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.