Renormalisation in dynamical systems

The PhD will be based in the School of Computing, Mathematics and Physics and will be supervised by Dr Andrew Burbanks, Dr Thomas Kecker and Maria Pickett.

This project uses a fascinating combination of functional analysis, complex analysis, and rigorous validated numerics to prove that certain types of behaviour are universal in dynamical systems used to model the World.

Project Highlights:

The work on this project will include:

• Investigating universal behaviour in nonlinear dynamical systems and estimating the corresponding universal constants
• Proving the existence of fixed-points of renormalisation operators via rigorous computer-assisted proofs
• Bounding the spectrum of the derivative of renormalisation operators at fixed points, in order to prove universality

Project description:

We will prove the existence of objects, known as renormalisation fixed points, that are crucial to understanding how physical systems undergo a transition from predictable to chaotic behaviour. Our approach is extremely powerful: results gained via renormalisation techniques are often universal – they apply to an enormous range of physical, biological, meteorological, ecological, chemical, and mathematical systems.

A classic example is provided by period doubling. In this scenario, a system undergoes cyclic behaviour that repeats over ever-longer time intervals leading eventually to chaos. Observations of mathematical models and physical experiments reveal that features of this transition are universal; the same qualitative features and quantitative measurements emerge across an enormous number of apparently unrelated models and experiments.

Renormalisation provides a means to explore and explain this universality by examining the properties of a renormalisation operator that we can think of as a simplifying transformation that preserves phenomena of interest.

More complicated examples occur when two systems each have their own intrinsic dynamics, but one system drives, or forces, the behaviour of the other. Such systems have their own universal features. A corresponding renormalisation analysis can reveal and explain the behaviour of a wide variety of physical and mathematical systems that have this structure.

Our central challenges are to prove that fixed points of renormalisation operators exist, to gain rigorous bounds on their properties, and to use this information to deduce new results about broad classes of systems. Historically, analytical proofs of such results have been extremely difficult to come by; a number of problems remain open after several decades. Instead, some of these questions have been settled via rigorous computer-assisted proofs. We will take this approach, which leads to existence proofs that are constructive, yielding rigorous bounds on the objects concerned, that may then be used in further analysis.

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 Doctoral College, covering a wide range of transferable skills (on organising projects, giving presentations, managing time, writing-up research, analysing data, and others).

General admissions criteria

You'll need a good first degree from an internationally recognised university (minimum upper second class or equivalent, depending on your chosen course) 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.

International students will require a study visa from UKVI to pursue the degree in the UK. If the research is in a sensitive or technological subject, the student may also need to secure an Academic Technology Approval Scheme (ATAS) certificate from the UK Foreign Office.

How to Apply

We’d 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 MAP10600529.

Funding Notes

Funding Availability: Self-funded PhD students only

PhD full-time and part-time courses are eligible for the UK Government Doctoral Loan (UK and EU students only - eligibility criteria apply).

Bench fees

Some PhD projects may include additional fees – known as bench fees – for equipment and other consumables, and these will be added to your standard tuition fee. Speak to the supervisory team during your interview about any additional fees you may have to pay. Please note, bench fees are not eligible for discounts and are non-refundable.