In nonlinear dynamics, we use lots of 
different representations

to understand what's going on. 
You've seen two:

the physical space, like when I have 
the camera on my pendulum,

and the time-domain plot on the axes x_n
versus n.

There are several others that are really
useful, and the goal of this segment is

to introduce you to one of them.

It's called the return map, it's also
known as the correlation plot,

and sometimes as the cobweb diagram.

It's a different way of plotting the
iterates of a 1D map

or other scalar data.

"Scalar" by the way means it's just a
single number, not a vector.

Instead of x_n versus n, we plot x_n+1
versus x_n.

The return map is useful because it brings
out the correlations

between successive points, hence the 
alternate "correlation plot" name.

"(First)" is because of this "1"; you 
could also plot a "second" return map

if you plotted x_n+2. You might do that 
if you were interested in figuring out

whether there was some sort of important
2-time-click correlation going on

in your data.

The time domain plot, in contrast, is
useful because it brings out the overall

temporal patterns of the iterates. 
The return map, which is what

I will call it most of the time, is also
really useful because it gives us

a graphical solution technique.

Here's the idea: imagine if we were
working with the logistic map.

That function on the right-hand side here
defines an upside-down parabola

on these axes. This is the function 
R(x_n)(1-x_n). If R increases,

that parabola will rear up a little
higher.

Now there's another very important 
feature on such a plot:

the line that defines the function
x_n+1 = x_n.

Now recall the definition of a fixed 
point. A fixed point x* is a point where

the dynamics don't move.

So the fixed points of this system
have to be on that green line,

and they also have to be on the 
blue parabolas.

So you can see, the crossings are where
the fixed points could be.

Whether or not a particular crossing is a 
stable, attracting fixed point depends on

the geometry of the blue parabola, and 
the green line, as we will see.

But first I want to show you how to
actually use this kind of plot as a

graphical solution technique. Imagine
that you're starting at, say, x_0= "here".

Then the act of evaluating the logistic
map is equivalent to walking straight up

to that blue curve. So that vertical line
is the evaluation of the function of the

logistic map R*x-0 (1-x_0). Then to 
figure out what x_n+1 is, you look at

how high that point is, which you can also
think about as walking over horizontally

to the green line. So here's x_1, and our
next task is to figure out x_2,

which is equivalent to going to this point
and walking vertically to the blue curve.

This is kind of hard to see; I'll make a
bigger drawing so you see what I mean.

And I've used the colors here to 
distinguish the vertical movements,

which are the evaluation of the function,
and the horizontal arrows,

which are effectively setting the result
equal to the next iterate.

Now you can continue this process, 
(you get the idea here, I hope),

and these points on the curve tell you 
where the iterates are.

Okay, thought question: is this a fixed
point? What do you think? I'd say it is,

because trajectories from initial
conditions nearby are converging to it.

So, it's a stable fixed point. Let me draw
another picture for a higher value of R.

And by the way, someone is popping bubble
wrap next door, so I apologize for the

punctuation in the soundtrack.

Remember I said that raising the R
parameter causes the parabola to rear up

a little bit higher. Here's the situation
at one of those higher R values.

Let's see what happens with that graphical
solution technique.

So what is it that's going on here? Well,
this fixed point is no longer stable.

It's kind of like the inverted point 
of my pendulum.

That is, if I started with an initial 
condition that was exactly, perfectly

equal to that x*, which is equivalent
to starting the pendulum

perfectly balanced at the inverted point, 
then the system would stay balanced.

So this is still a fixed point, it's just 
an unstable fixed point.

So what do you think it was about the
geometry of the blue curve and the

green line that made one of those fixed
points stable, and one unstable?

What do you think?

As a hint, it has to do with the slope
of the blue curve at the fixed point.

By the way, if we drew this sequence of 
orbits in the time domain,

we'd get something like this. This is 
that oscillatory convergence that we saw

in the app last week. This sequence is 
also oscillating,

but the amplitude of that oscillation 
is growing.

We also saw non-oscillatory convergence
last week. That would correspond to

an iteration sequence that looked like
this on a return map.

Finally, remember the two-cycle?

Here's what the curve shapes have to look 
like on the return map for that situation

to arise.

It's actually useful to think about a
second-return map when you're thinking

about 2-cycles like this. Here, we're
plotting x_n+2 against x_n, and if it's a

periodic orbit with period 2, then it 
should be a fixed point on these axes.

You can get the mathematical form for that
by composing the logistic map with itself

like I've just done. And here's what the 
shape of that looks like on the plot.

And we can draw the same green line on
this plot and think about what that point

circled in black means: that's a fixed
point of the 2-time-click map -

the logistic map composed with itself,
applied to x_n. That's a new map,

let's call it L^, and a fixed point of the
2-time-click map is a 2-cycle of the

1-time-click map.

The return map representation is really
useful. It not only helps you understand

why the iterates go where they go, 
that is, how the dynamics influence

the state of the system, but it also helps
you understand why bifurcations happen.

In the next segment, we'll dig into 
bifurcations a little bit more.