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# Prof Peter Ashwin

## PhD/Masters projects

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 many neurons 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 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 for differential equations (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.