Mathematics and Statistics

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Prof Peter Ashwin

Professor

 P.Ashwin@exeter.ac.uk

 (Streatham) 5225

 01392 725225

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Overview

I am a Professor of Mathematics specializing in dynamical systems theory and computational modelling. I have been at Exeter since 2000, having previously held teaching and research positions at Surrey, Warwick, Nice (France) and Marburg (Germany).

Research Interests

My main interests are in nonlinear dynamical systems and applications: bifurcation theory and dynamical systems, especially synchronization problems, symmetric chaotic dynamics and spatially extended systems and nonautonomous systems. Applications of dynamical systems that I have been studying include climate (bifurcations, tipping points), fluids (bifurcations and mixing), laser systems (synchronization), neural systems (perceptual rivalry, intermittent models), materials and electronic systems (digital signal processing), biophysical modelling (cell biology).

Particular themes running through my work include symmetries and intermittent behaviour, and the structure of attractors, in particular, riddled basins of attraction and associated phenomena. I also have interests in random dynamics/stochastically forced systems and in low dimensional dynamics/ergodic theory.

Current research projects include the mathematical modelling of paleoclimate transitions, molecular networks, phase change materials in data storage media, using heteroclinic networks to model functional dynamics in neural and other biomedical systems and tipping points in nonautonomous systems with applications in climate and finance; for more detail see my publications.

In 2012 I was a co-founder of the "Maths for Climate" network "CliMathNet" with EPSRC funding. This continues as an international network of researchers interested in developing and applying cuttiing-edge mathematics and statistics to the Climate system. During 2015-2018 I led the project "ReCoVER" with Prof Tim Lenton (Exeter).  I was Exeter coordinator of the EU Marie Curie ITN programme "CRITICS" looking at mathematics and applications of critical transitions (2015-2019). I was Co-I for a Leverhulme-funded project on ER-organelle interaction (PI: Imogen Sparkes) since 2015. I was Co-I for the EPSRC-funded "Centre for Predictive Modelling in Healthcare" (2016-2020), and the Exeter coordinator of the EU funded "TIPES: Tipping points in the Earth System" that launched in 2019. I am currently PI of the EPSRC funded project "Applied Nonlinear Nonautonomous Systems: Theory, Methods and Applications" 2020-2023 and am the Exeter PI for the EU Marie Curie ITN programme "CriticalEarth" running 2021-2025.

I am happy to supervise MMath, MSc or PhD projects in the areas of my research interests, or related areas of mathematics. Any available positions are advertised on the Centre for Systems Dynamics and Control blog.

Teaching Interests

I am currently teaching on:

  • MTH3019 Mathematics; History and Culture
  • MAGIC020 Dynamical Systems

I am also Director of the MAGIC mathematics PhD taught course centre, a consortium of UK universities.

Other Relevant Information

Since 2022 I am Director of Research and Impact within the Department of Mathematics and Statistics at Exeter.

I am Associate Editor for the journals Chaos: An interdisciplinary journal of nonlinear science, Dynamical Systems; an International Journal and Mathematical Neuroscience and Applications and a member of the EPSRC peer review college. I have been an editor for Journal of Mathematical Neuroscience and Frontiers in Applied Mathematics and Statistics. I served as Head of Mathematics and Computer Science at Exeter for the period 2010-14. I was secretary for the European Dynamics Days 2001-2008 and have contributed to activites of the SIAM activity groups on Dynamical Systems and Mathematics of Planet Earth. For 2021-24 I am serving on the Council of the London Mathematical Society.

Qualifications

MA (Camb), CAS (Camb), PhD (Warwick), Senior Fellow of Higher Education Academy

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Publications

Copyright Notice: Any articles made available for download are for personal use only. Any other use requires prior permission of the author and the copyright holder.

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Further information

Personal Homepage

Research links Journals/Newsletters Current PhD students as main supervisor
  • Ruth Chapman
  • Saleh Alharthi
  • Mohamed Fadera
  • Raphael Roemer
  • Said Amhemed Elahjel
Current postdocs/fellows
  • Julian Newman
  • Nicolas Verscheuren van Rees
Previous PhD students as main supervisor
  • John Terry
  • Xin-Chu Fu
  • Jon Borresen
  • David Hawker
  • Marcello Trovati
  • Abul Kalam Al-Azad
  • John Wordsworth
  • Ozkan Karabacak
  • Congping Lin
  • Jorge Vazquez Diosdado
  • Asma Ismail
  • Ummu Atiqah Mohd Roslan
  • Mary Thoubaan
  • Damian Smug
  • Hassan Alkhayuon
  • Saad Almuaddi
  • Jordan Moore
  • Ibrahim Alraddadi
  • Thoraya Alharthi
Previous postdoc/fellows
  • Xin-Chu Fu
  • Rob Sturman
  • Gabor Orosz
  • Konstantin Blyuss
  • Prasad Patnaik Behara
  • Chris Bick
  • Congping Lin
  • Jen Creaser
  • Camille Poignard
  • Tanja Zerenner
  • Lea Oljaca
Older links to events etc Last edit: P.Ashwin@exeter.ac.uk 2023

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.

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You can book a personal tutorial meeting with me via this link.

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