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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.

  • How do you determine invariant measures numerically?
  • Given a dynamical system (or orbit data only), what can be learned computationally about the rates of correlation decay within the system? are there barriers to efficient mixing?
  • What almost invariant structures exist within a system?
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 Apr 2014 , supervision team: 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). (Supervision team: Tim David, Rua Murray)
  • Sophia Di. Escape rates and metastability in open dynamical systems. (Math hons, 2013, now at MetService)
  • Andreas Kempe. Simple models of contagion in complex banking networks. (Math hons 2013, now studying Masters of Actuarial Science, Cass Business School, London)
  • Mohd Hafiz Mohd. Detailed analysis of biologically relevant reaction diffusion systems (Math PhD May 2013 onwards, supervision team: Rua Murray, Mike Plank, Will Godsoe)
  • 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, supervision team: Tim David, Rua Murray)
  • Jacky Sung. Reconstructing Distributions from Option Prices. (MSc, May 2013, now analyst at ACC)
  • James Williams. L2 quantization of area preserving maps. (Math hons 2011, now Fullbright Scholar at Yale University).

Sample projects: Honours/Masters/PhD

Rigourous analysis of a mechatronically measurable double pendulum. My colleague Raazesh Sainudiin has built a measurable double pendulum; see here. This device can be modelled with a chaotic low-dimensional system of ODEs, but it can also be used to produce large data streams. We are seeking a student to apply methods of symbolic dynamics and ergodic theory to develop new computational methods to match up the predictions of theory with real experimental data! This project will be cosupervised by Rua Murray and Raazesh Sainudiin.

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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