Causodynamics of waves and patterns. PhD Mathematics

The University of Exeter EPSRC DTP (Engineering and Physical Sciences Research Council Doctoral Training Partnership) is offering up to 4 fully funded doctoral studentships for 2019/20 entry. Students will be given sector-leading training and development with outstanding facilities and resources. Studentships will be awarded to outstanding applicants, the distribution will be overseen by the University’s EPSRC Strategy Group in partnership with the Doctoral College.


Supervisors:
Professor Vadim Biktashev, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences
Professor Peter Ashwin, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences

Project description:
Self-organization is a process whereby order arises due to interactions between parts of an initially disordered system. Self-organisation may be decentralised, when various parts of the system make comparable contribution to the emergent properties; alternatively, the system may spontaneously partition itself into a relatively small part serving as an organising centre, and the periphery, i.e. the rest. Processes in the organising centre dominate what happens to the rest of the system, while processes in the periphery are largely without consequences. Examples of organising centres include spontaneous pattern formation in social amoebae, cardiac arrhythmias and, speculatively, also epileptic seizures and hurricanes. In some cases distinguishing between the organising centre and the enslaved periphery may be intuitively evident, but in more complicated cases intuition may fail and rigorous methods are then required. Improving our understanding of these mechanisms from mathematical viewpoint, and learning methods to best diagnose and control such systems is therefore important both for fundamental science and for applications.

The proposed PhD project will be concerned with investigation of causal links underlying selforganization in models used to describe waves and patterns in mathematical biology, such as partial differential equations, particularly of "reaction-diffusion" type, and possibly also discrete network models. Specifically, the project will focus on sensitivity of established and transient solutions in such models to various sorts of small perturbations. This sensitivity will be determined by the method of ’causodynamics’, which involves integration of the adjoint linearized equations backwards in time. A few selected examples will be considered during the project. Tentatively, the initial stage
will consider Turing patterns and propagating pulses in one spatial dimension and spiral wave solutions in two spatial dimensions. At the next stage, spatio-temporal chaos (such as generated by Kuramoto-Sivashinsky equation) in one spatial dimension will be considered. Further examples will concentrate on more complicated regimes, relevant for modelling cardiac fibrillation, such as: competing spiral waves, ’mother rotor’ regimes, and two- and three-dimensional spiral and scroll wave turbulences of various sorts. The overall aim in each case will be to develop diagnostic criteria for organising centres, and to characterize the possibilities to control such regimes by
small perturbations.

Candidate requirements:
The project will require from the candidate some fundamental mathematical knowledge, including linear algebra, basic dynamical systems theory, asymptotic methods for ordinary and partial differential equations, and possibly some elements of functional analysis. It will also
require suitable IT skills, including numerical solution of ordinary and partial differential equations. Knowledge from appropriate subject areas, e.g. mathematical biology, will be beneficial but is not strictly necessary. The candidate should be prepared to learn necessary
disciplines and skills during the project if they do not possess them already.