So it’s quite easy to make hi-res plots of map functions with a GPU. The result is cool and science fictiony:

## More weird properties of chaos: non-mixing

We’ve done “chaos is not randomness” before. Here’s another interesting property to do with mixing.

Mixing is a property of dynamical systems whereby the state of the system in the distant future cannot be predicted from its initial state (or any given state a long way in the past). This is pretty much the same as the kind of mixing you get when you put milk in a cup of tea and swirl it around: obviously when you first put the milk in, it stays roughly where you put it, but after time it spreads out evenly. The even spread of the milk will be the same no matter where you put the milk in originally. More formally, if

is a “distribution” or density function of where the “particles” of milk are when you have just put them in the tea, and

is the distribution after seconds. “Mixing” is formally defined as

You don’t have to think about these distributions as probability distributions, but I find it easier if you do. For those that know probability, it is obvious that what the above is saying is that the distribution of milk after a long time is probabilistically independent of its distribution at the start.

In cups of tea, this happens (mostly) because of the “random” Brownian motion of the milk (possibly enhanced by someone swirling it with a spoon).

## Jelly challenge: decode the chaotic messages

Whilst hyperflunking across the interdimensional quantum vibration matrix, your spaceship detects three jumbled up signals. They sound like random noise, but you suspect they are in fact secret messages from Glycerol Soap Bomb, the ruler and Maximal Liapunov Exponent of the planet Cholesky Decomposition. The messages can be downloaded from the following locations:

## A secret message from another dimension

We’ve touched on the difference between chaos and randomness before. One strange property of chaotic systems is that they are able to synchronise to each other, so that in spite of their intrinsic tendency to vary wildly, a chaotic system can (actually quite easily) be persuaded to match the behaviour of another chaotic system. As this post will show, it is possible to use this property for a kind of secret message transmission.