It’s been quite a while since my last chaos theory post. I’ve

been caught up in other things, and I’ve needed to do some studying. Based

on a recommendation from a commenter, I’ve gotten another book on Chaos

theory, and it’s frankly vastly better than the two I was using before.

Anyway, I want to first return to dense periodic orbits in chaotic

systems, which is what I discussed in the previous chaos theory

post. There’s a glaring hole in that post. I didn’t so much get it

*wrong* as I did miss the fundamental point.

If you recall, the basic definition of a chaotic system is

a dynamic system with a specific set of properties:

- Sensitivity to initial conditions,
- Dense periodic orbits, and
- topological mixing

The property that we want to focus on right now is the

dense periodic orbits.

In a dynamical system, an *orbit* isn’t what we typically think of

as orbits. If you look at all of the paths through the phase space of a

system, you can divide it into partitions. If the system enters a state in any

partition, then every state that it ever goes through will be part of the same

partition. Each of those partitions is called an *orbit*. What

makes this so different from our intuitive notion of orbits is that

the intuitive orbit *repeats*. In a dynamical system, an

orbit is just a set of points, paths through the phase space of

the system. It may never do anything remotely close to repeating – but it’s

an orbit. For example, if I describe a system which is the state

of an object floating down a river, the path that it takes is

an orbit. But it obviously can’t repeat – the object isn’t going to

go back up to the beginining of the river.

An orbit that repeats is called a *periodic orbit*. So

our intuitive notion of orbits is really about *periodic*

orbits.

Periodic orbits are tightly connected to chaotic systems.

In a chaotic system, one of the basic properties is a particular

kind of unpredictability. Sensitivity to initial conditions

is what most people think of – but the orbital property is

actually more interesting.

A chaotic system has *dense periodic orbits*. Now, what

does that mean? I explained it once before, but I managed to

miss one of the most interesting bits of it.

The points of a chaotic system are *dense* around

the periodic orbits. In mathematical terms, that means that

every point in the attractor for the chaotic system is

*arbitrarily* close to some point on a periodic orbit. Pick

a point in the chaotic attractor, and pick a distance greater than zero.

No matter how small that distance is, there’s a periodic orbit

within that distance of the point in the attractor.

The last property of the chaotic system – the one which makes

the dense periodic orbits so interesting – is topological mixing. I’m

not going to go into detail about it here – that’s for the next post. But

what happens when you combine topological mixing with the density

around the periodic orbits is that you get an amazing kind of

unpredictability.

You can find stable states of the system, where everything

just cycles through an orbit. And you can find an instance of

the system that *appears* to be in that stable state. But

in fact, virtually *all* of the time, you’ll be wrong. The

most miniscule deviation, any unmeasurably small difference between

the theoretical stable state and the actual state of the system – and

at some point, you’re behavior will diverge. You could stay close to the

stable state for a very long time – and then, whammo! the system will

do something that appears to be completely insane.

What the density around periodic orbits means is that

even though *most* of the points in the phase space aren’t

part of periodic orbits, you can’t possibly distinguish them

from the ones that are. A point that appears to be stable

*probably* isn’t. And the difference between real stability

and apparent stability is unmeasurably, indistinguishably small.

It’s not just the *initial* conditions of the system

that are sensitive. The entire system is sensitive. Even if you

managed to get it into a stable state, the slightest pertubation,

the tiniest change, could cause a drastic change at some unpredictable

time in the future.

This is the real butterfly effect. A butterfly flaps its wings –

and the tiny movement of air caused by that pushes the weather system

that tiny bit off of a stable orbit, and winds up causing the

diversion that leads to a hurricane. The tiniest change at any

time can completely blow up.

It also gives us a handle on another property of chaotic systems

as models of real phenomena: we can’t reverse them. Knowing the

measured state of a chaotic system, we *cannot* tell how it

got there. Even if it appears to be in a stable state, if it’s part

of a chaotic system, it could have just “swung in” the chaotic

state from something very different. Or it could have been in what

appeared to be a stable state for a long time, and then suddenly

diverge. Density effectively means that we can’t distinguish

the stable case from either of the two chaotic cases.