In the last segment, I introduced the
notion of a return map:
plots of x_n+1 vs x_n.
These bring out the correlations between
successive points in a trajectory.
Now, a hard thing about this field when
you're first getting into it, we use lots
of different representations to
understand & explain dynamical systems,
and we often switch between these
representations quickly
and without explanation.
I'm going to try to narrate when I switch
between representations
and tell you what each one is good for.
So what I'm gonna do right now is try to
help you make the connection between the
time-domain plots, which are x_n vs n,
and return maps, which are x_n+1 vs x_n,
and I'm going to do that by showing you
lots of different side-by-side pictures
in those two domains, like I started to
at the end of the previous segment.
The app that you used in the last unit to
explore the logistic map
makes that very easy, and it even puts in
all those lines that I was drawing
in different colors to show where the
iterates are, and it makes them alot
straighter than I did.
The right-hand plot in this app shows what
happens if you iterate the logistic map
from this initial condition, using this
R-parameter value, for this many iterates.
And you can see from this plot that those
iterates reach a fixed point.
You probably recall, if we raise the R-
parameter, the fixed point will move up.
Here, this is R=2; that's R=2.1.
You also probably recall, if we increase
R further, say to 2.2, you start seeing
a little bit of overshoot. Let's increase
R a little bit further to make that
even more pronounced. There's R=2.6 and
you can really see the oscillatory
convergence on the right-hand side. Now
look in here: this is a cobweb plot,
this is x_n+1 vs x_n, like a drew a bunch
of last time, and you can start seeing
that this convergence is oscillatory.
Let's make it bigger.
There's R=2.8, and you're starting to see
the squared-off, inwards spiral
that I was drawing in the last segment.
You also recall, I hope, that when we
raised R further, there was a bifurcation
causing this fixed point to go away
(actually it just became unstable,
it didn't go away), and a
2-cycle to appear.
Here's what that looks like on the return
map.
As we raise R further, the 2 cycle gets
bigger and bigger. Here's 3.3, here's 3.4,
and at a certain point, we get a 4-cycle.
Now this gets a little bit hard to see
what the asymptotic behavior is, but you
can use this "Remove Iterate" button
right here to remove some of the iterates,
starting from the beginning.
So what I'm doing here is removing the
transient.
And if I remove enough of the transient
off the front of the trajectory,
all you see is the attractor, and you can
see very clearly that it's a 4-cycle,
and you can see very clearly from the
left-hand plot what it is about
the geometry of the yellow curve and the
blue line that makes that happen.
For R=3.65, the iterates are bouncing all
over the range of the function.
This is a chaotic attractor, and you can
also see from this picture on the left
why this is sometimes called a "cobweb
diagram."
At R=3.8, the trajectory is fully chaotic,
but look what happens at 3.83.
A 3-cycle. So there's been another
bifurcation between R=3.8 and R=3.83,
from a chaotic orbit to a periodic orbit.
So how can we capture all that richness
of behavior - the different attractors
and the bifurcations between them - on one
plot?
We can use another useful representation
called the "bifurcation diagram",
and it is plotted on the axes of x_n vs R.
Now how to think about this?
Each slice of this plot is one of those
time-domain plots that we built before,
viewed from the side, like this. That's
an attempt to draw a human eye looking
down the side of that plot.
Now with the bifurcation diagram, we're
interested in the attractor,
the asymptotic behavior. So when you're
doing this, you actually
remove the transient from the front
of the trajectory as well.
Now think if you were looking at this
plot from the side, that's your eye:
what are you gonna see? You're gonna see
a bunch of dots in one place, this high.
Remember, you're not gonna see these guys,
because we threw them out.
So what you're gonna see at the R=2 mark
of that bifurcation diagram is
all those dots right on top of each other.
This slice through the bifurcation diagram
corresponds to this plot, and that slice
of the bifurcation diagram
corresponds to this plot. So what is this
eye gonna see?
Again, we're throwing out the transient.
Viewed from the side, that right-hand plot
is gonna look like 2 dots ("dot-dot")
and then "dot-dot", and so on & so forth.
So I've drawn the picture on either side
of the bifurcation that caused
the fixed point to go away, and the 2-
cycle to be born.
What if I look in between there more
carefully?
This is what I'd see. Remember, the fixed
point moves up, then bifurcates
at this point into a 2-cycle, and that 2-
cycle gets wider and wider.
And if you do that really carefully with
a computer,
rather than a tablet and a stylus, you see
something really beautiful.
And next time, we're going to dig into
that structure in more detail.