Clouds are not spheres, mountains are not cones … Nature exhibits not simply a higher degree but an altogether different level of complexity. - Benoît Mandelbröt, The Fractal Geometry of Nature
Chaos (n): the inherent unpredictability in the behaviour of a complex natural system. - Merriam-Webster Dictionary
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.
Welcome to one of the most marvellous fields of modern mathematics.
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.
Some systems flip this premise around, with orderly effects emerging out of turbulent and chaotic causes.
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
Tiny variations … never repeat, and vastly affect the outcome. - Jeff Goldblum (“Ian Malcolm”), Jurassic Park.
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.
Accurate minute-by-minute forecasts added up into days, and then weeks.
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.
After a well-earned coffee break, he returned to discover something unexpected. 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. The starting weather conditions had been virtually identical. The two predictions were anything but.
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.
A nice way to see this “butterfly effect” for yourself is with a game of pool or billiards. No matter how consistent you are with the first shot (the break), the smallest of differences in the speed and angle with which you strike the white ball will cause the pack of billiards to scatter in wildly different directions every time.
The smallest of differences are producing large effects - the hallmark of a chaotic system.
It is worth noting that the laws of physics that determine how the billiard balls move are precise and unambiguous: they allow no room for randomness.
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 (thanks to ever-improving measuring instruments and models).
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 …
Though the dance of the planets has a lengthy prediction horizon, the effects of chaos cannot be ignored, for the intricate interplay of gravitation tugs among the planets has a large influence on the trajectories of the asteroids.
Keeping an eye on the asteroids is difficult but worthwhile, since such chaotic effects may one day fling an unwelcome surprise our way.
On the flip side, they can also divert external surprises such as steering comets away from a potential collision with Earth.
Attractive, strange behaviour
First, a highly regular motion_ towards the attractor … then a much more irregular motion on _it. - Ian Stewart, The Magical Maze.
Stability is desirable in many scenarios, such as flying. Commercial aircraft are aerodynamically stable, so that a small turbulent nudge (possibly butterfly-related) won’t push the plane out of a level flightpath.
Comfortingly, it takes a large change in the flight controls to effect a large change in the plane’s motion.
On the other hand, this stability is somewhat of an inconvenience to fighter pilots who prefer their aircraft to make rapid changes with minimal effort.
Modern fighter jets achieve great manoeuvrability by virtue of being aerodynamically unstable - the slightest nudge is enough to drastically alter their flightpath.
Consequently, they are equipped with on-board computers which constantly and delicately adjust the flight surfaces to cancel out the unwanted butterfly effects, leaving the pilot free to exploit his own.
If you can tease out the pattern’s underlying chaotic systems, you can effect a measure of control over randomness and turn instability into an asset.
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. Once there it clings to its attractor as it is buffeted to and fro in a literal sea of chaos, and quickly moves back to the surface if temporarily thrown above or dumped below the waves.
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.
The behaviour of the system can be observed by placing a point at the location representing the starting configuration and watching how that point moves through the phase space.
In phase space, a stable system will move predictably towards a very simple attractor (which will look like a single point in the phase space if the system settles down, or a simple loop if the system cycles between different configurations repeatedly).
A chaotic system will also move predictably towards its attractor in phase space - but instead of points or simple loops, we see “strange attractors” appear - complex and beautiful shapes (known as fractals) that twist and turn, intricately detailed at all possible scales.
The branch of fractal mathematics, pioneered by the French American mathematician Benoît Mandelbröt, allows us to come to grips with the preferred behaviour of this system, even as the incredibly intricate shape of the attractor prevents us from predicting exactly how the system will evolve once it reaches it.
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.
Fortunately, this intricate state of synchronisation is an attractor of the system - but it is not the only one. If the system is jolted somehow, it may find itself on an altogether different attractor called fibrillation, in which the cells constantly contract and relax in the wrong sequence.
The purpose of a defibrillator - the device that applies a large voltage of electricity across the heart - is not to “restart” the heart cells as such, but rather to give the chaotic system enough of a kick to move it off the fibrillating attractor and back to the healthy heartbeat attractor.
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
Universality made the difference between beautiful and useful. - James Gleick, 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.
Consider both a dripping tap and the supercooled liquid helium that the Large Hadron Collider uses as a coolant (which makes parts of the LHC colder than deep space).
Both are non-chaotic systems - at first - but as you slowly heat the helium, tiny convection cells will begin to form, and as you slowly open the tap, the dripping sounds will change in character.
Eventually the increases in temperature and water flow will cascade into the chaos of boiling helium and rushing water, respectively.
Amazingly, the transition from order to chaos in these systems is controlled by the exact same number - the Feigenbaum constant.
From dripping taps to the LHC, from a beating heart to the dance of the planets, chaos is all around us.
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.
See more Explainer articles on The Conversation.