Bulletin

Issue: May 2006

Research in Action

Making things simpler with the theory of complex systems

What do the intricate activities in the cortex of our brain, and the turbulence that rocks jetliners, have in common?

Both, say La Trobe University mathematicians, are ‘complex systems’ whose mysteries are being unravelled by the application of the theory of complex systems, a rapidly emerging mathematical science linking pure mathematics with a myriad of applications.

La Trobe, as one of the five university members of the Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS), is playing a leading role in Australia in the application of this theory to everyday situations.

Professor of Mathematics, Reinout Quispel, is one of MASCOS’ 13 Chief Investigators and La Trobe is attracting international postgraduate students and postdoctoral research fellows in this area of study.

One currently working with Professor Quispel is Dr Lennaert van Veen whose postdoctoral studies in complex systems is sponsored 75 per cent by MASCOS and 25 per cent by La Trobe.

Dr van Veen is applying the theory in three fields: turbulence, brain dynamics and population dynamics.

However Professor Quispel and Dr van Veen say the area of application is much wider and enables mathematicians to unravel the complexities of such things as the internet, air traffic control, irrigation, robotics, power distribution, communications, manufacturing and finance.

A problem for mathematicians has been to define complex systems in terms that a lay person can understand.

Both agree with a definition published in the journal Science which described complex systems as having an evolution very sensitive to initial conditions or to small perturbations, one in which the number of independent interacting components is large, or one in which there are multiple pathways by which the system can evolve.

Dr van Veen, who obtained his PhD from the University of Utrecht before spending two years at the University of Kyoto studying fluid dynamics, is using the theory of complex systems for a new approach to turbulence.

‘Using the increased compu-tational power now available, I am examining what kind of mathematical structures are involved in turbulence, what happens at the mathematical level,’ he said.

He hopes that information gained from the new approach will have practical applications in a number of areas including reducing friction in gas and fluid pipelines and aircraft design.

Dr van Veen is also collaborating with colleagues at Swinburne University who have developed a mathe-matical model of the human cortex in their study of brain dynamics.

He is working on mathematical aspects of the functioning of the cortex – the ‘walnut’ shaped part of the brain where many vital functions such as speech are processed.

‘From a mathematical perspective, the mathematical model for turbulence and that for the cortex of the brain are very similar,’ Dr van Veen said.

He is also working with Indonesian masters student Mr Ivanky Saputry on an area of population dynamics concerning the roles of predators and prey in fish populations which may eventually have practical bearing on the fishing industry.

Professor Quispel said that the research underway at La Trobe illustrated MASCOS’s mission. This is to stimulate research activity in the mathematical and statistical modelling of complex systems and to encourage the cross fertilisation and application of ideas and techniques from different areas of mathematics to the analysis of complex systems.

The Centre has defined seven research themes chosen because of their impact on knowledge creation or applicability to industry.

They are critical phenomena; Monte Carlo methods; dynamical systems; risk modelling; scientific computation; statistical modelling of complex systems; and modelling and control of complex systems.

In addition to its research program, the Centre organises and funds conferences, seminars and courses for students, researchers and industry. It also maintains a vigorous industry outreach program, collaborating with organisations such as medical research institutes, government departments, financial institutions and manufacturing companies.

The ARC is funding MASCOS to the tune of $11 million dollars between 2003 and 2007, the largest single grant for mathematics research ever provided in Australia.