Teaching
 
  Current courses:

          AMH9: Multiscale Methods and Stochastic Model Reduction


 
   Possible projects for Honours students:

     Dispersive regularization of fluid systems

It is the aim of this project to develop innovative numerical methods to integrate evolution equations which support shock solutions. Shock solutions arise as generic solutions in incompressible fluid dynamics and rarefied gas dynamics. We will develop models and techniques which approximate the essential dynamics on a coarse grid in such a way that the price is not paid by sacrificing the preservation of conservation laws of the underlying equations. The project will do this for the so called shallow-water equations. Prior knowledge of Matlab and a symbolic software package such as mathematica or Maple is desirable.



     Networks of coupled oscillators

Many biological systems are structured as a network. Examples range from microscopic systems such as genes and cells, to macroscopic systems such as fireflies or even an applauding audience at a concert.
Of paramount importance is the topography of such a network, ie how the nodes, let's say the fireflies, are connected and how they couple. Can they only see their nearest neighbours, or all of them. Are some fireflies brighter than others, and how would that affect the overall behaviour of a whole swarm of fireflies? For example, the famous 'only 6 degrees of separation'-law for the connectivity of human relationships is important in this context.
In this project we aim to understand the influence of the topography of such a network. Question such as: How should a network be constructed to allow for maximal synchronization will be addressed.
This project requires new creative ideas and good programming skills.



     Cardiac alternans

Cardiac alternans is the phenomenon where a short heart-beat duration is followed by a long one, followed by a short one and so forth. It is widely believed that these alternans are the precursors of fatal cardiac failures such as atrial fibrillation.
This project - aimed at students interested in mathematical biology - will investigate different models of the heart, and study the possibility of alternans within these models. Questions such as ``Are alternans a real instability, and on which physiological time scale does the possible instability evolve?'', or ``Is this a stable phenomenon?'' will be investigated.
This project is mainly computational, although in the case of quick progress, we can build an analytical toy-model which captures all the effects observed during the numerical simulations.



     Discrete stochastic processes and their asymptotic limits

A reliable and efficient method to distinguish between chaotic and non-chaotic behaviour in noise-contaminated but essentially deterministic data has far reaching applications. For example, the onset of atrial fibrillation is accompanied by non-chaotic electrocardiogram data, hence the ability to detect periodicity is of great clinical significance.
Recently, we proposed a new method to detect chaos which applies directly to the time series data and does not require complicated phase space reconstruction. Moreover, the dimension and origin of the dynamical system are irrelevant. The input is the time series data and the output is zero or one depending on whether the dynamics is non-chaotic or chaotic. This zero-one test for chaos is universally applicable to any deterministic dynamical system, in particular to ordinary and partial differential equations.
In this project you aim to understand the influence of the sampling time on this test. The sampling time is the time between two distinct time series points - it is like the time between two measurements. In the test you try to detect Brownian motion. However, we are dealing here with a discrete approximation of Brownian motion. How does the approximation affect the predictions, such as diffusion coefficients? For example, it is obvious, that the sampling time has to be small enough. Otherwise, the asymptotic formulae do not make sense anymore. However, what if the sampling time gets smaller and smaller? The theory underlying the test for chaos states, that in the chaotic case, an auxiliary variable will behave asymptotically like Brownian motion. What does asymptotically really mean in this context? In this project you try to understand whether we can quantify the optimal range of sampling times. Discrete Brownian motion as the ones you will study in this project also occurs in the analysis of financial data.
This project requires new creative ideas and good programming skills. Knowledge of stochastic calculus and Brownian motion would be an advantage.



     Critical pulse propagation in excitable media

Many systems in biology and chemistry are so called excitable media. Examples are nerve fibres and heart tissue. Excitable media can support wave-trains and spiral waves. Propagation failure of wave-trains is often associated with clinical conditions such as atrial fibrillation in cardiology. The proposed project investigates propagation failure in 1D using a novel perturbation technique. This work is both of analytical and of computational nature. The proposed work will look at issues of propagation failure which have so far not been explored.
This project requires sound analytical and programming skills.

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