I am an applied analyst, and my research field is nonlinear dynamical systems, in particular the theory of renormalization and its applications to mathematical physics. Renormalization is the theory that underlies universal metrical behaviour in critical phenomena such as the transition to chaotic dynamics.

Typically renormalization theory leads to functional equations and the study of these equations forms the backbone of my research work. In addition to research in nonlinear dynamical systems, I have an interest in the application of mathematics in energy systems and in the study of asset bubbles in financial systems.

Some further details of my research interests are given below.

Quasiperiodic renormalization

One of my principal interests is the application of renormalization techniques to the study of correlations in quasiperiodic quantum systems, strange nonchaotic attractors and the barrier billiard system. An early highlight of this research was the analysis of the so-called Ketoja-Satija orchid flower, a renormalization strange set, which governs the correlations of the fluctuations of the wave function for a generalized Harper equation with golden-mean frequency ratio. The SIAM orchid flower was used for the poster for the Snowbird meeting in 2007.

Poster for 2007 SIAM Conference on Applications of Dynamical Systems in Snowbird, Utah, USA, feating the Ketoja-Satija orchd flower.
Poster for 2007 the SIAM Snowbird Conference on Applications of Dynamical Systems, featuring the Ketoja-Satija orchid flower.

One exciting byproduct of this project has been the analysis by my former student Paul Verschueren of the so-called Sudler sine product for golden-mean rotation number.  The product has application to diverse fields such as partition theory, Páde approximation & continued fractions, KAM theory and string theory, as well the renormalization theory of strange nonchaotic dynamical systems.

Energy Research

My research in energy systems has followed two strands. The first, in which I have collaborated with colleagues and students at the OU has used System Dynamics modelling to analyse the long-term impact of renewable-energy policy options on the electricity system of São Miguel in the Azores, an exemplar ‘small island’ system.

The second strand has focussed on the interaction of system stability, energy storage and market price, studying, for example the optimal charging/discharging strategy for a storage battery operator and studying the use of storage to stabilise the electricity system, focussing on adaptive control via a price signal sensitive to the mismatch between conventional generation and load (net of renewables) while incentivising storage providers via arbitrage opportunities.

Webpage advertising the Workshop on the Mathematics of Demand Side Management and Energy Storage, held at the Open University, in June 2015.
Webpage advertising the Workshop on the Mathematics of Demand Side Management and Energy Storage, held at the Open University, in June 2015.

I also have been keen to promote energy research. I helped organise the Energy Systems Week at the Newton Institute in 2011, a multidisciplinary event that brought together mathematicians and economists with representatives of industry, regulatory authorities and government. At the Open University, I have worked on the Open Energy initiative with colleagues in Engineering (most notably William Nuttall). I was the principal local organiser of three meetings on energy research in the period 2013 – 2015 and, in 2016, I co-organised a joint India-UK workshop on energy management at the International Centre for Mathematical Sciences in Edinburgh.