Chaos: butterflies, fractals and the double pendulum

You might think that physics is all maths, equations and graphs. And in some cases, you’d be right! But chaos theory – one of my favourite areas of physics – is one of the exceptions to the rule.

Well, OK, maybe that’s not entirely true. But chaos theory, in essence, is all about how a few (fairly) simple equations can become one very complicated, unpredictable mess – and normally it’s the other way round!

It all started with one guy called Edward Lorenz. He was working on weather prediction models on a very basic computer, inputting some weather equations and some initial values (like 5mm rainfall and 10 degrees Celsius, for example) and letting the computer calculate the weather pattern that resulted. He stopped the simulation at one point, went back to the start and put in almost the same numbers again (one of them was rounded up) – and suddenly the computer’s output was completely different! This phenomena, known as sensitivity to initial conditions, is one of the fundamental parts of chaos theory: one tiny, tiny change can make a massive difference. That’s where the “butterfly effect” comes from: a butterfly flapping its wings can create a hurricane on the other side of the world.

A butterfly flying above some pink flowers

This one butterfly could create a distant hurricane. Image: fox_kiyo

This is the reason why complex systems – like the weather – are so difficult to predict. The more weather sensors, the better the prediction – except even introducing the sensors means that the weather’s disrupted! – and we can’t put sensors everywhere. It is still annoying when the Met Office gets it really wrong though!

But that’s not all there is to chaos theory. Another big part is all about fractals, first discovered by Benoit Mandelbrot. The word ‘fractal’ is kind of a shortening of ‘fractional pattern’: these patterns are basically repeated over and over within themselves, so if you keep zooming in you see the original pattern repeated over and over. You may find it hard to believe, but these beautiful pictures are in fact based on maths! Here’s a really good video that demonstrates fractals:

My final, and favourite, part of chaos theory is the double pendulum. The butterfly effect is mind-boggling, and demonstrates just how complicated our world is; fractals are beautiful and mathematical at the same time (which is pretty mind-boggling in itself!) but double pendulums are a wonderful example of chaos theory, how something that seems so simple can be so unpredictable.

A double pendulum is pretty much what it sounds like: you attach one pendulum to the end of another, making it a double-jointed pendulum. But unlike a normal pendulum, the double pendulum is sensitive to initial conditions: a raindrop 10 metres away can make a difference to the path it takes! So when you let it go, it does all sorts of weird stuff, as you can see about 45 seconds into this video:

Now obviously there’s a lot of computer simulation and/or mechanics equations behind all the stuff I’ve talked about here: but what I love about chaos theory is that it can be demonstrated in really cool ways, without any need to use the equations to explain what’s going on. And in case you were wondering – yes, I do own my own double pendulum!



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