In the previous two subunits we've looked
at two dimensional differential equations.

We generalized the idea of a 
phase line to a phase plane.

And we saw that two dimensional 
differential equations are capable of

somewhat more interesting behavior than
1D differential equations.

In particular, 2D differential equations
can show cycles or oscillations, and

sometimes those oscillations
are attractors.

In this subunit, we'll continue our
journey into higher dimensions by looking

at two dimensional iterated functions.

We've looked at one dimensional iterated
functions a lot, mainly the logistic

equation. That's one dimensional because
we're keeping track of just one variable,

usually x over time. In this subunit we'll
look at two dimensional iterated functions

where we're keeping track of two variables
at a time. We'll see how that works and

how to visualize orbits in two dimensions.

The two dimensional iterated function that
we'll look at is something known as the

Hénon map. It's given by these two
formulas.

This is an iterated function, like the
logistic equation, except now there are

two variables, x and y. So an orbit, or
itinerary, would be a list of two numbers,

not just a list of one numbers.

So this says to get the next x value, you
take the current y value, add 1 to it

and then subtract a(x_n)^2. 
"a" in this case is a parameter and "b"

is a parameter too.

And then this says that the next y 
value is b times the current x value.

In order to get numbers out of this, we
need to choose values for parameters

a and b, like we needed to choose a value
for the parameter r in logistic equation.

For this example, I'll choose a=0.9 and
b=0.3. So those are my parameter values.

We also need to choose initial conditions.

There are two initial conditions now
because there are two variables, x and y.

For my x initial value I'll choose 0.2 and
for my y initial value I'll choose 0.4.

Let's calculate the first iterate;
that'll be x_1 and y_1.

So to do that we follow the equation. 
The equation determines the next value,

and the next value and the 
next value and so on.

This is, as usual, a deterministic
dynamical system.

So x_1=y_0+1 minus a(x_n)^2,
a=0.9 and x_n=0.2.

For y_0 I should've put in 0.4.
I'll write this again.

x_1=0.4+1-0.9(0.2)^2

Let's do this on a calculator.

(0.2)^2=0.04. Let's multiply that by 0.9,
0.036. So this becomes 0.4+1-0.036.

Now let's do this on the calculator. 
0.4+1-0.036=1.364. So that's x_1.

Let's also get y_1. We need to keep track
of two variables now, not just one.

I think I have enough room to do this,
let's see. So this equation tells me that

y_1=b times x_0. b=0.3, x_0=0.2.
So y_1=(0.3)(0.2), you can do this on a

calculator or in your head depending
how nimble mentally you're feeling.

And this will give you 0.06.

So I'm just following the rule, like we
always do with iterative functions;

we're just applying the same rule over
and over again.

In this case I just did it once. And I got
that x_1=1.364 and that y_1=0.06.

These two taken together are
my initial conditions.

This is my first iterate, x and y.

So as you've probably guessed, we'll want
to do this many, many times and soon we'll

turn to a computer to do this. Before we
do so, I suggest doing the quiz that

follows this lecture to make sure you see
how this two dimensional iterative

function business is working.