## The End of Defining Chaos: Mixing it all together

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.

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:

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.

## More about Dense Periodic Orbits

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:

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.

## Orbits, Periodic Orbits, and Dense Orbits - Oh My!

Another one of the fundamental properties of a chaotic system is
dense periodic orbits. It's a bit of an odd one: a chaotic
system doesn't have to have periodic orbits at all. But if it
does, then they have to be dense.

The dense periodic orbit rule is, in many ways, very similar to the
sensitivity to initial conditions. But personally, I find it rather more
interesting a way of describing key concept. The idea is, when you've got a
dense periodic orbit, it's an odd thing. It's a repeating system, which will
cycle through the same behavior, over and over again. But when you look at a
state of the system, you can't tell which fixed path it's on. In fact,
miniscule differences in the position, differences so small that you can't
measure them, can put you onto dramatically different paths. There's
the similarity with the initial conditions rule: you've got the same
basic idea of tiny changes producing dramatic results.

## Chaos and Initial Conditions

One thing that I wanted to do when writing about Chaos is take
a bit of time to really home in on each of the basic properties of
chaos, and take a more detailed look at what they mean.

To refresh your memory, for a dynamical system to be chaotic, it needs
to have three basic properties:

1. Sensitivity to initial conditions,
2. Dense periodic orbits, and
3. topological mixing

The phrase "sensitivity to initial conditions" is actually a fairly poor
description of what we really want to say about chaotic systems. Lots of
things are sensitive to initial conditions, but are definitely not
chaotic.

Before I get into it, I want to explain why I'm obsessing
over this condition. It is, in many ways, the least important
condition of chaos! But here I am obsessing over it.

As I said in the first post in the series, it's the most widely known
property of chaos. But I hate the way that it's usually
described. It's just wrong. What chaos means by sensitivity
to initial conditions is really quite different from the more general
concept of sensitivity to initial conditions.

## Back to Chaos: Bifurcation and Predictable Unpredictability

So I'm trying to ease back into the chaos theory posts. I thought that one good
way of doing that was to take a look at one of the class chaos examples, which
demonstrates just how simple a chaotic system can be. It really doesn't take much
at all to push a system from being nice and smoothly predictable to being completely
crazy.

This example comes from mathematical biology, and it generates a
graph commonly known as the logistical map. The question behind
the graph is, how can I predict what the stable population of a particular species will be over time?

## Chaotic Systems and Escape

One of the things that confused my when I started reading about chaos is easy to
explain using what we've covered about attractors. (The image to the side was created by Jean-Francois Colonna, and is part of his slide-show here)

Here's the problem: We know that things like N-body gravitational systems are chaotic - and a common example of that is how a gravity-based orbital system that appears stable for a long time can suddenly go through a transition where one body is violently ejected, with enough velocity to permanently escape the orbital system.

But when we look at the definition of chaos, we see the requirement for dense periodic orbits. But if a body is ejected from a gravitational system, ejection of a body from a gravitational system is a demonstration of chaos, how can that system have periodic orbits?

## Strange Attractors and the Structure of Chaos

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

## Defining Dynamical Systems

In my first chaos post, I kept talking about dynamical systems without bothering to define them. Most people who read this blog probably have at least an informal idea of what a dynamical system is. But today I'm going to do a quick walkthrough of what a dynamical system is, and what the basic relation of dynamical systems is to chaos theory.

The formal definitions of dynamical systems are dependent on the notion of phase space. But before going all formal, we can walk through the basic concept informally.

The basic idea is pretty simple. A dynamical system is a system that changes
over time, and whose behavior can be (in theory) described a function that takes
time as a parameter. So, for example, if you have a gravitational system which
has three bodies interacting gravitationally, that's a dynamical system. If you
know the initial masses, positions, and velocities of the planets, the positions of all three bodies at any future point in time is a function of the time.

## Chaos

One mathematical topic that I find fascinating, but which I've never had
a chance to study formally is chaos. I've been sort of non-motivated about
blog-writing lately due to so many demands on my time, which has left me feeling somewhat guilty towards those of you who follow this blog. So I decided to take this topic about which I know very little, and use the blog as an excuse to
learn something about it. That gives you something interesting to read, and
it gives me something to motivate me to write.

I'll start off with a non-mathematical reason for why it interests me.
Chaos is a very simple idea with very complex implications. The simplicity of
the concept makes it incredibly ripe for idiots like Michael Crichton to
believe that he understands it, even though he doesn't have a clue. There's an astonishingly huge quantity of totally bogus rubbish out there, where the authors are clueless folks who sincerely believe that their stuff is based on chaos theory - because they've heard the basic idea, and believed that they
understood it. It's a wonderful example of my old mantra: the worst math is no math. If you take a simple mathematical concept, and render it into informal non-mathematical words, and then try to reason from the informal stuff, what
you get is garbage.

So, speaking mathematically, what is chaos?

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