Chaos in Nature
Introduction, and Non-Chaos Maths in Nature
Only recently have we began to discover how much mathematics governs nature. There are many examples of both chaotic and non-chaotic maths in nature. On the non-chaotic side, there is, for example, the fibonachi sequence. This is a simple iterative sequence, given by: fn = fn-1 + fn-2 , where f0 = f1 = 0.
This sequence of numbers occurs in many natural features, such as the number of leaves or petals on certain plants. There is also the 'golden ratio', which is also observed in many natural features. The golden ratio can be described as follows:
We define a rectangle whose sides are in this ratio to be a 'golden rectangle'. Then if we take a golden rectangle, and remove the largest square possible from one end (shaded), we are left with another golden rectangle.
However, much of the most interesting, and important, maths in nature is chaotic. One example is the self repitition that many plants, such as ferns, exhibit. Self repitition means that a small part of the structure, enlarged, looks like the whole structure. This is very closely linked to some chaotic maths, and fractals. We continue, however, with fluids.
Fluids and Turbulence
The way that water flows has interested people for many centuries. For example, Leonardo da Vinci produced many pictures of the flow of water. He was very concerned to try and draw the water exactly as it flowed in reality, though this was a diffucult thing to attempt. The Royal Library of Windsor has many of these pictures.
But why is water so difficult to draw, and indeed to model? It is due to the complexities of fluid motion, and at the heart of the matter is chaos. First though, a simple experiment, to show that the behaviour of water is far from deterministic. For this experiment you will need nothing but a tap.
The Dripping Tap
Turn your tap on, just enough for it to drip regularly. The drips come at regular intervals. Now increase the level, slightly. You should be able to get a different, but regular rhythm, something like drip-drip...drip-drip...drip-drip etc. Now, if you turn the tap a tiny bit more, so that the drips have almost formed a stream of water, you should be able to get to a point where there is no pattern to the falling drips. A small increase in the amount of water being allowed out, but a big change in what's happening. This is a typical example, on a very simplistic level, of chaos.
Back to Fluid dynamics
First, a quick idea of what turbulence is. It has the same meaning here as in normal use eg. "turbulent seas", but we must be slightly more precise. Turbulence is the state a fluid reaches after smooth laminar flow has broken up. Laminar flow is where a fluid slides over itself in layers. Going back to the tap, laminar flow is when the water is flowing out of the tap in a steady stream, maybe with some extra, twisted structure. Turbulence is when the tap is on full, with spray everywhere, and no obvious structure at all.
The classical motion of a viscous fluid was described by Claude Navier and Sir George Stokes, based on ideas by Euler. The equations are deterministic, and therefore predictable, but turbulence is very irregular. So turbulence could not be correctly predicted from these equations. Many people since worked on turbulence, and the first modification of the theory came from Lev Landau and Eberhard Hopf. Their theory was one of wobbles, accumulating and causing the transition to turbulence. For 30 years this was the preferred theory. Then, David Ruelle and Floris Takens used the mathematics of Topology to prove that the Landau-Hopf theory was impossible, and therefore wrong. This was a controversial view at the time, but it was right. Harry Swinney and Jerry Gollub used some of the ideas of Ruelle and Takens, and some sophisticated laser equipment to study the water flow, as it became turbulent. They got very good results, just what they were looking for. The water went through a number of neat transitions, all predicted, and observed by a single, sharp frequency. But as they searched for the next transition, all they found was the gradual emergence of a broad band of frequencies. As Swinney and Gollub put it: "What we found was, it became chaotic." Since then, chaos theory and the technique of renormalisation has been used to further develop and refine the theory of turbulence.
The Weather
That weather is chaotic shouldn't be the biggest surprise in the world. However, before the advent of chaos, things were different. It was once thought that, when technology became advanced enough, we would be able to predict the weather months, even years in advance, and that these predictions would be very accurate. It turns out, now that we have the technology, predicting the weather 4 days in advance is about as good as it gets, and that these predictions are often wrong. Edward Lorenz discovered the reasoning behind this, some 40 years ago...
The Butterfly Effect
One of the most famous effects of Chaos Theory is known as the butterfly effect, concerning weather. It was discovered, and named, by Edward Lorenz, a mathematician turned meteorologist, in winter 1961. The butterfly effect is the theory that a single butterfly, flapping its wings in one part of the world, could cause a tornado in another entirely different part of the world. Lorenz made his discovery entirely by accident. Lorenz had a number of equations, based on work by Saltzman, that predicted simple weather systems. Crucially, he had an early computer, a Royal McBee LGP-300, capable of about one iteration of his weather system equations per second. He used the computer to simulate small weather environments. The output of this computer was simply a string of numbers. Now, Lorenz had already calculated the solution to one system, but wanted to see how it evolved further. He therefore, to save time, started the calculations of the computer halfway through - the computer would repeat the second half of the data as a check, and then continue. Lorenz left his computer to do the work, but when he came back he found that the new calculations started off the same, but soon became wildely different. The reason for this was that inside the computer program, values were stored to 6 decimal places, but Lorenz had only entered the first three, as printed out by his computer. Lorenz had entered 0.506 instead of 0.506127, and the traditional way of thinking was that this tiny difference - a butterfly flapping its wings - should not have such a devastating effect. Lorenz realised that his equations were not behaving normally, and in fact were displaying signs of Chaos.
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Will Bolam 2001