# Posters in the London Underground

March 2000: Maths Predicts
(click on the image for larger version)

The weather is notoriously hard to predict. Ever since Michael Fish famously declared on national television in October 1987 that there was going to be no hurricane, the day before the worst storms since 1703, people have been wary of weather reports. But in fact our ability to forecast the weather has improved immeasurably in the past few decades: mathematical researchers have been working with meteorologists, oceanographers and physicists since the end of World War II on the problem.

There are many difficulties in weather prediction. When it's raining in your town, it is quite possible for it to be dry (or even sunny!) just a few miles away. No TV presenter can show that level of detail on a weather map, and whatever the presenter says somebody will complain that it wasn't right. Another difficulty is that the weather is chaotic - which means that tiny changes in the atmosphere today can result in completely different weather patterns in a few days' time. This is known as the "Butterfly Effect": if a butterfly decides to flap its wings in Florida Springs then it could cause a hurricane in Spain a week later. This is one of the hallmarks of a chaotic system. The phenomemon of chaos is still not completely understood and mathematicians work on it even today.

In 1963 the metereologist Edward Lorenz, working at the Massachusetts Institute of Technology in the USA, invented and studied a simplified model of thermal convection, which can be seen as a very basic model of the weather. This model consists of only the three "differential equations" as shown on the poster:

It would take too long to explain here what the variables x, y and z describe physically, but a readable account of the derivation of these equations can be found in the book by Peitgen, Jürgens and Saupe. The greek letters (sigma), (rho) and (beta) are parameters of the system, and Lorenz used the now classic values of 10, 28 and 8/3 respectively. To his surprise he found that the equations behaved in an unpredictable way: the smallest changes in the starting conditions lead to very different evolution of the system after only a short time. Despite its simplicity, the system is chaotic.

In spite of this chaotic nature there is a remarkable structure in the equations: it is possible to find a so-called "strange attractor", which is shown on the poster as the yellow spiralling set of points. Whatever initial conditions you use, the system of equations is attracted to this set; but the motion on the attractor is very unpredictable and continually mixes around. The Lorenz attractor (as it is called today) turned out to be a prototype for chaos in other dynamical systems. Similar chaotic attractors have been found in many areas of study, for example, in mechanical, electronic and optical systems.

So Lorenz had showed that even seemingly simple systems can have astonishingly complicated dynamics. And that means that the weather is at least as complicated.

The study of the exact nature of chaos in the Lorenz equations remains an active field of research. In order to bring out the structure of the system it helps to find the "stable manifold" of the system, which is a collection of the special points which, when used as initial conditions, lead to the system ending up at another special point x = y = z = 0 (which happens to be a point of unstable equilibrium of the system). The stable manifold is difficult to compute, but a new method has been found (see the links below) which has enabled us to show it on the poster as the blue surface. It is calculated by "growing" it in concentric rings, which can be seen on the poster in different shades of blue. The poster also illustrates the importance of visualisation tools in modern mathematics.

The study of chaos is important in many fields other than the weather: the movement of share prices in the Stock Exchange, and turbulence in fluids, for instance. Much more work has yet to be done!

References:
• H.-O. Peitgen, H. Jürgens and D. Saupe: Chaos and Fractals, Springer Verlag 1992.
• J. Gleick: Chaos, the Making of a New Science, Heinemann 1987.