Funding

Self-funded

Project code

SMAP7480423

Department

School of Mathematics and Physics

Start dates

October, February and April

Application deadline

Applications accepted all year round

Applications are invited for a self-funded, 3-year full-time or 6-year part time PhD project.

The PhD will be based in the School of Mathematics and Physics and will be supervised by Dr Andrew Burbanks.

The work on this project could involve:

  • An in-depth review will lead to the study of pertinent problems from the analytical side and a series of numerical experiments will find approximate fixed points and evaluate their properties. This will lay the groundwork for the viability of proofs (Milestone 1).
  • Training in relevant functional analysis techniques (calculus in Banach spaces) and computational methods, will lead to adapting the existing rigorous computer framework to a new problem (Milestone 2).
  • The framework will be used to establish a rigorous existence proof with bounds on the properties of the fixed point, leading to new applications to fractal dimension, noisy systems, and others (Milestone 3).
  • The computer framework could also be generalised, resulting in the release of efficient computational tools for use in the wider subject.

Dynamical systems are used to model real-world phenomena. Even some of the simplest systems display complicated behaviour. One example is the period-doubling route to chaos, in which the system cycles-through repeating behaviour where the length of the cycle doubles as a physical parameter is varied.

This behaviour is universal, which means that the same phenomenon and the same numerical constants - governing the rate at which a system becomes chaotic - are common across a wide variety of systems, including those that model real physical, biological, meteorological, ecological, and chemical, events.

Universal behaviour is determined by the fixed points of a “renormalisation operator” that simplifies the dynamics. Much work has been done to approximate these special points numerically, to prove that they exist mathematically, and to explore their properties in isolated systems.

The team at the ϳԹ has made breakthroughs recently in adapting these techniques to produce new proofs for isolated systems and coupled systems (in which two or more subsystems interact).

We created computational frameworks to overcome two fundamental hurdles: (1) the spaces where these fixed points lie are infinite-dimensional, and (2) computer arithmetic has limited precision. We combined the resulting rigorous computer-assisted calculations with the circle of ideas around Newton’s Method and the Contraction Mapping Theorem to give existence proofs that are constructive, giving bounds on the fixed point.

This project would advance the state of the art in computer-assisted proofs in renormalisation, by tackling more complicated examples of coupled systems relevant to real-world applications.

A successful candidate will join an active research collaboration and will gain valuable mathematical and computational experience. In addition, the student can undertake training courses, from the University Graduate School, covering a wide range of transferable skills (on organising projects, giving presentations, managing time, writing-up research, analysing data, and others).

Entry requirements

You'll need a good first degree from an internationally recognised university or a Master’s degree in an appropriate subject. In exceptional cases, we may consider equivalent professional experience and/or qualifications. English language proficiency at a minimum of IELTS band 6.5 with no component score below 6.0.

You should have a strong background in Mathematics, preferably including Applied Mathematics, Mathematical Modelling, and Dynamics. You should have experience in using programming to solve problems numerically. The project will involve a combination of pure mathematical work in addition to computational work and programming. You should have the motivation to study the necessary foundations in Dynamics, Functional Analysis, and Rigorous Computation, and the ability to work both independently and in collaboration with the supervision team. Successful candidates will be expected to undertake relevant training during the PhD and to present their work at conferences and in research papers.

How to apply

We encourage you to contact Dr Andrew Burbanks (Andrew.burbanks@port.ac.uk) to discuss your interest before you apply, quoting the project code.

When you are ready to apply, please follow the 'Apply now' link on the Mathematics PhD subject area page and select the link for the relevant intake. Make sure you submit a personal statement, proof of your degrees and grades, details of two referees, proof of your English language proficiency and an up-to-date CV. Our ‘How to Apply’ page offers further guidance on the PhD application process. 

When applying please quote project code:SMAP7480423.