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
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: sn+1 = f(sn). 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 fN(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.