Chaos Theory Demystified

To most of us, the term feedback conjures up memories of the deafening screech you hear from a speaker placed too close to a microphone. Let’s think about what causes this: the hum from a speaker gets picked up by a microphone; the microphone feeds the signal to an amplifier, which makes the hum louder; the louder hum comes out of the speaker and is picked up by the microphone and amplified again. Soon an imperceptible hum has become a deafening screech. The general definition of feedback in a system is that the output “feeds back” into the input. Positive feedback occurs when the system acts as an amplifier, so that small changes in input become bigger changes in output, which become bigger changes in input, which…you get the picture!

The weather is full of these kinds of feedback loops. For example, a cloudless day over England causes the sun to warm the ground; the warm earth heats the surrounding air, which then expands, creating a region of high pressure; high pressure over England causes clouds blowing in from the Atlantic to be deflected to continental Europe, so England has more cloudless, sunny days.

If you could know the present state of the weather with infinite precision at every point in the Earth’s atmosphere, then you would, in theory, be able to predict the future precisely. However, there are uncertainties in any scientific measurement and you can’t put a sensor at every point in the atmosphere (you’d need an infinite number of them). In a system with feedback, tiny changes can grow bigger and bigger over time. So an infinitesimal change in the present can cause a massive change in the future. The weather beyond about a week is therefore so unpredictable it might as well be random. This is the technical definition of chaos: seemingly random behaviour in a deterministic system arising from extreme sensitivity to initial conditions.

This is famously encapsulated by the butterfly effect: a butterfly flapping its wings could theoretically cause (or prevent) a hurricane a month later on the other side of the Earth. Incidentally, one of the most worrying predictions from climate scientists is that global warming could trigger the melting of permafrost at the poles, releasing vast quantities of methane into the atmosphere. Methane is a far more powerful greenhouse gas than carbon dioxide, and so climate change could go on accelerating through positive feedback even if humans stopped burning fossil fuels.

So why had many of the world’s finest minds neglected the possibility that, through feedback, immeasurably small differences in initial conditions could lead to massive changes in outcome? The answer is that they had never dealt with feedback before, because any equation with feedback immediately becomes non-linear, and therefore insoluble without a computer. With a linear equation, a small discrepancy in input causes only a small change in output; and so, with an approximate knowledge of the present conditions of a system, you can make an approximate prediction about how it will behave in the future. However, as physicists began to use computers to explore more and more non-linear systems, they found countless other examples of chaos.

Physicists not only had to learn to analyse new phenomena, but they had to learn to analyse phenomena in completely new ways. Theoretical physicists are often fiercely proud of the objective rigour and precision of their discipline. They tend to look down their noses at the subjectivity of many of the “applied” sciences. A population biologist attempting to work out whether or not a species was likely to go extinct or a vulcanologist wondering whether an eruption was imminent might use his or her knowledge of similar situations in the past as the basis for an educated guess. A physicist trying to find the trajectory of a particle under the influence of a force would derive an equation, solve it and plug in the appropriate numbers. Provided he did not make a mistake, his answer would be 100% guaranteed to be right.

When dealing with chaos, however, the physicists could not solve the equations: they simply had to put the data into a computer and watch what came out. When they looked at how chaotic phenomena changed over time, what they saw seemed completely disordered. They knew, however, that time was irrelevant to the systems under study. If you drop a ball from your hand, the way it bounces will depend on the height you drop it from, whether there is any wind, how hard the ground is etc. It doesn’t depend on whether you drop it today, tomorrow or next year. The important thing is how the variables that make up a feedback loop change in response to each other.

Edward Lorenz had become a meteorologist because pilots needed weather forecasts during WW2, but at heart he was a mathematician. He knew that real weather systems would have too many variables to represent on a single graph, but he wanted to illustrate chaos graphically anyway so he could see what the graph would look like. So he took the equations for the movement of heat in the atmosphere and simplified them until he had only three variables left, which he called X, Y and Z, all dependent on each other and on nothing else. They no longer had any connection to atmospheric physics, but they did show chaos. Then he chose arbitrary starting values of the three variables and let his computer work out how they would evolve over time. The graph below shows the three variables plotted on a three-dimensional graph. The graph is bizarre, baroque and beautiful, but far from disordered.

Detailed analysis of the graph would be extremely complicated and somewhat pointless, so let’s note just two key features:

1. Two lines so close together they are almost indistinguishable can suddenly veer in completely different directions. This illustrates the butterfly effect (incidentally, the resemblance of the graph to a butterfly is amusing but coincidental.)
2. The graph goes on forever without intersecting itself (this is not obvious when you look at a two dimensional projection of it). If two lines did join, they would never separate, because the exact state of the system at any one time is specified by the position of the point on the graph (remember X, Y and Z depend on each other and on nothing else). If a deterministic system were to be in exactly the same state on two different occasions, it would evolve in the same way both times.

Mathematicians went on to discover countless other simple systems of equations that showed chaos. When plotted on graphs, some of them seem to look like arrowheads, some of them saddles, others lumps of plasticene randomly reshaped by a four year old; but all of them have the two features listed above. And all of them are the best possible demonstration that beauty is just as present in science as it is in any art.

For more information

Chaos - 5 easy lessons
http://order.ph.utexas.edu/chaos/

Solve Nonlinear Equations Online!
http://www.engs-comp.com/solvnonlinearequ/index.shtml