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.