Chaos Theory Demystified

Throughout that time, the goal of physicists has been to describe orderly behaviour in an apparently disorderly world. In 1687, Isaac Newton’s law of universal gravitation showed that the force that causes apples to fall from trees is the same force that keeps the Moon in orbit around the Earth, and the Earth in orbit around the Sun - the same equation governs all of them. Ever since then, when a physicist is trying to understand some phenomenon, what he is almost certainly striving for is an equation that governs its behaviour – a recipe that, if he knows what it is doing now, will tell him what it will do at any point in the future. A phenomenon that is mathematically predictable like this is called deterministic.

In the nineteenth century, physicists were convinced that the universe was completely deterministic. Their belief was that some day soon all the equations that governed the universe would be known; and that by knowing the state of the universe at present, it would be possible, in theory, to predict the future. This discovery had to be revised at the beginning of the twentieth century when physicists discovered that subatomic particles do not obey deterministic equations. Nevertheless, for other systems, the dream remained alive.

In the 1950s and ’60s, that dream burned brightest in the arena of weather forecasting. The weather is a deterministic system, and all the equations governing the weather, from the movement of clouds to the flow of heat, were known. The problem was that they fell into a very large group of equations, found everywhere in physics, called non-linear differential equations, that cannot be solved using elegant mathematical reasoning. Instead, you have to do the same simple calculation millions and millions of times – a process technically called numerical analysis, but which physicists are more likely to refer to as brute force. This is hardly feasible for a human mathematician. These equations were locks that history’s finest minds had been unable to pick. Now, armed with the previously unheard of calculating speeds of the first computers (even if they are pathetic compared to modern PCs), physicists relished the chance to kick down the doors. What they would find behind them would be the deaths of some of their most basic assumptions about how the Universe worked, and the birth of a new branch of science.

In 1960, one promising young meteorologist named Edward Lorenz constructed a drastically simplified computer model of the weather. Lorenz’s model did not have clouds, rain or seasons - it simply attempted to represent the way air moves around within the atmosphere. Lorenz input twelve values into his computer model which represented the initial temperature, pressure etc. and watched transfixed as the wind changed direction and speed, the temperature rose and fell, sometimes gradually, sometimes suddenly. One day, he decided to look at a particular part of a weather sequence again, and so he copied out the 12 values of temperature, pressure etc. from the point at which wanted to restart the sequence and set the simulation to start again from these values. Since he believed that the future of the weather was exactly determined by conditions in the present, he assumed that the simulation would reproduce his previous printout precisely. Returning from a coffee break, however, he was shocked to find that as time progressed within the model his two printouts rapidly diverged, until soon there was no indication that they had started from the same point.

After checking to see that he had not made a mistake and that the machine was not broken, he realised something incredible: the computer had stored the figures to a slightly greater degree of accuracy than it had printed out. Lorenz had copied the data from the printouts into the computer to restart the sequence. The differences in pressure, wind speed etc. between the original starting point and the starting point of the re-run had been tiny – so small that no equipment could have detected them in real weather; and yet these accidental, infinitesimal changes had altered the weather several months later beyond recognition. The weather might be deterministic in theory, but in practice it was totally unpredictable beyond a few days. Long range forecasting was doomed. The reason for this is simple: feedback