I’ve found a really interesting chaotic system. The canonical, textbook example of a chaotic system is:
A[n+1] = fpart(2*A[n])
Which keeps getting data down from successive binary digits. Start that with a number like π, and you’ll get (almost) random behavior forever. Its behavior is orderly only when it reaches a number either very close to zero or very close to one, in which case it slowly launches itself away from there, until it gets to some threshold and becomes chaotic again.
Anyway, my system is the same idea, but cooler because it has connections to continued fractions.
A[n+1] = 1/fpart(k A[n])
If you look at the behavior for the initial condition π (and parameter k=1), you’ll see something not unlike the first system. But start it at e for various parameters. Here’s e at k=1:
Pretty cool, huh? I’m going to sit down with mathematica and explore some of those properties.