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.