate models, as presented in this thesis, should likewise lead to an understanding of the mechanisms of climatic change.

I derive and present two low-order climate models. They are analysed by means of singular perturbation theory and bifurcation theory. The focus is on mechanisms by which the ocean, i.e. the slow subsystem, can play an active role in the coupled dynamics. This turns out to happen in the vicinity of bifurcation points of the fast subsystem. Both low-order climate models include the Lorenz-84 model for the atmosphere. Therefore I have described the derivation and bifurcation analysis of this model.

The thesis is based on the following papers:

  1. Baroclinic flow and the Lorenz-84 model, 2002, Int. J. Bifur. Chaos, to appear.
  2. Active and passive ocean regimes in a low-order climate model, 2001, Tellus 53A, p. 616-628.
  3. Overturning and wind driven circulation in a low-order ocean-atmosphere model, 2002, Dyn. Atmos. Oceans, submitted.

Source Lennaert van Veen

 


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Last Updated: 1 May 2002.
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