The goal of this segment is to explore
what happens to the attractors
in the dynamics of the logistic map,
as the parameter R changes.
By way of review, here's what happens to
the logistic map if you started at the
initial condition 0.2 and the parameter
value 2 and you plot 50 iterates.
Very rapidly the trajectory reaches a fixed point.
And I showed you, changing the initial
condition, doesn't really change the
position of the fixed point, it just
changes the transient from the
initial condition to the fixed point.
All the initial conditions that I just
explored are in the basin of attraction
of that attractor. They are all raindrops
running to the same ocean.
Now I'm gonna do is raise the R parameter
slowly and show you what happens.
I'm going back where we started with
the initial condition 0.2 and the R
parameter value of 2 , and raise R to 2.2,
then 2.4, 2.6, you start seeing that
oscillatory convergence that I mentioned
the last segment, 2.8, still converging
in an oscillatory fashion but the
transient has got longer.
It's taken longer to converge to that
fixed point. 2.9 the transient is
getting much longer.
This oscillatory convergence, by the way,
played a role in the quiz problem from the
previous segment.
Aha! Something different. Looks like,
maybe the fixed point is no longer
there. But it looks like, this might be
getting closer, so let's try a few more
iterates.
Mmmm. Still looks like it might be
converges.
Let's raise R a little bit more and see
what happens.
Still looks like it might be converging
to a fixed point, but again, this
transient is really long. Let's go a little
bit higher and see what happens.
Now, it looks, for sure, like that
fixed point is gone and something
else is happening. Remember that I said
that were several types of attractors,
this is another one, it's called a periodic
orbit or a limit cycle. These are synonyms.
Here the period is two. You can imagine
this, as 1 year with lots of rabbits in
my backyard, and the next year with
lots of foxes, and then the next year
with lots of rabbits, and then the next
year with lots of foxes. Which actually
happens in real populations, just like
the fixed point we explored for lower
R values, is an attracting fixed point.
This is an attracting periodic orbit.
You can see that, right here, there is a
convergence to the periodic orbit.
If we started from a different parameter
value, we would reach the same periodic
orbit. Just from a different direction.
By the way, an important point here,
many plotting tools, like to draw lines
between points. That is inappropriate when
you 're working with maps.
A map is a discrete time system. Time
makes no sense, in between iterates.
You can even think about, what it might
be x of 0.5, it doesn't exist. Do not
connect your dots, when you're plotting
iterates of a map.
Ok. What do you think will happen
if we raise R further. Aha!
At this value, it looks like the two
cycle is no longer in existence.
The fixed point is certainly gone
and we've converged to something
called a four-cycle. This is also a
limit cycle or periodic orbit. The
period is four.
Here's what those two periodic
orbits might sound like, if you hooked
the vertical axis, of the logistic map
to your piano.
These changes from my fixed point to
a two cycle, and from a two cycle to a
four cycle, as we change the R parameter,
are called bifurcations.
That English word suggests forking in two,
but it's mathematics use is more general.
It means that there's been a change in
the topology of the attractor. We 'll get
back to the topology in attractors
later. For now, suffice it to say that
a bifurcation is a qualitative change in
the attractor, not just that a fixed
point moves, but that it vanishes
or that the period of a periodic
orbit changes. Now I sipped a word in
there, parameter. R is a
bifurcation parameter in the logistic map
in effects that dynamics in a fundamental
way, causing those qualitative changes.
Let's keep increasing the value of that
parameter and see what we see.
There is no pattern that jumps out
you here. Things don't appear to
converge to anything, not to a fixed point
not to a periodic orbit.
Let's do a longer plot. Again, doesn't
look like there is any convergence.
As you do more and more iterates
though, you start to see some
patterns, especially for example right
here, looks like something is
repeating for a while. Maybe it's almost
a periodic orbit, but then it's not.
The patterns that you're starting to see
here, are a chaotic attractor. And we
gonna spend a lot of time talking about
features and properties of chaotic
attractors. Sometimes, by the way, they're
called strange attractors.
Recall the experiments that we did with the
other attractors, the fixed point and the
periodic orbits and varying initial
conditions, when we did that the
trajectory always converge to the same
attractor. It does that here too, but it
can be pretty hard to see in a time domain
plot, like this.
The issue is that the points are tracing
out the same subset of states,
but in a different order. It's kind of
like dropping a woodchip in to an
eddy, like I talked about the very first
segment.
If you drop a woodchip, in different
parts of an eddy, the woodchip
would trace out the same structure
but in a different order.
The structure of a chaotic attractor
is a very deep and important part
of non linear dynamics. Another
important feature is sensitive
dependence on initial conditions. What
you may have heard called the
butterfly effect. Again I explained this
in the very first segment.
Here is a slide that summarizes some
of the important vocabulary and
concepts that I used in this first unit
of this course.
You should make sure that you understand
what each of these words means and how
it manifests in the logistic map dynamics.
Now ,the slides that I used in this
lectures are lecture aids , they are not
lecture notes. And that maybe a bit of
a challenge, particularly in view of
the fact that there is no a textbook for
this course.
Now, as I explained in the previous unit,
this course pulls together stuff from
lectures I've heard, my own thoughts,
different textbooks, papers I've read
so there is no single textbook, no
single source, that's a map out for
this course like there are some of
the other complexity explorer
courses.
These videos are your main source
of information in this course.
They're designed to be short, self-
contained, descriptions of
individual topics.
If you want to dig more deeply, or there's
something that you just not getting, take
a look at the links on the Supplementary
Materials page, right here.
This will be flashed out a lot more as
the course goes on,
and for each unit,
and each segment of each unit, I will post
links to things that you might need to
understand that unit, to do the homework,
and also I will post links that point to
what I would suggest you look at,
if there is some background
you're missing, or what's the next
thing to read if you wanna know more
about this.