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In nonlinear dynamics, we use lots of
different representations
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to understand what's going on.
You've seen two:
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the physical space, like when I have
the camera on my pendulum,
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and the time-domain plot on the axes x_n
versus n.
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There are several others that are really
useful, and the goal of this segment is
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to introduce you to one of them.
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It's called the return map, it's also
known as the correlation plot,
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and sometimes as the cobweb diagram.
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It's a different way of plotting the
iterates of a 1D map
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or other scalar data.
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"Scalar" by the way means it's just a
single number, not a vector.
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Instead of x_n versus n, we plot x_n+1
versus x_n.
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The return map is useful because it brings
out the correlations
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between successive points, hence the
alternate "correlation plot" name.
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"(First)" is because of this "1"; you
could also plot a "second" return map
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if you plotted x_n+2. You might do that
if you were interested in figuring out
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whether there was some sort of important
2-time-click correlation going on
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in your data.
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The time domain plot, in contrast, is
useful because it brings out the overall
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temporal patterns of the iterates.
The return map, which is what
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I will call it most of the time, is also
really useful because it gives us
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a graphical solution technique.
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Here's the idea: imagine if we were
working with the logistic map.
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That function on the right-hand side here
defines an upside-down parabola
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on these axes. This is the function
R(x_n)(1-x_n). If R increases,
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that parabola will rear up a little
higher.
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Now there's another very important
feature on such a plot:
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the line that defines the function
x_n+1 = x_n.
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Now recall the definition of a fixed
point. A fixed point x* is a point where
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the dynamics don't move.
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So the fixed points of this system
have to be on that green line,
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and they also have to be on the
blue parabolas.
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So you can see, the crossings are where
the fixed points could be.
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Whether or not a particular crossing is a
stable, attracting fixed point depends on
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the geometry of the blue parabola, and
the green line, as we will see.
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But first I want to show you how to
actually use this kind of plot as a
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graphical solution technique. Imagine
that you're starting at, say, x_0= "here".
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Then the act of evaluating the logistic
map is equivalent to walking straight up
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to that blue curve. So that vertical line
is the evaluation of the function of the
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logistic map R*x-0 (1-x_0). Then to
figure out what x_n+1 is, you look at
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how high that point is, which you can also
think about as walking over horizontally
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to the green line. So here's x_1, and our
next task is to figure out x_2,
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which is equivalent to going to this point
and walking vertically to the blue curve.
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This is kind of hard to see; I'll make a
bigger drawing so you see what I mean.
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And I've used the colors here to
distinguish the vertical movements,
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which are the evaluation of the function,
and the horizontal arrows,
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which are effectively setting the result
equal to the next iterate.
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Now you can continue this process,
(you get the idea here, I hope),
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and these points on the curve tell you
where the iterates are.
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Okay, thought question: is this a fixed
point? What do you think? I'd say it is,
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because trajectories from initial
conditions nearby are converging to it.
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So, it's a stable fixed point. Let me draw
another picture for a higher value of R.
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And by the way, someone is popping bubble
wrap next door, so I apologize for the
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punctuation in the soundtrack.
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Remember I said that raising the R
parameter causes the parabola to rear up
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a little bit higher. Here's the situation
at one of those higher R values.
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Let's see what happens with that graphical
solution technique.
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So what is it that's going on here? Well,
this fixed point is no longer stable.
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It's kind of like the inverted point
of my pendulum.
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That is, if I started with an initial
condition that was exactly, perfectly
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equal to that x*, which is equivalent
to starting the pendulum
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perfectly balanced at the inverted point,
then the system would stay balanced.
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So this is still a fixed point, it's just
an unstable fixed point.
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So what do you think it was about the
geometry of the blue curve and the
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green line that made one of those fixed
points stable, and one unstable?
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What do you think?
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As a hint, it has to do with the slope
of the blue curve at the fixed point.
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By the way, if we drew this sequence of
orbits in the time domain,
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we'd get something like this. This is
that oscillatory convergence that we saw
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in the app last week. This sequence is
also oscillating,
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but the amplitude of that oscillation
is growing.
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We also saw non-oscillatory convergence
last week. That would correspond to
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an iteration sequence that looked like
this on a return map.
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Finally, remember the two-cycle?
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Here's what the curve shapes have to look
like on the return map for that situation
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to arise.
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It's actually useful to think about a
second-return map when you're thinking
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about 2-cycles like this. Here, we're
plotting x_n+2 against x_n, and if it's a
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periodic orbit with period 2, then it
should be a fixed point on these axes.
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You can get the mathematical form for that
by composing the logistic map with itself
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like I've just done. And here's what the
shape of that looks like on the plot.
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And we can draw the same green line on
this plot and think about what that point
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circled in black means: that's a fixed
point of the 2-time-click map -
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the logistic map composed with itself,
applied to x_n. That's a new map,
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let's call it L^, and a fixed point of the
2-time-click map is a 2-cycle of the
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1-time-click map.
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The return map representation is really
useful. It not only helps you understand
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why the iterates go where they go,
that is, how the dynamics influence
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the state of the system, but it also helps
you understand why bifurcations happen.
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In the next segment, we'll dig into
bifurcations a little bit more.