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Dynamical systems and ergodic theory

Dynamical systems is the study of any mathematical system where spatial structure evolves with time. Modern approaches may be geometric, topological, probabilistic or computational. Much research combines several of these.

Ergodic theory is the study of dynamical systems from a probabilistic point of view: rather than tracking individual orbits one wishes to make statements about entire families of initial conditions. I am particularly interested in numerical methods that allow the insights of ergodic theory to be applied.

I can supervise a range of projects in theoretical and/or computational ergodic theory. A few ideas are listed below. I am also able to co-supervise projects in applied dynamics (particularly Bioengineering) and financial mathematics.

Advice for potential International Masters and PhD students

I am available to supervise motivated students, and especially welcome enquiries from well-prepared and enthusiastic candidates from all nationalities.

If you are considering studying in New Zealand, and would like to study under my supervision, please make sure that your initial contact includes ALL of the following information:

  • A description of your University education, including graduate coursework undertaken - for the upper level courses, supply a few sentences describing the contents of each course, and which chapters of textbooks you used
  • A brief of outline of the research problems you are interested in tackling - please be reasonably specific (up to one page of explanation)
  • For PhD applicants: a copy of your Masters thesis (if you are presently working on a Masters thesis, a brief description of its contents)
  • A clear statement about what scholarship support you have already obtained, or applied for
Please note: The decision about admission is made by the University of Canterbury, not me! All students must meet admission requirements. Before seeking a supervisor, please check the Postgraduate website for admission information.

Current and recent students

  • Janice Asuncion (Mech Eng). Long Term Dynamics of Freight Transportation and Production Driven by Fuel and Emission Constraints (PhD, completed 2014, supervisors: Susan Krumdieck, Rua Murray, Shannon Page - Lincoln)
  • Kenneth Churcher. Large-scale networks of nonlinear oscillators: parameter gradients, spatio-temporal decoherence and chaos. (UC Summer Scholar 2011-2012, now a game developer, supervisors: Tim David, Rua Murray)
  • Sophia Di. Escape rates and metastability in open dynamical systems. (Math hons, 2013)
  • Ken Fisher. Excitability in random dynamical systems. (Math hons, 2017)
  • Patricio Gallardo. (Mech Eng) Energy transition for freight movements: data mining, modelling tools and policy scenarios - with case study in Ecuador (PhD, Feb 2017-, supervisors: Susan Krumdieck, Rua Murray)
  • Michelle Goodman. (Bioengineering) Ionic Concentration Dynamics and Wave Propagation in Spatial Media. (PhD Feb 2015-, supervisors: Tim David, Rua Murray, Paul Docherty)
  • Benjamin Jeffrey. Indentification of coherent structures within a flow using the Ulam matrix method. (Math hons 2016, supervisors: Rua Murray, Miguel Moyers-Gonzalez, Phil Wilson)
  • Andreas Kempe. Simple models of contagion in complex banking networks. (Math hons 2013, then Masters of Actuarial Science, Cass Business School, London)
  • Ben Litchfield.Implementation of chaos-based communications systems in real time (Math/SERC MSc completed 2017, supervisors: Rua Murray, Graeme Woodward, Kelvin Barnsdale, Branislav Jovic)
  • Mohd Hafiz Mohd. Detailed analysis of biologically relevant reaction diffusion systems (PhD completed 2016, supervisors: Rua Murray, Mike Plank, Will Godsoe, now at USM Malaysia)
  • Julie Mugford. Coupled cell dynamics. (Math hons, 2013, supervisors: Rua Murray, Tim David, Mike Plank)
  • Jimmy Shek (BlueFern HPC). Homogenised models of smooth muscle and endothelial cells (Masters in Bioengineering, 2012-2013, supervisors: Tim David, Rua Murray)
  • Jacky Sung. Reconstructing Distributions from Option Prices. (Math MSc, completed 2013, now analyst at ACC)
  • James Williams. L2 quantization of area preserving maps. (Math hons 2011, Fullbright Scholar and PhD from Yale University).

Sample projects: Honours/Masters/PhD

Optimisation methods in transient dynamics. The traditional focus of dynamical systems and ergodic theory is on determining asymptotic behaviour (ie infinite time). However, many interesting features are not amenable to this kind of analysis, and important features persist only for finite times (eg "coherent structures" blocking mixing in ocean systems, cells in your body responding to changing ion concentrations). This project will involve numerical work to test out some new optimisation based methods for computing "locally invariant" structures.

Mathematical physiology and high-performance computing. Modern physiological modelling presents a myriad of fascinating and difficult mathematical and computational challenges. I am available to co-supervise projects in Bioengineering. See Bioengineering at UC.

Sample projects: Honours/Masters

Escape from a dynamical system. Many dynamical systems have holes! The system may be born with these (in the case of a system of hard scatterers, or a billiard table --- so called ``open systems"), or holes may be introduced as leaks from a closed dynamical system. Typical orbits escape from open systems with a characteristic exponential rate, depending on delicate nonlinear phenomena. The details of this project are flexible, but will include some exciting mathematics!

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