So lets 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.
I 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-they 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 any differences
in concentration, leading to a homogeneous
or uniform distribution of the food coloring
or whatever it is that is 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 spacial 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,say, chemicals flowing in
on this side and flowing out on this side
Then we get a smooth distribution and
interpolation between those two values.
So, 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.
So 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 you
But then this says there's some added term
that's an interaction,
typically between "u" an "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 a different interaction,
a function "g", but it's a function of u an 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
spatialy extended dynamical system
So it we give the initial condition,
initial values of u and v everywhere in the latice
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
then 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
give reference to the same point
So another way to say this is that there
are no 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 is typically the case:
u is usually an activator-
it's something that catalyzes its own growth
so the presence of u give 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.
"v" is some sort of an inhibitor
It's something that typically would also grow
in the presence of u, but 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
I'll show you an example of that 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.
And, in some sense, the initial values as well
It will also depend on the