Chaos made simple
This shows a lot of tiny particles moving around. If you were one of these particles, it would be hard to predict where you'd go. See why? It's because each time you approach the crossing, it's hard to tell whether you'll go into the left loop or the right one.
predict which way you'll go: it's not random. But to predict it, you need to know your position quite accurately. And each time you go around, it gets worse. You'd need to know your position extremely
accurately to predict which way you go - left or right - after a dozen round trips.
This effect is called deterministic chaos
. Deterministic chaos happens when something is so sensitive to small changes in conditions that its motion is very hard to predict in practice
, even though it's not actually random.
This particular example of deterministic chaos is one of the first and most famous. It's the Lorenz attractor
, invented by Edward Lorenz as a very simplified model of the weather in 1963.
The equations for the Lorentz attractor are not very complicated if you know calculus. They say how the x, y and z coordinates of a point change with time:
dx/dt = 10(x-y)
dy/dt = x(28-z) - y
dz/dt = xy - 8z/3
You are not
supposed to be able to look at these equations and say "Ah yes! I see why these give chaos!" Don't worry: if you get nothing out of these equations, it doesn't mean you're "not a math person" - just as not being able to easily paint the Mona Lisa doesn't mean you're "not an art person". Lorenz had to solve them using a computer to discover chaos. I personally have no intuition as to why they work... though I could get
such intuition if I spent a week reading about it.
The weird numbers here are adjustable, but these choices are the ones Lorenz originally used. I don't know what choices David Szakaly
used in his animation. Can you find out?
If you imagine a tiny drop of water flowing around as shown in this picture, each time it goes around it will get stretched in one direction. It will get squashed
in another direction, and be neither squashed nor stretched
in a third direction.
The stretching is what causes the unpredictability: small changes in the initial position will get amplified. I believe the squashing is what keeps the two loops in this picture quite flat. Particles moving around these loops are strongly attracted to move along a flat 'conveyor belt'. That's why it's called the Lorentz attractor
With the particular equations I wrote down, the drop will get stretched in one direction by a factor of about 2.47... but squashed in another direction by a factor of about 2 million! At least that's what this physicist at the University of Wisconsin says:
J. C. Sprott, Lyapunov exponent and dimension of the Lorenz attractorhttp://sprott.physics.wisc.edu/chaos/lorenzle.htm
He has software for calculating these numbers - or more precisely their logarithms, which are called Lyapunov exponents
. He gets 0.906, 0, and -14.572 for the Lyapunov exponents.
For more nice animations of the Lorentz attractor, see:http://visualizingmath.tumblr.com/post/121710431091/a-sample-solution-in-the-lorenz-attractor-when
David Szakaly has a blog called dvdp
full of astounding images:http://dvdp.tumblr.com/
and presumably this one of the Lorenz attractor is buried in there somewhere, though I'm feeling too lazy to do an image search and find it.