Hello, and welcome to Unit 6 of the course.
This unit is on the phenomenon of universality.
In the previous unit,
we looked at the bifurcation diagram for the logistic map,
and we saw repeated transitions
from periodic behavior to aperiodic behavior.
And, along the way,
as this transition was occurring,
we saw period doubling--
say, the period went from one to two, two to four,
four to eight, eight to sixteen, and so on.
These period doublings on the bifurcation diagram
appeared as those pitchforks or sideways "U's"
that we saw again and again and again
as we zoomed in and out of the diagram.
In this unit, we'll see that some features
of these period doublings--
the period doubling transition to chaos--
are universal. And what that means is,
is that there are certain properties that are
the same for many, many different functions,
not just the logistic equation.
Moreover, we'll see that some features
of period doubling in real, physical phenomena
are also the same as in the simple logistic equation.
It's an amazing result--seems almost magical--
and I think it's one of the most incredible realizations
that comes from dynamical systems.
So, we'll get started by looking at the
bifurcation diagram again for the logistic equation,
and then we'll look at the bifurcation diagram
for some other equation and note some similarities.
So, let's get started.
So, here's our old friend, the logistic equation.
"f of x" is r x(1-x)
Here's a graph of the right-hand side of this function, and as we've seen,
it's just an upside-down parabola.
It has its maximum at .5 and, changing r,
(r is a parameter)
has the effect of stretching this function up.
So these would stay pinned, but this would get stretched up this way as I increase r.
In the last unit, we spent quite a while looking at bifurcation diagrams for this.
And, to just remind you what that looked like, here's the bifurcation diagram--
the full one from 0 to 4. Most of the action occurs in here.
Here's a zoomed-in version. This is 3 to 4. And we see period 2 turning into 4,
4 into 8, and so on, regions of aperiodic behavior, these dark, filled-in regions.
And then, periodic windows. So we got pretty familiar with the shape,
and you zoomed in and out of this quite a bit.
So, that's the logistic equation.
But we could ask, what about other equations?
Could we make bifurcation diagrams for them?
And, what do they look like?
So, let's do that.
First example I'll do is the cubic equation,
and that's almost the same as the logistic equation, but I just square the x here.
So there's the cubic equation. The only difference is a square here--there isn't a square there.
It's called cubic because if you multiplied this out, you get an x cubed term, hence "cubic".
Here's a plot of the function.
Note that it's no longer symmetric; it increases,
has a peak, goes down, but the peak is over here at about 0.65, or something.
As before, r is a parameter that has the effect of stretching this function up.
The larger r is, the steeper and taller this would become.
We can make a bifurcation diagram for this,
just like we did for the logistic equation.
And here's the result of doing that.
So here's a bifurcation diagram for the cubic equation,
and it's kinda similar. It's not identical to that for the logistic equation,
but the same general features are evident:
period doubling (1 to 2, 2 to 4), regions of aperiodic behavior interrupted by windows.
Let's see if I can get these both in the screen, here.
So, very similar. Not quite identical; they're shaped a little bit differently, but very similar bifurcation diagrams.
And that's perhaps a little weird because the two functions, well, they both have a single peak,
and they're both smooth, but this one is asymmetric and this one is symmetric,
so different-looking functions are giving rise to very similar-looking bifurcation diagrams.
So let's look at one more function, and I'll look at the sine function.
That's (f of x) is r times sine (pi x over 2)
There's a sine function. I think this is for r = 1 here.
Note that the x value now goes from 0 to 2, instead of from 0 to 1,
as for all of these functions, increasing r stretches this function up,
so it will make this steeper, this higher, but it will keep the zeros here,
and we can make a bifurcation diagram for this
and let's see, if I do that, I get this.
Again, this looks pretty familiar. This is a lot like the logistic equation.
Period 1 to 2, 2 to 4, regions of aperiodicity interrupted by periodic windows.
See, just for comparisons' sake, there is the bifurcation diagram.
On the lower left, this is the logistic equation.
On the upper right, this is the sine equation.
So, again we're seeing similar, very similar bifurcation diagrams for--on the surface--quite different functions.
This is a second-order polynomial; it's just a parabola.
This is a trigonometric function. They're giving rise to very similar bifurcation diagrams.
So that's the puzzle that we'll begin this unit with:
How is it that these three equations, and indeed, many others, give rise to similar bifurcation diagrams?
And, are these bifurcation diagrams merely sorta lookalike, or is there anything that's exactly alike about them?
To do so, in the next lecture I'll go back to the logistic equation and we'll look
in much more detail at this period doubling wrapped in chaos.
Finally, let me just mention (you've probably suspected as much) that this cubic equation
made the bifurcation diagram that appeared in the homework for Unit 5, and this sine equation
made the bifurcation diagram that appeared in the test for Unit 5.
So, you haven't seen these formulas before, but you have seen the bifurcation diagrams before.