Peter Ashwin

If you are interested in a PhD or M-level (Math, MSci, MSc) project supervised by myself, it is best to email me for a chat. Below are a list of some topics or areas where I could offer projects. Do also have a look at my publications to see other possible topics that may not be listed below.

**Synchronization and pattern formation in
chaotic systems** (PhD/M-level)

The brain is
composed of a vast number of cells each of which has
comparatively simple dynamics. One of the fundamental problems in
neurophysiology is to understand how such a system can organise
itself to permit information processing and storage. This project
aims to look at very simple models of such coupled systems in an attempt to understand and classify the possible
types of behaviour of such systems, with a view to applying them to
more physically relevant models studied by researchers in
neurophysiology and physics.

**Spatio-temporal chaos** (PhD/M-level)

We
normally think of waves as patterns that propagate in space. Spiral
waves are such waves where one end of the wave is pinned at a `spiral
core' and the wave rotates around this. Such waves have been observed
to arise in many systems, from the behaviour of heart muscle during
heart attacks to the oxidation of carbon monoxide on catalytic
converters. This project will aim to develop a better understanding
of the existence, stability and bifurcation of such waves through the
use of dynamical system theory and symmetries.

**Nonlinear dynamics of climate models** (PhD/M-level)

Climate
systems or subsystems are often highly nonlinear with a range of
feedbacks present. This project, join with members of Exeter Climate
Systems, will look at some aspects of these models, ranging from
"tipping points" to coupled global circulation models.

**Numerical approximation of random
attractors** (PhD/M-level)

If a system is
forced by a random noise input, one might think that only statistical
models will be useful. By viewing the noise as coming from a
deterministic dynamical system we can apply a variety of techniques
of `random dynamical systems'. This project will examine the
existence of and aim to develop new theory for the behaviour of
so-called random attractors in numerical approximations of randomly
forced system.

**Dynamics in the presence of
discontinuities** (PhD/M-level)

The dynamics of
systems where all are equations are smooth is at a high level of
sophistication. By contrast, those of systems with discontinuities
are poorly understood, partly because there are many ways in which
this can happen. However there are very basic problems that remain
unsolved, for example: consider a triangle in which we play
`billiard', i.e. we draw a line inside the triangle and reflect at
each boundary it hits. It is unknown whether all triangles have a
periodic trajectory, i.e. a trajectory that repeats exactly! Similar
problems arise in the mechanics of impacting systems and digital
signal processing. There is plenty of scope in this project to
specialise on applications or to work on theoretical problems. This
project will work with the supervisor and interact with colleagues in
Exeter, San Francisco and Marseille at developing a theory for
understanding such maps.

**Perceptual rivalry ** (PhD/M-level)

For
experiments where people are shown differing images appear in each
eye, the brain attempts to make sense of the contradictory
information by alternating perception between the two different
images rather than necessarily trying to fuse them. This is a simple
test system where one can begin to understand decision making
processes within the brain and there are a variety of mathematical
models available to explain the cognitive processes involved. This
project will look at some mathematical models of processes including multi-state perceptual rivalry.

**Chaotic attractors and riddled
basins ** (PhD/M-level)

It is well known that
nonlinear iterated mappings can behave in a seemingly unpredictable
way; the phenomenon of chao attractors. Basins of attraction for
chaotic attractors can display fascinating and complicated fractal
geometry, including what has been called riddled basins. This
project aims to look at some of the theory and numerical examples of
riddled basin attractors.

**Bifurcation theory ** (PhD/M-level)

Bifurcation
theory is a powerful theory for understanding the behaviour of
systems of nonlinear differential equations on varying a parameter.
By studying the change in solutions one can better understand
fundamental instabilities in many systems, focussing on problems in coupled networks of nonlinear dynamical systems.

**Billiards in polygons ** (PhD/M-level)

We investigate the mathematics
of idealized billiards within a polygon. This is a simple model for
a one-particle gas in two dimensions where a particle travels in
straight lines between bouncing off the walls. The dynamics of the
billiard system depends critically on the shape of the polygon and
can be surprisingly non-trivial.

**Fractal dimension and measure**
(M-level)

The images of fractals (sets with dimension that is not an integer)
are used often in popular culture and for example advertisements;
they also have serious uses in science and technology. In fact there
are many different types of fractal set that can be characterised in
many different ways. This project examines some definitions and
fundamentals underlying fractal geometry and dimension before moving
on to generate and analyse examples of fractals.

**Evolutionary Game Theory ** (M-level)

This
project will look at the fundamentals and some applications of
evolutionary game theory. This has arisen from applications in
biology and economics where interactions between different species
or agents lead to them changing their strategy according to a number
of models. This project will look in particular at the `replicator
equations' and issues of whether a strategy will disappear as a
result of the evolutionary game.

**Topics in the History of
Mathematics** (M-level)

A possible topic would be a biography of a
mathematician or group of mathematicians, or the development of a
particular concept or method within mathematics. You will be
expected to read original (primary) and secondary sources related to
the topic.

P.Ashwin@ex.ac.uk Last modified: 6th June 2013