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.