Possible research student projects
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?
Current students
- Janice Asuncion (Mech Eng). Long Term Dynamics of Freight Transportation and Production Driven by Fuel and Emission Constraints (PhD Apr 2010 onwards, supervision team: Susan Krumdieck, Eli van Houten, Rua Murray)
- Kenneth Churcher (UC Summer Scholar 2011-2012). Large-scale networks of nonlinear oscillators: parameter gradients, spatio-temporal decoherence and chaos. (Supervision team: Tim David, Rua Murray)
- Jacky Sung (Math&Stat). New mathematical and statistical models of financial processes. (PhD April 2011 onwards, supervision team: Rua Murray, Marco Reale, Carl Scarrott)
- James Williams (Math&Stat). L2 quantization of area preserving maps. (Honours 2011).
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.
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!