The phone rang about 2 on Thursday afternoon, just as I was about to settle down with my book draft for a long, dull afternoon of revisions. If I was implicitly fantasizing about something to get our adrenaline pumping, I got it. Our social worker called and asked if we would consider taking a 17 month old boy with severe speech delays and special needs. Oh, we'd need to come pick him up downtown before 4:30.
Yikes. My first inclination was to say "no" since we've wanted to take a sibling group, but there was something about this that just felt right to both Eric and I. We had planned to…

# Chaos

I kind of wandered off on y'all - we just spent four days visiting and celebrating Thanksanukah ;-) with biological and chosen family. We didn't go to my parents' place for Thanksgiving this year, so we headed out and ate turkey and latkes together, spent four glorious days goofing off, and are home refreshed.
Of course, in the meantime I realized I was *supposed* to have written my Anyway Project Update before I left (it was kind of like packing my toothbrush - somehow some things just get left behind - and there's not "Anyway Project Update Store" in Beverly MA, the way there are…

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…

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 itwrong as I did miss the fundamental point.
If you recall, the basic definition of a chaotic system is
a dynamic…

Another one of the fundamental properties of a chaotic system isdense 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…

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:
Sensitivity to initial conditions,
Dense periodic orbits, and
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.…

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?
If there was an unlimited…

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…

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
which has two different answers.
First, attractors provide an interesting way…

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…

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…