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.