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What is Chaos Theory?

Clouds are not spheres, mountains are not cones… Nature exhibits not simply a higher degree but an altogether different level of complexity. - Benoît MandelbrötThe Fractal Geometry of Nature

Chaos Theory is a delicious contradiction - a science of predicting the behaviour of “inherently unpredictable” systems. It is a mathematical toolkit that allows us to extract beautifully ordered structures from a sea of chaos - a window into the complex workings of such diverse natural systems as the beating of the human heart and the trajectories of asteroids.

At the centre of Chaos Theory is the fascinating idea that order and chaos are not always diametrically opposed. Chaotic systems are an intimate mix of the two: from the outside they display unpredictable and chaotic behaviour, but expose the inner workings and you discover a perfectly deterministic set of equations ticking like clockwork.

How can order on a small scale produce chaos on a larger scale? And how can we tell the difference between pure randomness and orderly patterns that are cloaked in chaos? The answers can be found in three common features shared by most chaotic systems.

Butterflies make all the difference

In 1961, a meteorologist by the name of Edward Lorenz made a profound discovery. Lorenz was utilising the new-found power of computers in an attempt to more accurately predict the weather. He created a mathematical model which, when supplied with a set of numbers representing the current weather, could predict the weather a few minutes in advance. Once this computer program was up and running, Lorenz could produce long-term forecasts by feeding the predicted weather back into the computer over and over again, with each run forecasting further into the future.

One day, Lorenz decided to rerun one of his forecasts. In the interests of saving time he decided not to start from scratch; instead he took the computer’s prediction from halfway through the first run and used that as the starting point. Although the computer’s new predictions started out the same as before, the two sets of predictions soon began diverging drastically. What had gone wrong?

Lorenz soon realised that while the computer was printing out the predictions to three decimal places, it was actually crunching the numbers internally using six decimal places. So while Lorenz had started the second run with the number 0.506, the original run had used the number 0.506127. A difference of one part in a thousand: the same sort of difference that a flap of a butterfly’s wing might make to the breeze on your face.

Lorenz had found the seeds of chaos. In systems that behave nicely - without chaotic effects - small differences only produce small effects. In this case, Lorenz’s equations were causing errors to steadily grow over time. This meant that tiny errors in the measurement of the current weather would not stay tiny, but relentlessly increased in size each time they were fed back into the computer until they had completely swamped the predictions.

Lorenz famously illustrated this effect with the analogy of a butterfly flapping its wings and thereby causing the formation of a hurricane half a world away. What at first glance appears to be random behaviour is completely deterministic - it only seems random because imperceptible changes are making all the difference.

The rate at which these tiny differences stack up provides each chaotic system with a prediction horizon - a length of time beyond which we can no longer accurately forecast its behaviour. In the case of the weather, the prediction horizon is nowadays about one week. Some 50 years ago it was 18 hours. Two weeks is believed to be the limit we could ever achieve however much better computers and software get.

Surprisingly, the solar system is a chaotic system too - with a prediction horizon of a hundred million years. It was the first chaotic system to be discovered, long before there was a Chaos Theory.

In 1887, the French mathematician Henri Poincaré showed that while Newton’s theory of gravity could perfectly predict how two planetary bodies would orbit under their mutual attraction, adding a third body to the mix rendered the equations unsolvable. The best we can do for three bodies is to predict their movements moment by moment, and feed those predictions back into our equations.

Keeping an eye on the asteroids is difficult but worthwhile, since such chaotic effects may one day fling an unwelcome surprise our way.

Attractive, strange behaviour

The key to unlocking the hidden structure of a chaotic system is in determining its preferred set of behaviours - known to mathematicians as its attractor.

The mathematician Ian Stewart used the following example to illustrate an attractor: Imagine taking a ping-pong ball far out into the ocean and letting it go. If released above the water it will fall, and if released underwater it will float. No matter where it starts, the ball will immediately move in a very predictable way towards its attractor - the ocean surface.

Though we may not be able to predict exactly how a chaotic system will behave moment to moment, knowing the attractor allows us to narrow down the possibilities. It also allows us to accurately predict how the system will respond if it is jolted off its attractor.

Mathematicians use the concept of a “phase space” to describe the possible behaviours of a system geometrically. Phase space is not (always) like regular space - each location in phase space corresponds to a different configuration of the system.

Phase space may seem fairly abstract, but one important application lies in understanding your heartbeat. The millions of cells that make up your heart are constantly contracting and relaxing separately as part of an intricate chaotic system with complicated attractors. These millions of cells must work in sync, contracting in just the right sequence at just the right time to produce a healthy heartbeat.

The main benefit to having a chaotic heart is that tiny variations in the way those millions of cells contract serves to distribute the load more evenly, reducing wear and tear on your heart and allowing it to pump decades longer than would otherwise be possible.

The cascade into chaos

Chaos Theory is not solely the providence of mathematicians. It is notable for drawing together specialists from many diverse fields - physicists and biologists, computer scientists and economists.

Not only can chaotic systems be found almost anywhere you care to look, they share many common features independently of where they came from.

Chaos Theory has turned everyone’s attention back to things we once thought we understood, and shown us that nature is far more complex and surprising than we had ever imagined.

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