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:

- Exactly where you start has a huge impact on where you’ll end up.
- 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. - 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.