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Chaos TheoryIntroduction

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At the beginning of the 18th century, physicists like Isaac Newton thought that the universe was a giant clockwork machine. If you had precise information about every object right now, you could use the laws of physics to predict what would happen in the future.

One great example is the pendulum. You’ve already seen how you can use differential equations to find an equation that calculates the position of the pendulum at any time in the future.

We often say that the motion of the pendulum is deterministic: it precisely follows Newton’s laws of gravity, and there is no randomness or probability. Watch the pendulum swinging, and try to predict what it’ll do next!

Now let’s try to make things a bit more interesting by attaching a second pendulum to the first one – this is called a double pendulum.

Once again, watch its motion, and try to predict what it will do next. You can even close your eyes for a few seconds … was your prediction correct?

The double pendulum still follows Newton’s deterministic laws of gravity, but the motion seems to be completely irregularbehave as if there were no gravityalways repeat itself.

This becomes every more apparent if we look at multiple pendulums. Let’s add three more pendulums behind the first one, each with a tiny, imperceptible change in their initial angle (less than 0.1°). Press play again and watch what happens!

At the beginning the four pendulums move along the same path – but after a few seconds they separatejoin upalternate and end up taking completely different paths.

Mathematicians call this behaviour chaotic: even if you know the physical laws of the system, it is impossible to predict what might happen in the future. Tiny changes in the initial conditions are quickly amplified and lead to a completely different motion.

It is important to realise that chaos is very different from randomness. The double pendulum contains no element of chance or probability. It follows the precise, deterministic laws of gravity, and nevertheless moves completely unpredictably.

Chaos appears in many unexpected places in nature and mathematics. In this course, we will explore some of these examples, and discover how mathematics can help us understand them.

The Butterfly Effect

In 1962, the mathematician Edward Lorenz was working at MIT, where he was developing computer simulations of the atmosphere to predict the weather.

Lorenz had developed an algorithm that took current weather data like temperature, humidity and wind speed as inputs. It then calculated how these values would change a few minutes into the future. By repeating the process over and over again, he was able to predict the weather days, or even weeks into the future.

Edward Lorenz used an LGP-30, one of the first off-the-shelf computers.

One day, Lorenz decided to run parts of his simulation a second time, with the same input data. To his surprise, the weather predictions created in both cases were completely different!

At first, Lorenz thought there must be a bug in the code – but he quickly worked out what was actually going on. When running the simulation a second time, he had rounded some of the input values to fewer decimal places (for example 0.506 instead of 0.506127). Even though the difference is tiny, less than 0.1%, it seems to have been enough to cause the simulation to make completely different forecasts.

Lorenz called this the butterfly effect: even the flap of a butterfly’s wings in Brazil might change the weather conditions enough to set off a tornado in Texas, at some point in the future.

Usually, you’d expect a small change to the input parameters to also lead to a small change in the output. For chaotic systems, like the weather or the double pendulum, the opposite is true. Even tiny changes can grow quickly, and lead to huge differences.

This is why weather prediction only works for a few days into the future. After that, slightly inaccurate weather data can lead to completely different forecasts.

There is an important difference between knowing the physical laws of a system, and being able to predict its behaviour. Newton’s laws of gravity tell us exactly how the double pendulum should move. The differential equations of fluid dynamics should, in principle, allow us to calculate the weather at any time in the future.

Unfortunately, for this to work in practice, we would have to measure the initial state of the pendulum with infinite precision, or know the exact position of every particle in the atmosphere – and that is clearly impossible. Over time, tiny measurement errors will completely throw off our predictions. In other words, we can never have a perfect weather forecast!

NASA uses satellites, planes and thousands of ground-based weather stations to take “snapshots” of the atmosphere: measuring important indicators like wind speed, rainfall, humidity, air pressure, and ocean currents.

Of course, a butterfly doesn’t actually cause a tornado – but it might change the conditions of the atmosphere just enough so that a tornado happens now rather than at some other time in the future.

You could think about it as a series of dominoes with increasing size. A small action that topples the first domino later leads to the largest domino falling:

The world’s largest domino toppling.

More Applications

You’ve already seen that the double pendulum and the weather behave chaotically, but there are countless other examples in nature and technology:

The Belousov–Zhabotinsky reaction occurs when you mix bromate with an acid. (Shown at 20x speed, © Tim Kench)

Stock markets, inflation and other parts of the economy are chaotic, making them very difficult to predict.

Scientists hope that explaining the origin of chaotic heartbeats (ventricular fibrillation) can lead to a cure.

Population dynamics often is chaotic – from the number of humans in a city, to rodent, fish or insect populations.

Variable stars pulsate by altering their brightness over time. Some of these stars pulsate in a chaotic pattern.

Computers can use chaotic systems to generate pseudo-random numbers, or to securely encrypt images.

Chaotic behaviour has also been found in electronic circuits, evolutionary biology as well as dripping water taps, and scientists around the world are researching many other applications.

But in all these examples, the existence of chaos means that there are fundamental limits to how well we can predict the future. This is often called a prediction horizon, beyond which it is impossible to make any accurate forecasts.

The weather forecast currently has a prediction horizon of approximately one week, and even with much better technology in the future, we might never surpass two weeks of accuracy.

Instead, scientists can use probability to predict averages: for example how often they’d expect certain weather events (like a tornado) to occur in a year. These averages can be very accurate, even if we don’t know when exactly a tornado might happen.

Let’s summarise the key properties of chaos that we have discovered so far:

  • A chaotic system follows precise, deterministicprobabilisticunpredictable laws, often described by one or more differential equations.
  • Chaotic systems are highly sensitive to change. TinyLarge initial differences multiply over time and can lead to hugesmallrandom differences in the result.
  • The behaviour is unpredictable and non-repeating. It might even look random, but only because it depends on imperceptible changes or measurement errors.

Chaos theory tells us that we can never predict the future, or know in advance what outcome our actions might have. This can be worrying if you consider how much in our world depends on mathematical predictions: from modelling the wind flowing around airplanes or skyscrapers, to assessing the impact of climate change.

On the other hand, chaos theory and the butterfly effect were such simple and powerful ideas, that they soon started to appear in books, music lyrics, and even in movies:

Extract from the movie Jurassic Park (© Universal Pictures, 1993)