What is the use of thinking about chaos? Chris Budd introduces nonlinear dynamics.
In the past few years a whole new science has been created. It is often called, rather loosely, "chaos theory". It is claimed by some (usually those who do not understand it well) that chaos theory helps us to understand the unpredictable things that we see all around us, such as the stock market. Is it really possible that science has advanced this far?
Well, to be honest, no, not really. However, chaos theory (or, to give it its real title, nonlinear dynamics) really does give us a way of understanding many complex and unpredictable natural phenomena. Such understanding has many important applications. It also seems to touch a deep chord in the human psyche. (Books about chaos theory seem to appear as often in the New Age section of bookshops as they do in mathematics.) We like things to appear random; unpredictable and erratic behaviour is often much more fun than boring, predictable behaviour. Paradoxes can fascinate our pupils where logical sequences leave them cold.
In 1692 Sir Isaac Newton published the Principia, comprehensively describing for the first time the fundamental laws of motion of the universe. Before Newton the world was seen as an unpredictable place in which events seemed to happen more or less at random. After Newton, it suddenly became apparent that some at least of the phenomena of the world could not only be explained, but also its future motions predicted. Crucial to this was the understanding that predictions could be made by using maths. This achievement cannot be over-emphasised. Maths is an abstract creation of the human mind which has little, if anything, to do with reality. We have all heard maths pupils complain: "What is the possible use of all this?". Since Newton, this question can be swiftly answered: the whole physical universe works along mathematical lines.
An early success in using maths to make predictions came with the discovery of the planet Neptune. A discrepancy between the actual motion of the planet Uranus and that predicted by Newton's laws led 18th-century astronomers and mathematicians to predict the existence of a further planet pulling Uranus away from its predicted orbit. After an international race to spot the planet, Neptune was found in its predicted location.
More triumphs of mathematical prediction came in the 19th and 20th centuries. James Clerk Maxwell proved the existence of radio waves through purely mathematical reasoning. Just think of the effect of this discovery on the human race. Albert Einstein discovered the theory of relativity, which neatly plugged some gaps in Newton's theories and then allowed mathematicians to learn about the creation of the universe in the big bang and discover such exotic creatures as black holes and worm holes.
So where does chaos fit into this? At the end of the 19th century the King of Sweden offered a prize to anyone who could use Newton's laws to say whether the solar system would stay as it is for a few more billion years. Even today, armed with powerful computers, this is a very difficult question to answer, and the best we can come up with is "maybe". At that time computers did not exist.
The great French mathematician Henri Poincare thought he had solved the question. But, at the last minute, he realised his solution was wrong. He had made the profound discovery that the solar system could behave very erratically. In certain special cases its motion was chaotic and unpredictable. Everyone was profoundly shocked. How could Newton's laws, the epitome of predictability and understanding, have unpredictable answers? By trying to make sense of this, chaos theory was born. (Incidentally, Poincare went on to win the prize - and become known as the father of nonlinear dynamics and chaos theory.) To demonstrate chaos in the classroom we do not need to go all the way out to the further reaches of the solar system. We can learn a great deal from the simple pendulum. At the end of the 17th century, Galileo, watching a pendulum swing back and forth in Pisa cathedral, discovered that it had a very predictable and regular motion: its swing always took the same time and two pendulums of the same length had the same period.
Really this is remarkable, if you think of the number of things that might affect the swing of the pendulum. These could include the motion of the air, irregularities in the shape of the pendulum or differences in the strength of the push that you give it to start it off. Yet these really do seem to have very little effect on the period of the pendulum swing: the motion is repeatable, predictable and understandable, hence its effectiveness in telling the time in a grandfather clock. This seems rather different from chaos.
However, to make the pendulum behave chaotically we simply have to add another pendulum beneath it, making a double pendulum. To picture this, imagine the first pendulum as the top half of your leg; now add another pendulum jointed to the first in exactly the same way as your lower leg joints to your upper leg at your knee (figure 1 shows the double pendulum in motion). It helps if the top pendulum is about twice the weight of the bottom one and the knee joint is really smooth. If you set this up and start it swinging, something remarkable happens.
For small swings the double pendulum behaves very regularly (just like the grandfather clock), but for larger swings its behaviour is completely irregular.
The double pendulum captivates any audience with its unpredictability. It toys with them, asking "what will I do next?". This is the true essence of chaos - a simple mechanical system which we feel we should understand, yet which outsmarts us. The crazy thing about this is that we really do understand the double pendulum. We can write down equations for it based directly on Newton's laws of motion, and these equations can be solved on a computer to say what the motion of the pendulum should be.
However, the computer itself predicts that the pendulum should move in an essentially random and unpredictable fashion (figure 2 shows the chaotic motion of the bottom part of the pendulum). Even though we can "compute" the motion of the pendulum, we still cannot "predict" what the pendulum should do even a short time after we release it. Now, very small disturbances to the way we start the pendulum, or the effect of the slightest air current, rapidly get amplified and make huge differences to the final motion.
