Ed Lorenz was a meteorologist working on a computer simulation of convection in the atmosphere around 1961. As you can imagine, the computer simulation was extremely basic compared to the simulations that are carried out today. He started with three interlocking mathematical formulas:

dx/dt=-10x+10y
dy/dt=30x-y-xz
dz/dt=-3z+xy

We do not need to concern ourselves with the details of the formulae, but we can note that it mentions three variables or qualities of the weather, x, y, and z. The first equation is about how the variable x changes over time, the second is about how the variable y changes over time and the third is about how the variable z changes over time. Because there is a y term in the x equation, x and z terms in the y equation and x and y in the z equation, each equation depends on the others. In the same way as Henri Poincaré’s three-body problem seemed simple, the interactions between the elements generated complex outcomes. Ed Lorenz’s equations also generated complex outcomes.

There is a story that has entered the folklore of Chaos Theory about Ed Lorenz’s discoveries. Computers back then computed data much slower than today. He decided to go out for a break and set up the computer to complete its run while he was away. He had been using the number 0.506127 as a part of his calculations. To speed up his calculations he reduced the number to 0.506000 assuming that the small difference between the two would only cause a negligible difference in the results. He was very surprised to find that even though the graphs followed a similar path at the beginning, the two graphs soon rapidly diverged to become completely different. This meant that only small changes occurring at the beginning point in a chaotic, far-from-equilibrium system can produce very different outcomes. This is why we can only predict the weather with any accuracy for a few days. If we just made an extremely small error in describing the system at the beginning, our predictions will soon be wildly incorrect. The Lyapunov exponent, named after the Russian mathematician Aleksandr Lyapunov, measures how quickly any two very similar states diverge over time. Some systems will diverge quickly while others will be more stable and predictable for longer. If the divergence from nearby states is small over time, the system will be more stable and predictable.

The length of time for which a chaotic system is accurately predictable is known as the Lyapunov time. A system with a longer Lyapunov time is more stable. Some electrical circuits have Lyapunov times measured in milliseconds, weather predictions are in days and our solar system has a Lyapunov time of around five million years. Even our solar system is chaotic and in time will become unpredictable.

We can lengthen the Lyapunov time by obtaining more accurate information on the system. The more accurately we can measure the initial starting point, the longer the Lyapunov time we can expect. But the difficulty is that gaining that information becomes more difficult exponentially. If each time we look at the system there are ten possibilities that could appear, after four measurements we have 10,000 possibilities to choose between and after the fifth 100,000. It does not take long before we are swamped with the possibilities to choose from.

When Ed Lorenz’s three equations are graphed on a three-dimensional chart (one dimension for each of the x,y and z variables), a very interesting picture emerges. Because the variables are interactive, there are many states that the system cannot move towards. For example, when the temperature is increased, the pressure will also be affected, so there are combinations of temperature and pressure that cannot co-exist. If the system does somehow get to such an unstable state, feedback loops will pull it back to a viable state. This graph therefore charts what is known as an attractor, because the system is pulled, as if by some sort of magnet into certain viable states.

The graph of the Lorenz attractor traces out a line. Any point on the line represents a state that the whole system can be in. The graph charts a single line that coils and winds about in three dimensions in a shape somewhat reminiscent of a butterfly.

You will notice therefore that even very small changes in the initial conditions of a complex system can drastically alter its later state. This is known as sensitive dependence on initial conditions. While in theory, a chaotic system might be deterministic (i.e. could be calculated if we had enough accurate data), we find that in the real world we cannot predict a chaotic system. No matter how accurately we measure the initial conditions of a complex system, we can always measure it more accurately and that difference, however small, will lead to a significantly different outcome. It means we can never absolutely predict the outcome of any real-world complex system. Even the planets in our solar system, which appear to be deterministic, will eventually become unpredictable.

If I were to place a boulder in a river near the place where the water sprang from the earth, I could very likely alter the place where the river would reach the sea by several kilometres. That same boulder placed near the river mouth would have no effect.

There have been critical points in history when small events have totally changed the course of history, such as the assassination of the Archduke Ferdinand precipitating the First World War or Richard III of England dying at the battle of Bosworth Field in 1485. It is commemorated in the rhyme:

For want of a nail the shoe was lost.
For want of a shoe the horse was lost.
For want of a horse the rider was lost.
For want of a rider the battle was lost.
For want of a battle the kingdom was lost.
And all for the want of a horseshoe nail.

This makes the point that small seemingly insignificant events can have an enormous impact on the outcome of events (even if the whole horse episode was an invention of Shakespeare).