Sorry for the slowness of the blog; I fell behind in writing my book, which is on a rather strict schedule, and until I got close to catching up, I didn't have time to do the research
necessary to write the next chaos article. (And no one has sent me any particularly
interesting bad math, so I haven't had anything to use for a quick rip.)
Anyway... Where we left off last was talking about attractors. The natural question
is, why do we really care about attractors when we're talking about chaos? That's a question
which has two different answers.
First, attractors provide an interesting way of looking at chaos. If you look
at a chaotic system with an attractor, it gives you a way of understanding the chaos. If
you start with a point in the attractor basin of the system, and then plot it over time, you'll
get a trace that shows you the shape of the attractor - and by doing that, you get a nice view
of the structure of the system.
Second, chaotic attractors are strange. In fact, that's their name: strange attractors: a strange attractor is an attractor whose structure has fractal dimension,
and most chaotic systems have fractal-dimension attractors.
Let's go back to the first answer, to look at it in a bit more depth. Why do we want
to look in the basin in order to find the structure of the chaotic system?
If you pick a point in the attractor itself, there's no guarantee of what it's going to do. It
might jump around inside the attractor randomly; it might be a fixed point which just sits in one
place and never moves. But there's no straightforward way of figuring out what the attractor looks
like starting from a point inside of it. To return to (and strain horribly) the metaphor I used in
the last post, the attractor is the part of the black hole past the even horizon: nothing inside of
it can tell you anything about what it looks like from the outside. What happens inside of a black hole? How are the things that were dragged into it moving around relative to one another, or are they moving around? We can't really tell from the outside.
But the basin is a different matter. If you start at a point in the attractor basin,
you've got something that's basically orbital. You know that every path starting from a point
in the basin will, over time, get arbitrarily close to the attractor. It will circle and cycle around. It's never going to escape from that area around the attractor - it's doomed to approach it. So if you start at a point in the basin around a strange attractor, you'll get a path that tells you something about the attractor.
Attractors can also vividly demonstrate something else about chaotic systems: they're not necessarily chaotic everywhere. Lots of systems have the potential for chaos: that is, they've got sub-regions of their phase-space where they behave chaotically, but they also have regions where they don't. Gravitational dynamics is a pretty good example of that: there are plenty of N-body systems that are pretty much stable. We can computationally roll back the history of the major bodies in our solar system for hundreds of millions of years, and still have extremely accurate descriptions of where things were. But there are regions of the phase space of an N-body system where it's chaotic. And those regions are the attractors and attractor basins of strange attractors in the phase space.
A beautiful example of this is the first well-studied strange attractor. The guy who
invented chaos theory as we know it was named Edward Lorenz. He was a meteorologist who was
studying weather using computational fluid flow. He'd implemented a simulation, and as part
of an accident resulting from trying to reproduce a computation, but entering less precise
values for the starting conditions, he got dramatically different results. Puzzling out why,
he laid the foundations of chaos theory. In the course of studying it, he took the particular equations that he was using in the original simulation, and tried to simplify them to get the simplest system that he could that still showed the non-linear behavior.
The result is one of the most well-known images of modern math: the Lorenz attractor. It's
sort of a bent figure-eight. It's dimensionality isn't (to my knowledge) known precisely - but it's a hair above two (the best estimate I could find in a quick search was in the 2.08 range). It's not a particularly complex system - but it's fascinating. If you look at the paths in the Lorenz attractor, you'll see that things follow an orbital path - but there's no good way to tell when two paths that are very close together will suddenly diverge, and one will pass on the far inside
of the attractor basin, and the other will fly to the outer edge. You can't watch a simulation for long without seeing that happen.
While searching for information about this kind of stuff, I came across a wonderful demo, which
relates to something else that I promised to write about. There's a fantastic open-source
mathematical software system called sage. Sage is sort of like
Mathematica, but open-source and based on Python. It's a really wonderful system, which I really
will write about at some point. On the Sage blog, they posted a simple Sage program for drawing the Lorenz attractor. Follow that
link, and you can see the code, and experiment with different parameters. It's a wonderful
way to get a real sense of it. The image at the top of this post was generated by that Sage
program, with tweaked parameters.