So as we've seen, dynamical systems are
capable of producing chaotic behavior; aperiodic,
unpredictable, irregular outcomes. But there's
a lot more in the world than just chaos; than
just unpredictability. And happily so. There's
patterns, there's structure, there's organization -
yes and chaos, and randomness, and unpredictability,
and the butterfly too, but the world around us
is a really interesting mix of patterns and
irregularities. We might ask: how do these
patterns emerge, where do they come from,
how do they form? Since many of them seem
to be formed without any design or with any
direction. Where in Newton's laws - the basic
laws of electromagnetism, Maxwell's equations...
the world is fundamentally made up of simple
particles - how do we produce these
incredibly complex phenomena like you
and me and all of our surroundings? And so,
dynamical systems, I think, doesn't answer that question
of course, but sheds some light on it because
it says that iterated systems like these iterated
functions or dynamical systems like differential
equations or the partial differential equations
that we studied in this unit - that active
repetition, of feedback on itself, of doing the same
thing over and over again has some surprising
creativity built into it. That that active repitition
you can take a simple process and simple
ingredients and get things out that maybe
aren't so simple. So maybe this can begin to
shed light, or is important intuition and
realization, as we think about the patterns
and structures around us and how they might
arise. Let me give a few examples to illustrate
what I mean by patterns emerging without
direction. So, the first example is the building
in which my office is in at College of the Atlantic.
It's a building known as "The Turrets" because
it has lots of turrets on it. And this building
was designed, like most buildings. Somebody
designed it, in fact I know who it was...I didn't
know him personally, it was a long time ago.
Bruce Pryce was the architect of this building here in
Maine. Apparently he is a famous architect.
He built Le Chateau Frontenac, the Hotel
Frontenac in Quebec City, a very famous landmark
hotel that if you've ever traveled there,
you've surely seen. It's a big hotel on the bluff.
So The Turrets, it's a complex structure, it's patterned,
it's a great building; a beautiful old building,
my office is the second window on the left there.
It's a great location. This is certainly a pattern -
some sort of structure. My office...well my
office is chaos...the building is not chaos though.
And it's not that surprising because somebody
built this to not be chaos. On the other hand,
if you look outside of my office window and
you crane your neck just a little built or turn around
from where that picture of turrets was, you'll see
this tree. It's a Copper Beech tree, and it's
just this amazingly beautiful tree, I don't
know how old it is, I don't think they're
native to here. In the summer they have
almost purple leaves - it's copper, they're
darker than usual leaves, but the structure of the
tree in the snow which is sort of what it looks
like now, because it's been snowing for months
and months, is this amazing structure. Again,
you would look at this tree and you wouldn't
say chaos - yes there's some irregularity. So who
made the tree? That's a little less clear. It wasn't
designed by an architect the way the building was.
There may be some designs in DNA, but there
even then it's more of an organic process -
something that, as the tree is growing, the
tree probably isn't checking some master plans;
some blueprints. As The Turrets as it was being built,
somebody as they were putting bricks together
was checking blueprints or a master plan.
So, here's an example, the tree, a pattern that, in a
sense, wasn't built the way my office building was.
Here's another example. These are altostratus
clouds. These pictures are from a friend and
colleague, Sarah Hall, who teaches geology
here at College of the Atlantic. She took
this picture and one can see nice structure,
in fact Sarah tells me these are from convection
rolls up in the atmosphere where the clouds
are and they form these structures. So again,
you might ask "well who made this?"
That question seems almost silly I hope.
Nobody made those clouds,
the clouds just did there own thing and they're
not thinking about patterns at all, they're just
obeying the simple laws of physics just like
the reaction-diffusion equations obey the
simple laws of physics that we write down, or
simple laws of chemistry, and we get these
patterns in a reaction-diffusion system
not that different from these patterns that
we see in clouds. So here, it is a case of
clear self-organization. It's not organization from
above; the cloud is not referring to any
almanac of clouds, or master plan, or
cloud blueprint. It's just doing it's own thing,
following the laws of physics, differential
equations, and iterating itself, and these
patterns emerge; they appear. One last example,
College of the Atlantic is right next to
Acadia National Park which has Cadillac Mountain,
which is actually not a mountain by most standards.
It's not that tall, but they view from it
is still amazing. This is a view from atop
Cadillac Mountain. The College...I don't think
you can see it in this picture, but it's down
on the coast. But again, you would look out
at this landscape and say, "well who made that?"
And again, it seems like sort of a silly question.
All sorts of geological processes made the coast
and the coast looking off to another peninsula,
and then there are a bunch of islands,
and one can see if you look carefully, a swamp
area with a little bit of a wandering stream.
Anyway, so all of this pattern...you wouldn't
look at this and say "oh it's just chaos".
All of this pattern and structure must come from
somewhere. Dynamical systems suggests that
not that patterns are easy to make, but maybe
some patterns actually are easy to make, and
that they can just arise on their own without
a designer or a master plan.
So I guess one lesson I would take
from this is as follows. If, in the study of
complex systems, we see some sort of patterned
or structured outcome or behavior, maybe
in a biological system or a social system, our impulse
might be to ask, especially in a social system,
"who built that, who did that,
who made that, why is that happening",
assuming that there is some kind of purpose
or design. And indeed it could be that there
was some intentionality... whatever the system is,
some social system... in producing that outcome.
But the study of dynamical systems points out,
that one can also get structured or patterned
outcome without any design, or purpose,
or intentionality. Things can self-organize.
Sometimes we can get order,
or patterns, or structures for free.
So again, one of the themes of the course -
is that simple iterated systems/simple dynamical
systems, do not necessarily have simple
behavior. Deterministic systems can behave
unpredictably and aperiodically and simple
systems can also produce patterns and outcomes of
surprising richness and complexity.
So this brings us to the end of unit 9
and it's almost the end of the course.
In unit 10, which will be the last unit, I'll
summarize some of what we've covered so far
and try to draw some conclusions and pull together
some themes. As part of this, I'll conduct
a few interviews with other researchers and
teachers in the areas of dynamical systems
and chaos to get some different perspectives
and draw out some ideas.
So we'll see you next week in unit 10.