The last major property of a chaotic system is topological mixing. You can
think of mixing as being, in some sense, the opposite of the dense periodic
orbits property. Intuitively, the dense orbits tell you that things that are
arbitrarily close together for arbitrarily long periods of time can have
vastly different behaviors. Mixing means that things that are arbitrarily far
apart will eventually wind up looking nearly the same – if only for a little
while.

Let’s start with a formal definition.

As you can guess from the name, topological mixing is a property defined
using topology. In topology, we generally define things in terms of open sets
and neighborhoods. I don’t want to go too deep into detail – but an
open set captures the notion of a collection of points with a well-defined boundary
that is not part of the set. So, for example, in a simple 2-dimensional
euclidean space, the contents of a circle are one kind of open set; the boundary is
the circle itself.

Now, imagine that you’ve got a dynamical system whose phase space is
defined as a topological space. The system is defined by a recurrence
relation: s_n+1 = f(s_n). Now, suppose that in this
dynamical system, we can expand the state function so that it works as a
continous map over sets. So if we have an open set of points A, then we can
talk about the set of points that that open set will be mapped to by f. Speaking
informally, we can say that if B=f(A), B is the space of points that could be mapped
to by points in A.

The phase space is topologically mixing if, for any two open spaces A
and B, there is some integer N such that f^N(A) ∩ B &neq; 0. That is, no matter where you start,
no matter how far away you are from some other point, eventually,
you’ll wind up arbitrarily close to that other point. (Note: I originally left out the quantification of N.)

Now, let’s put that together with the other basic properties of
a chaotic system. In informal terms, what it means is:

1. Exactly where you start has a huge impact on where you’ll end up.
2. No matter how close together two points are, no matter how long their
   trajectories are close together, at any time, they can
   suddenly go in completely different directions.
3. No matter how far apart two points are, no matter how long
   their trajectories stay far apart, eventually, they’ll
   wind up in almost the same place.

All of this is a fancy and complicated way of saying that in a chaotic
system, you never know what the heck is going to happen. No matter how long
the system’s behavior appears to be perfectly stable and predictable, there’s
absolutely no guarantee that the behavior is actually in a periodic orbit. It
could, at any time, diverge into something totally unpredictable.

Anyway – I’ve spent more than enough time on the definition; I think I’ve
pretty well driven this into the ground. But I hope that in doing so, I’ve
gotten across the degree of unpredictability of a chaotic system. There’s a
reason that chaotic systems are considered to be a nightmare for numerical
analysis of dynamical systems. It means that the most miniscule errors
in any aspect of anything will produce drastic divergence.

So when you build a model of a chaotic system, you know that it’s going to
break down. No matter how careful you are, even if you had impossibly perfect measurements,
just the nature of numerical computation – the limited precision and roundoff
errors of numerical representations – mean that your model is going to break.

From here, I’m going to move from defining things to analyzing things. Chaotic
systems are a nightmare for modeling. But there are ways of recognizing when
a systems behavior is going to become chaotic. What I’m going to do next is look
at how we can describe and analyze systems in order to recognize and predict
when they’ll become chaotic.

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:

1. Sensitivity to initial conditions,
2. Dense periodic orbits, and
3. 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 beginning 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 minuscule deviation, any unmeasurably small difference between the theoretical stable state and the actual state of the system – and at some point, your 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 perturbation, 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.