In the last segment, I showed you how return maps work,
how to go back and forth between them and the time domain,
and how they help you understand the dynamics, as well as
how to understand bifurcations in the dynamics as the parameter value changes.
I finished up with a 3rd representation, the bifurcation diagram.
Here's a bifurcation diagram of the logistic map.
On the vertical axis is a set of iterates of the logistic map,
at some parameter value R, which is graphed on the horizontal axis.
Just to remind you of the correspondence between this kind of plot
and the time domain, and the return map, I'm gonna draw a few pictures.
Here's a time domain plot of an orbit of the logistic map
at a low value of the parameter R that is converging to a fixed point.
On a return map, this orbit would look like this.
To construct a bifurcation diagram, you remove the transient;
that is, you iterate a bunch of times, and throw those points away,
and then you iterate a bunch more times, and you plot those points
as if you were looking at that top plot edge-on from the side.
In this case, those points would all fall on top of each other, there.
So again, each vertical slice of the bifrucation diagram
is one time-domain plot like this, with the transient removed,
viewed from the side.
If we turn R up a little bit, the time- domain plot will look like this,
the return map will look like this,
and the point on the bifurcation diagram will look like 2 dots.
Again, three different representations bring out three different things:
the time-domain plot on the top left
brings out the overall behavior of the iterates;
the return map on the lower left brings out the geometry of why
the iterates go where they go, & also the correlation between successive iterates;
the bifurcation plot brings out what changes about the asymptotic behavior
of the trajectory as R changes, including bifurcations.
Now if you repeat the procedure that we just went through at a much finer grain,
but using a computer instead of tablet and a stylus, what you'll see is this.
There's actually one more step in there which we'll circle back to
at the end of this segment.
Now, you can see all sorts of structure in this plot.
That's the main focus of this segment.
First of all, you see the fixed point coming along for low R, here,
and then bifurcating into a 2-cycle right here,
bifurcating into a 4-cycle right here, and then eventually,
getting into a chaotic regime. That's what this gray banded behavior is.
That's what this right-hand plot would look like if you looked at it edge-on
from the right-hand side of the screen.
Within the chaotic regimes, you also see these "veils":
areas where the attractor is darker than in other areas.
Those veils are related to what are called "unstable periodic orbits",
and we'll talk more about them later.
As we've seen, there's this bifurcation sequence
from a fixed point, to a 2-cycle, to a 4- cycle, to an 8-cycle, & so on & so forth.
That's called a "period-doubling cascade" for the obvious reason.
I also showed you in the last segment that there were regions of order
within the chaos; that is, for some R-value, there was chaos,
but then if you raised R a little bit, you went back into a periodic regime.
This particular periodic regime starts out with a 3-cycle, and then goes to a 6-cycle
and a 12-cycle, and so on and so forth.
So it's another period-doubling bifurcation sequence.
You may remember, in the very first segment of this course,
I showed you the title page of a paper called "Period-3 Implies Chaos".
The fact there is a period-3 orbit in this map is very, very significant.
And if people are interested in that, I can record an auxiliary video about that.
Another interesting thing to note about this structure
is that it contains small copies of itself.
If you were to zoom in on that piece of the structure inside the red circle,
it would look like the whole structure.
That is, this is a fractal object.
I'm sure many of you have heard about fractals.
Fractals are sets that have non-integer Hausdorff dimension
(mathematically, that's the formal term).
Informally, they're "self-similar".
The second row of images here show something called the Koch curve.
The way you construct this fractal is by taking an equilateral triangle,
and then taking 3 equilateral triangles, 1/3 the size in the sense of edge-length,
and sticking them to each exposed face of that thing.
Then you iterate; you take little triangles
and stick them to the sides of each of those pointy faces, and keep going.
Eventually, you'll get this beautiful structure that looks alot like a snowflake
Fractals play an interesting role in mathematics.
There are also lots of examples of frac- tals & fractal-like structures in nature.
Here's an example.
Fractals are also useful analogs for nature in computer graphics.
Here's a beautiful fractal called the Mandelbrot set,
and this video is showing you that if you zoom in on the Mandelbrot set,
you keep seeing more and more structure;
in fact, you keep seeing structure that is self-similar.
There's a whole new Mandelbrot set way down in the tendrils of the old one.
And you can keep zooming in and zooming in,
and you'll keep seeing self-similar structure.
I've included a link to that video on the supplementary materials section
of the Complexity Explorer website for this course,
right here, under the section for this segment of this unit.
Remember, this is where you should go
for links to materials that you might need to do the homework,
like this Logistic Map app,
for materials like this paper, which you would look at
if you wanted to learn more about the concepts that I talked about
in that segment.
And I've also included some links to tutorial materials
and other sorts of things that might help you if you need some background to fill in
And here's an important thing: the connection between fractals and chaos.
There is a connection, but it is not an "if-and-only-if".
Many chaotic systems have some fractal structure,
but it is by no means the case that all chaotic systems have fractal structure;
that is, there are chaotic systems that do not have fractal structure,
there are certainly tons of fractals that have nothing to do with chaos,
but the popular science press has conflated these two topics.
If you want to learn more about fractals,
you can take a look at Dave Feldman's course on the Complexity Explorer MOOC.
One last point here, relating to transient length:
remember that for some R-values, the transient was really long?
How do you think that will manifest in a bifurcation diagram?
That is, there is some fixed point here, but the trajectory is taking
a really long time to get there.
What that will look like on a slice of the bifurcation diagram is this.
That's hard to see, but I'm trying to draw a series of points coming up from the axis
and slowly getting closer and closer and closer, but taking forever to get there.
So if we want to see the asymptotic behavior,
we want to throw out the transient, but how many points do we need to throw out
if we want to get rid of the transient here?
To get rid of the transient, we actually need another step in our code here.
Really what we need to do is iterate a whole bunch of times,
but not plot those points,
and then from the ending point of that orbit,
iterate a bunch more times, and plot those points.
That amounts to omitting the transient.
But the question is, these words: how do you pick how many points to iterate
to get rid of the transient, and how do you pick how many points to plot
so that you get a really nice picture? Those are both tricky.
You want the red bunch number to be large enough so that you see the structure
but not so large that the finite size of the plotted points obscures the structure.
And you want to throw out enough points so the transient has really died out,
but how long is that? There's no way to know, really.
And they tend to get longer just before a bifurcation.
In practice, what you do is increase the number of points that you throw away
before plotting, until the periodic orbits are crisp on your plots.
That amount of thrown-away points is overkill far away from the bifurcations,
of course, where the transient is short,
but otherwise, your orbits will thicken up near the bifurcation point.
All of that will play a role in the next segment, where we'll dig into the pattern
behind the shrinking widths and heights of the pitchforks in the bifurcation plot.