Unlike the simple pendulum, the double pendulum's motion is not repeatable, predictable or easily understood. I strongly recommend that all maths and physics teachers make a double pendulum. It demonstrates excellent scientific principles and can lead to many investigations. It also has a mesmerising property with a motion that lies somewhere between science and art. A splendid classroom pacifier!
All chaotic systems have this basic property of being unpredictable and unrepeatable, with the smallest changes having enormous effects later on. This is often called the "butterfly effect". Edward Lorenz, one of the chaos pioneers in the 1960s, captured the essence of this concept by remarking that the flap of a butterfly's wings in Borneo could lead to a hurricane in Florida. (In fact the butterfly would have to have very large wings for this to happen, but you get the idea.) Now, imagine an island with a population of rabbits x in the current year, what will the population of the rabbits be next year? We might think that there might be some rule (f) so next year's population is given by f(x). A popular rule is f(x)=ax(1-x) where a is a number which is small if the rabbits are breeding slowly, and larger if they breed likeI rabbits. Rules like this occur everywhere in science - and in many other aspects of life.
For example, it is possible that similar rules could tell us the numbers of people suffering from a disease every year or the way that tomorrow's weather depends on what has happened today. So how does our population of rabbits change from one year to the next?
To find this out, we apply the rule over and over again, so that if xn is the population of rabbits in the year n then the population xn+1 of rabbits in the year n+1 is given by xn+1 = f(xn) with f given above. It is easy to use a programmable calculator or a computer to see how the population changes. As a teaching tool it has almost unlimited potential, as students can experiment with different starting populations of rabbits x1 and different breeding rates a. (Always take x1 between 0 and 1 and a between 0 and 4.) Try it and see. If a is less than 3 then the populations settle down in a very predictable manner, and are insensitive to the starting point x1 - just as the motion of the simple pendulum was predictable.
However, for larger values of a the populations get ever more complex, and as a approaches 4 they are chaotic. Very small changes to x1 greatly affect what you see. (Figure 3 shows the chaotic population changes.) So, chaotic motion really does exist, both in nature and in maths. Why should this concern us? First, as the pendulum shows, seemingly complicated behaviour may have some simple explanation underlying it. This discovery tempts us to think that we may be able to understand other complicated things (such as the stock market) in terms of simple rules.
Sadly, this is not always the case. Many things are complicated because they are made up of the interactions of many hard-to-control events, such as clouds or (probably) the stock market. Chaos theory will not help too much here, although it has proved useful in understanding some irregular events such as disease epidemics.
Second, chaos theory is important because it shows a limit to how well we can predict (and hence control) the physical world. This is actually quite worrying when you think how much of the modern world relies on our making predictions from scientific formulae. Every time I drive my car over a bridge I rely on the car (and the bridge) obeying well-defined and predictable physical laws. Fortunately, not all physical systems are chaotic.
However, there are two chaotic systems which affect us greatly. The first is the weather. Although weather equations are pretty well understood and are solved by meteorological computers every day, it is impossible to take into account all the factors influencing the weather (remember the butterfly).
No set of data is perfect, nor are computers perfect at solving the equations. The effects of these small errors build up remarkably quickly. After about 10 days it is essentially impossible to forecast weather with any degree of accuracy. This fundamental long-term unpredictability of the weather makes assessing climate change very difficult.
Chaos in the solar system is also key. While the motion of the planets is very predictable, the motion of many of the asteroids is not. Although asteroids do obey Newton's laws, they may well have orbits which move erratically about space. Such an erratic asteroid is thought to have hit the Earth 65 million years ago and wiped out the dinosaurs.
The consequences of a similar incident occurring today are unthinkable; the nature of chaotic motion means that such events are virtually impossible to predict until too late.
Chaos raises many interesting philosophical issues and can lead to many fruitful mathematical investigations. But is it of any use? Yes, it is. The heart is thought to behave chaotically when it goes into fibrillation (quivering) after a heart attack. Intense study is going on to see whether chaos theory can help predict this and to design pacemakers to restore a fibrillating heart to its normal regularity.
Car brakes squealing occur in a random, chaotic pattern: in this case, randomness is less destructive than regular mechanical wearing. Modern encryption uses chaotic signals. Fractals (closely related to chaotic maps) are very important in computer graphics. Lasers, power systems, fluid motions, disease epidemics, car suspensions, capsizing ships and particle accelerators, all depend on chaos.
However, it cannot be used to explain the random behaviour of a child in a classroom - only a good teacher can do that.
* For more information, try the website http:nrich.maths.org.uk for a feast of mathematical articles, including many about chaos. The book Does God Play Dice, by Ian Stewart(Penguin, pound;8.99) gives a highly readable account about the theory and applications of chaos.
Chris Budd is professor of applied mathematics in the Department of Mathematical Sciences,University of Bath, Claverton Down, Bath BA2 7AY and chair of mathematics at the Royal Institution of the UK, Albemarle Street, London.He has recently published, with ChristopherSangwin, Mathematics Galore! (Oxford University Press, pound;14.95)