ARC Complex Open Systems Research Network

Prof Anthony Roberts

 

Professor Dept of Mathematics & Computing. Address: A.J. Roberts Dept of Mathematics & Computing University of Southern Queensland Toowoomba, Queensland 4350 Australia Phone: +61 7 4631 5539 Fax: +61 7 4631 5550 Email: aroberts@usq.edu.au Webpage: http://www.sci.usq.edu.au/staff/aroberts Research Node: QLD-2

 

Role in Network

COSNet Management Committee Member - Node Coordinator (Alternate) QLD-2
Academic - Researcher

COSNet Research Themes

1. Irreversibility and Emergence in Nonequilibrium Systems
2. Turbulence and Coherent Structures, Control and Computation
3. Dynamics and Statistics of Multi-Scale Systems

COSNet Application Areas

1. Complex Physical Systems
3. Complex Computational Systems

 

Research Topics

I am the world leader in using and further developing a branch of modern dynamical systems theory, in conjunction with new computer algebra algorithms, to derive mathematical models of complex systems. My research in this field developed upon earlier significant research into the mathematical analysis of fluid dynamics. Also I have wide ranging interests including multifractals, traditional numerical methods, parallel computer science, industrial applications, applied probability and game theory.

The most significant achievement is my development of theory and techniques, based on centre manifold theory, for the rational and complete low-dimensional modelling of complicated dynamical systems. These techniques have been applied to a variety of physical problems (such as turbulent floods, thin film flows, fractionation, hermoviscoelasticity, quasi-geostrophic approximation and numerical discretisation) and lead to many new insights (such as the rational provision of initial condition, boundary conditions and forcing). Further, the systematic basis of the techniques allows us to devise new models in ways previously unimagined (such as embedding the centre manifold and approximating inertial manifolds).

Much of this work is a decade ahead of others: for example, Temam et al (2002) recently used normal forms to explore the quasi-geostrophic approximation, whereas Cox & I did so circa 1990. My research has been recognised by invitations to lecture, for example, in the ANU Dynamic Summer, at Oberwolfach, at the University of Marburg, the program at Woods Hole, and at the Fields Institute in Toronto.  

Publications

See the mostly complete list of publications.
1. W. R. Young, A. J. Roberts, and G. Stuhne. Reproductive pair correlations, brownian bugs and plankton patches. Nature, 412:328--331, 2001.
2. R. Valery Roy, A. J. Roberts, and M. E. Simpson. A lubrication model of coating flows over a curved substrate in space. J. Fluid Mech., 454:235--261, 2002
3. Z. Mei, A. J. Roberts, and Zhenquan Li. Modelling the dynamics of turbulent floods. SIAM J. Appl. Math. 63:423--458, 2003.
4. A. J. Roberts. The utility of an invariant manifold description of the evolution of a dynamical system. SIAM J.Math. Anal., 20:1447--1458, 1989.
5. A. J. Roberts. Highly nonlinear short-crested water waves. J. Fluid Mech., 135:301--321, 1983.  

End-User Applications

I make some of my developed software available to all for all purposes. Software (computer routines to do various
taks such as estimate fractal dimensions, solve DAEs, etc, mainly in matlab)

 

Professional or Public Outreach Activities

An introduction to the elements of using LaTeX. This is a set of web documents shows how to transform
plain text into a beautifully typeset document in LaTeX. It introduces what I consider are the most important fundamentals of LaTeX2e.

Lego fractals may be constructed to stun your friends.

I was the director of the 1993 Applied Mathematics Conference, held in Hahndorf, South Australia.
organised the 1996 miniconference for the Queensland branch of ANZIAM.
was the Programme Coordinator for the 34th Applied Mathematics Conferenceheld in Coolangatta in 1998.
Initiated and helped Markus Hegland organise the Summer school in Computational, Industrial and Applied Mathematics held at anu in January, 2002.
Addressed the ANZAAS '91 congress on visualising chaos.
Gave two talks on mathematics and chaos to the demonstrators of Adelaide's then new Investigator Science and Technology Centre. One talk was about the items in the Mathematica exhibition and some of their subtleties. The other talk was on Chaos in nature.
Have given a number of public talks on fractals, the substance of which was published in Australian Science.
Have written another article for Australian Science on Modelling and Simulation.
Have presented several activities to high school students as part of the Toowoomba Mathematics Camp.
Have talked to the Toowoomba branch of the Australian Association of Mathematics Teachers about the nature and role of research in mathematics.
Have talked to a regional meeting of Science teachers about Roger Penrose's fascinating thesis on the nature of human intelligence.
Successfully requested the questacon travelling maths exhibition to visit Toowoomba in 1998  

Complex Systems Software

• Holistic discretisation of dynamical Partial Differential Equations obtain the spatial discretisation of a dynamical PDE using dynamical systems theory.


• Book of multiscale modeliing of dynamics Dynamically generate a book on using modern dynamical techniques of slow manifolds, normal forms and computer algebra. Illustrates multiple scales, stochastic system modelling and numerical discretisations.


• Submit your data to analyse for multifractality Since it is notoriously difficult to analyse data for multifractal structure, submit your 2D data for me to find its maximum likelihood quarternary multiplicative multifractal approximation.


• Normal form of your system of SDEs or ODEs with multiscale fast and slow variables Submit your ODE or (Stratonovich) SDE and find its normal form that separates the dynamics into its true slow and fast variables.

 

Membership/Fellowship of Key Organisations

member of the Division of Applied Mathematics of the Australian Mathematical Society
member of the Australian Society for Operations Research.
was a member of the Council of the Australian Mathematical Society from July 1993 to 1996.
on the Council of the Austral Math Soc from July 1997 in my role as Electronic Editor for the Society.
served on the executive committee of the Division in 1989--90, and am still serving on the occasional supercomputer sub-committee of the division.
fellow of Trinity College, Cambridge,  

Awards and Distinctions

Awarded USQ Award for Excellence in Research
Member ARC College of Experts 2005--2007
Electronic Editor Australian Mathematical Society 1997-->