Research in Applied Mathematics

This is the webpage of researchers in applied mathematics at Massey University in Palmerston North. Our research covers a diverse range of both mathematical methods and applications. Below are some specific topics. Our group also has research in the areas of Numerical Analysis, Combinatorics and topology, Biological physics, and Mathematical physics.


Modelling cells and fluids
Contact: Richard Brown

My research is in a diverse range of application areas, but the common thread is finding interesting practical problems which benefit from the application of mathematical techniques. As an applied mathematician, I draw on tools from different areas of mathematics: stochastic processes, differential equations, linear algebra, geometry, and optimisation.

A selection of applications I have worked on include using glider flight data to reconstruct lee wave windflow patterns near mountain ranges, using computer vision techniques to reconstruct breast surface motion for the DIET breast cancer screening system, and using mathematical models to study the effect of different management strategies on controlling invasive weed species.

Presently I'm working on the modelling of autoregulation of blood flow in the human brain. This project has numerous aspects to it, from the development and analysis of differential equation models of biochemical cells, to developing large-scale computational algorithms for simulating the behaviour of blood vessel networks, both at PC and supercomputer scale.

Some recent research:

  1. Habitat fragmentation: Simple models for local persistence and the spread of invasive species.
    Simulation of random biocontrol of an plant invasion
  2. A pointwise smooth surface stereo reconstruction algorithm without correspondences.
    3D reconstruction of a viscoelastic breast phantom used in the development of the DIET breast cancer screening system
  3. Numerical techniques for dynamic resistive networks.
    The image below shows a simplified model of a vascular tree embedded in cortical tissue.

Modelling powders and fluids
Contact: Luke Fullard

My research interest lies in the application of mathematics to solve real life problems. I especially enjoy modelling the flow of fluids and powders. My research interests include:

The following plots come from recent work on modelling powder in a conical hopper. These show the time it takes for the powder to discharge from the hopper at different locations.


Nonsmooth dynamical systems
Contact: David Simpson

Nonsmooth dynamical systems are mathematical models lacking global continuity or differentiability. Such systems are increasingly being used to model the complex dynamics of phenomena involving abrupt events, such as impacts and stick-slip friction in mechanical systems, rapid switching in circuits and control systems, and decisions in ecological systems and social sciences. Many core aspects of classical dynamical systems theory (such as centre manifold reduction) simply does not apply to nonsmooth systems, and nonsmooth dynamics is currently a highly active area of research.

Some possible thesis topics:

  1. The fractal structure of mode-locking regions for piecewise-linear maps.
    Oscillators connected by weak interactive forces may synchronise or mode-lock. This was reported by Christiaan Huygens in the 17th century for two pendulum clocks attached to a single wooden beam, and is nowadays recognised as an important phenomenon in various fields, particularly mechanics and neuroscience. Regions of parameter space for which the system mode-locks to a certain period are known as Arnold tongues. This name is a due to the theoretical work of Vladimir Arnold, and, with some imagination, the observation that such regions often have a tongue shape.
    For systems modelled by piecewise-linear continuous maps, Arnold tongues typically have points of zero width (called shrinking points) and a curious chain structure. This is seen for the figure below. Here different tongues are shaded with different colours. Such a chain structure has been described in models of neurons, DC/DC power converters, and business cycles. For the most part, the behaviour of a system near a single shrinking point is well understood, but many important problems relating to the global structure of Arnold tongues in piecewise-linear maps remain unsolved. For instance, are shrinking points distributed according to rules that are universal and perhaps fractal?
  2. Arbitrarily many coexisting attractors in piecewise-linear maps.
    Dynamical systems with more than one attracting solution have the problem that the long-time behaviour of the system depends heavily on the initial data. This issue is exasperated when many attractors are present. For continuous, piecewise linear maps (which arise in models of diverse phenomena, such as predator-prey systems, mechanical oscillators with friction, and relay control systems), there are some isolated results but a general theory is lacking.
    In particular, it has recently been shown that two-dimensional, continuous, piecewise-linear maps may have infinitely many attractors. Here is an example:
    Iterations of this map can approach one of infinitely many periodic solutions depending on the starting values. Here is a plot of the basins of attraction for a similar example (different colours indicate different basins):
  3. Noisy sliding motion.
    Sliding motion is evolution on the discontinuity surface of a system of discontinuous differential equations. Such dynamics arises in control theory: so-called "sliding-mode control" specifically utilises sliding motion to achieve a control objective. The goal of this project is to understand exactly how sliding motion is affected by noise or randomness of different types.

Some recent research:

  1. A classification of border-collision bifurcations that is independent of dimension.
    Unlike smooth maps, xi+1= f(xi), piecewise-smooth maps, such the as two-dimensional map given above, can exhibit so-called border-collision bifurcations at which the dynamics changes in a possibly extremely complicated way. For instance infinitely attractors can be created (see above). With more dimensions (i.e. more dependent variables) more complications seem possible. Yet if one looks only at fixed points and period-2 solutions then border-collision bifurcations can be grouped into exactly four cases, see:
  2. The combined effects of smoothing and noise on discontinuous differential equations.
    Discontinuous, piecewise-smooth differential equations are used to model a wide variety of systems that, in reality, are neither deterministic or nonsmooth, when examined on an appropriately fine scale. Thus a smooth (but highly stiff) and stochastic model should provide a more accurate representation of the true system. In a general setting we showed that although the smoothing of a piecewise-smooth system can introduce spurious dynamics, such dynamics is washed out by the noise. Thus the predictions of the smoothed stochastic model and the original nonsmooth deterministic model are roughly the same. This justifies the use of the original model which is simpler and much easier to work with. We applied the results to a model of a dry-friction oscillator, see:
  3. Stochastic grazing bifurcations.
    For impacting systems, so-called "grazing bifurcations" correspond to the onset of recurring impacts. We showed that different assumptions on sources of randomness and uncertainty lead to different dynamical behaviour near grazing bifurcations and explained this via three different stochastic return maps, see:
    • D.J.W. Simpson and R. Kuske. The Influence of Localised Randomness on Regular Grazing Bifurcations with Applications to Impacting Dynamics. To appear: J. Vib. Contr.

Numerical PDEs
Contact: Igor Boglaev

Development of monotone alternating direction implicit (ADI) schemes for numerical solving systems of nonlinear parabolic equations. Development of uniformly convergent monotone ADI schemes for numerical solving singularly perturbed systems of nonlinear parabolic equations. Development of inexact monotone iterative methods for numerical solving nonlinear elliptic and parabolic problems.

Some possible thesis topics:

  1. Novel numerical methods for solving nonlinear partial differential equations.
  2. Uniformly convergent numerical methods for numerical solving singularly perturbed differential equations.

Some recent research:

  1. I. Boglaev. A uniform monotone alternating direction (ADI) scheme for nonlinear singularly perturbed parabolic problems. J. Comput. Appl. Math., 272:148-161, 2014. doi:10.1016/j.cam.2014.05.007
  2. I. Boglaev. Inexact block monotone methods for solving nonlinear elliptic problems. J. Comput. Appl. Math., 269:109-117, 2014. doi:10.1016/j.cam.2014.03.029

Applied analysis
Contact: Bruce van Brunt


This page maintained by David Simpson. Last modified on May 30, 2016.