So let's summarize Unit Nine.
As usual I'll begin with a topical summary
of what we covered in this unit
and then speak a little bit more broadly
and generally about key themes and ideas.
So, this unit was on pattern formation.
It began by pointing out that we've seen
that dynamical systems are capable of
exhibiting chaos; that's unpredictable,
aperiodic behavior, butterfly effect.
But there's a lot more to the story.
Dynamical systems can do much more
than produce chaos or disorder,
they can also produce patterns,
structure, organization, and so on.
And in this unit we looked at one
particular example of a pattern forming
dynamical system, a class of systems known
as reaction-diffusion systems.
So first we talked about diffusion,
and I tried actually to demonstrate it
by dropping a few drops of food coloring
in a glass of water.
Diffusion is the tendency of a substance
to spread out as a result of random motion
so the random ink molecules or food
coloring molecules bounce around and
eventually they'll spread themselves
out evenly throughout the glass of water.
Another way to say this is that chemicals
tend to move from regions of high
concentration to low concentration and the
result of this is that diffusion tends to
smooth out differences in concentration,
leading to a homogeneous, or uniform
distribution of the food coloring
or whatever it is that's diffusing.
So diffusion is described mathematically
by the diffusion equation.
Here it is.
In this equation D is the
diffusion constant.
It's related to how fast the
substance diffuses.
Different substances will diffuse at
different rates in different media.
Often this is something that one could
measure experimentally.
This is the rate of change of
concentration, and this is the Laplacian.
This is a certain type of spatial
derivative of the system.
So I should say that, u, here is the
concentration of the chemical and
it's a function of x and y.
We might have, say, on this sheet of paper
different concentrations at different
points on the page.
So this describes the diffusion of
substances once
you know D and the initial conditions.
In the equilibrium state, when this stops
changing, if I set this equal to zero,
then I'm, in effect, setting this quantity
equal to zero, and that has the effect of
picking out the most boring
function possible.
It'll just be a flat, level distribution
if the boundary conditions allow it.
If there's more chemicals flowing
in on this side and out on this side
then we get a smooth distribution and
interpolation between those two values.
Again, the main point is that diffusion
leads to boring functions;
functions that are as
homogeneous as possible.
But things get more interesting
in reaction-diffusion systems.
In this we now have two different
types of chemicals,
often called "u" and "v" or "a" and "b".
The details may vary, and u will be the
concentration of one chemical as a
function of x and y, so different
concentrations at different points in
space, same story for v.
And both of these chemicals diffuse they
spread out across the surface
but they also interact with each other.
So the equation of motion, what determines
their values, the rule of the dynamical
system are these two equations.
So, this is just diffusion for u.
But then this says there's some added term
that's an interaction, typically between
u and v, so it's diffusion, plus something
else that depends on u and v.
Same thing for the v equation, v will
diffuse plus there's some other
different interaction, a function g,
but it's a function of u and v.
So it's diffusion, plus something else.
This is a deterministic, dynamical system
there's no element of chance,
no stochasticity in the rules, and it's a
spatially-extended dynamical system.
So if we give the initial condition,
initial values of u and v everywhere
in the lattice or the rectangle,
it doesn't have to be a lattice,
and we specify what the conditions are at
the edges, then that determines the future
values everywhere on this piece of paper.
Another point that is important is that
this rule is local. Let me explain that.
This use of local is not an entirely
standard one, but I think it's justified.
The idea is that if I want to determine
the next value of u so I know the present
value of u and v at some particular point
on our surface, if I want to figure out
the next value using something like
Euler's Method, all I need to know is the
value of u here, the value of v here, and
the derivatives of u and v at this point.
So what I mean by local is if I want to
know the next value of u, how u changes,
at this point, I don't need to know
the value of u over here.
So that these equations are all local and,
in any given equation, x and y
gives reference to the same point.
So another way to say this is that there
aren't any long range interactions here.
The next value of u is determined by
the present value of u, and a little bit
about it's curvature given by the
Laplacian at that point.
So we have a local, deterministic,
spatially-extended dynamical system.
So in reaction-diffusion systems or
specifically, activator-inhibitor systems
the following setup typically is the case:
u is usually an activator.
It's something that catalyzes its own
growth, so the presence
of u gives rise to more u.
Rabbit growth, exponentially growing
rabbits, are an example of this.
We don't usually talk about rabbits
catalyzing their own growth,
but in a sense that's what they do.
Now v is some sort of an inhibitor.
It's something that typically would
grow in the presence of u,
but it would also inhibit u.
It prevents u from growing too much.
In the example I gave, these were foxes,
foxes grow in the presence of u;
foxes eat rabbits, there get to be more
foxes but the foxes inhibit the rabbits.
They prevent them from
growing and growing.
So we have these two things
and that v diffuses faster than u.
This can lead to stable
spatial structures.
We saw some examples of that. I'll show
you one again in a second.
The particular shapes determined depend
on a number of different things:
the relative diffusion rate, how much
faster one diffuses than the other,
and also the geometry of the system.
In some sense, the initial values as well.
Of course it will also depend on the
particular functions you choose for f, g.
There are lots of different possibilities
and one can do three
components, u, v, and w.
There are lots of different models here
and the mathematics of analyzing them
analytically gets pretty complex
pretty quickly.
In the additional reading section for this
unit I have some suggestions for places
you can go to learn more.
So these are reaction-diffusion systems,
one particular type of them.
Here's some results. We experimented
with the excellent program at
Experimentarium Digitale.
Here's the url. There's a link to this
on the links to program page.
This was the Guépard setting.
So this makes Cheetah spots.
There it is. It doesn't show up quite that
well in black and white.
It's a more compelling picture on the
screen.
Here the diffusion rate of u is 3.5, that
may be hard to see, and b was 16.
And that program specifies what the f and
g functions are as well.
The main point is that even though we have
a diffusive system, where things should be
spreading out, you wouldn't expect to have
a higher density of u here than here and
have that be a permanent situation.
It should diffuse away, that's what
diffusion does; it smooths things out.
But, if we have a reaction-diffusion
systems where things interact in the way
I describe, one can get a variety of
different stable spatial structures.
These systems form patterns stable spatial
structures, seemingly out of nowhere.
This isn't just a mathematical result,
lots of physical systems do this.
I showed a little bit of a video of a
Belousov Zhabotinsky experiment taking
place on a Petri dish. Here's a link to
that YouTube video. This link is also in
the additional resources section.
This video is by Stephen Morris of the
University of Toronto who we'll be talking
to next week.
So this looks a little different than the
cheetah spots but it's the same general
thing; reaction-diffusion, we get these
propagating wave fronts that move out and
then the wave fronts collide with each
other and interact in interesting ways.
So to summarize once more,
pattern formation.
There's more to the study of dynamical
systems than just chaos. Lot's more.
It's very often the case that simple,
spatially-extended dynamical systems
with local rules, like these reaction-
diffusion systems, are capable of
producing stable, global patterns
and structures.
This begins to give some insight into how
patterns might emerge in an otherwise
structureless world.
The reaction-diffusion systems that we
study here are just one of many, many
examples of pattern forming systems.
There's lots of different classes of
models; the partial differential equations
of reaction-diffusion systems is
just one of them.
In general, there's a certain creativity
to these simple dynamical systems that
can not only produce chaos but can also
produce these interesting structures.