Here is the webpage where you can find the next program I would like to use or to show you.

That will let you explore the Hénon attractor.

This page is not on the Complexity Explorer site, it is at the CNRS site in France (https://experiences.math.cnrs.fr/L-attracteur-de-Henon.html).

And, accordingly, it is in French.

But I don't think you will need to read any French to understand what is on the page.

Up here is a little bit of history of the Hénon map and the Hénon attractor.

You can use Google Translate to translate in English or any other language if you want.

But if you scroll down the page, you get to the javascript program here.

And this is what I want to focus on.

As before on the program on Complexity explorer, you can choose 'a' and 'b'.

And then it will plot the attractor for you.

It defaults to the standard values a=1.4 and b=0.3, that is what Michel Hénon studied originally.

And we see again this attractor shape, this bend curve 'U' sort of shape.

If you click on this, you can zoom in.

And every time you click, it zooms in by a factor of 4.

So, let us see what is going on here in this region that appears solid.

We zoomed in and, something to note is that when we do that you can see the individual dots being plotted.

And they are being plotted in the order that they arise in iteration.

So you can see, and I said this before, and maybe this obvious when you looked at the time series plots,

that this complex geometric shape, which is smooth, is made by points that bounce around.

It is not the case that the orbit slides along this way like we saw, say, for the Lotka-Volterra equation lines in phase-base.

Here, at this (x,y) point, there are dots that jump around.

I will zoom in again and you can see the individual dots appearing on the screen.

I will zoom in once more, and now, since we are looking at such a small piece of the large shape,

you can see this phonomenon of points being kind of filled in one at a time.

Clearly, the orbit is not sliding up and down these lines.

That is one point: [the map,] the functions are making the shape by bouncing the points around.

Another thing to notice, which you probably already noticed, is that this is a fractal.

[That one sees] That any time you see a solid line and you zoom in on it, it actually turns into 2 lines, or perhaps into 4 lines or 8 lines.

The structure is something that is folded over on itself, again and again and again... like a pastry dough or something like that.

Any time you think you have a single flake of dough or flake of the attractor,

if you zoom in, you actually realize that that was not one flake, that that actually is two flakes.

And these two flakes each are made up of two flakes and so on.

We zoom in now, more and more, and we keep seeing this structure, where single lines turn into two lines.

And wee zoomed in many, many times now (I lost track of how many times), but we still see that there is some structure here.

You zoom in more and more, and there continues to be this sort of structure.

Let me zoom back out. You can do that by clicking here.

And there is the full attractor again we were zoomed way in, somewhere over here.

Let me zoom in here a couple of times, just for fun.

You can see, the shape is bend within itself again and again.

You can see more and more of those bends.

You can see here, this looks like it is actually two lines, not one.

That is double also, not single.

So there is an incredible intricate structure to this shape.

The motion on it is chaotic.

But it is an attractor because orbits are pulled towards it.

You can explore and zoom in on different regions and play around with this.

The last thing I want to show you on this program is taking a look at different parameter values.

When we go back here, you can change the parameter values with this slider bar or by typing in.

I am going to type it in, I am going to a=0.8 and then I hit 'enter'.

And that was the example we started this subunit with.

This was the periodic example of period 2:

There is one point.

And there is another point.

This must have been about 1.2 and -0.1 or something.

There are different scales on the x- and y-axis.

So, just seeing again this, sort of, period-2 behaviour.

We take the slider bar and move it down to a= 0.1.

And then I can step up things by 0.1 at a time.

Here we see just a single fixed point, at a = 0.1.

And I am going to increase a by just hitting the right arrow key on my keyboard (to a = 0.2).

And I notice that the single fixed point is moving, but it is still one single fixed point.

Somehow like what we saw in the logistic equation: it made one fixed point but that fixed point moves around.

Increasing a (to a = 0.39), and there we have period 2.

So, there is period 1 and there is period 2.

And at the transition one sees a little bit of smudging or smearing just like we did in the bifurcation diagram for the logistic equation.

I will increase it more - sometimes the one is off the screen here, but it is going to come back.

They are both back now.

Period 2, this is at a = 0.8, that is the one we started before.

Increasing this more and we will see this turn into period 4.

There it is (at a = 0.92): one, two, three, four.

It is going to split, this is period doubling again.

One could calculate the a values at which this occurs and it would agree with Feigenbaum's number.

There is period 8.

And then we cannot really resolve period 16, it splits into some chaotic region, some chaotic behaviour.

Strange attractors all along here.

The shapes are a little different.

Increasing up (a=1.23). Oh, now we are in a periodic window.

This looks like period 5, I think.

Maybe it is period 6, or maybe that is a smudge on my monitor, I cannot tell.

But it looks period 5, oh no: 1, 2, 3, 4, 5, 6, 7 - that is period 7.

a = 1.25 is period 7, that is a period 7 window.

That doubles, doubles, and then back to chaos (for a = 1.28).

And we can increase our way up to a = 1.4.

So, one could form a bifurcation diagram for a system like this.

[...] Either do a really complicated 3D plot that is difficult to visualize or restrict oneslef to just plotting either the x final states or the y final states.

In any event, this program, I think, is pretty easy to use:

it is easy to explore it with different parameter values, it is easy to zoom in.

This can be a fun tool to explore the Hénon map a little bit further.

So, we have seen our first example of a strange attractor.

The attractor for the Hénon map with the standard values a=1.4 and b=0.3, and for many other values is a strange attractor

that has this sideways U structure that is folded over itself again, and again, and again...

Let me just underscore what a 'strange attractor' is:

A strange attractor, first of all, is an attractor.

That means that almost all orbits are going to be pulled into that attractor.

We will see that general shaping no matter what the initial conditions are.

Remember that attracting behaviour is stable, [a step?] that we would expect to observe.

The 'strange' part is that the behaviour on the attractor is not periodic but chaotic.

It has sensitive dependence on initial conditions and is aperiodic.

It stays on the attractor, on that U shape thing, bouncing around.

But the motion is aperiodic, it has sensitive dependence on initial conditions.

It is chaotic, because it is a deterministic system and the orbits are clearly bounded.

So, this is a form of stable chaotic behaviour.

The strange attractor, sort of, is a merging of what one before might have formerly thought to be opposites.

It is a predictable shape, no matter what the initial condition is, we are going to see that shape.

But the motion on the attractor is unpredictable, strongly unpredictable in the sense of the butterfly effect.

It is unstable in the sense that, again it has a butterfly effect,

so unstable meaning that if we have two initial conditions that start very close, they get pushed apart very quickly.

Butterfly effect indicates a certain type of instability.

On the other hand it is stable: because even though they get pushed apart, they are going to stay on the attractor.

If an external influence comes and pulls something off the attractor, it will get pulled to it very quickly and then it will continue moving on the attractor in a chaotic and unpredictable fashion.

So, strange attractors combine order and disorder, predictability and unpredictability.

Another thing I just want to highlight for you about the example we just did is that the Hénon attractor is, in some ways of looking at it, a very complex shape.

It is this sideways U that is folded over itself again, and again, and again...

Never the last, that complex or complicated shape is made by a very, very simple iterated function.

Again, it is just a parabola, one of the functions is linear (y_{n+1} = b <i> x_n), the other one is a parabola (x_{n+1} = y_n + 1 - a </i> x_n^2), and it is able to make that incredibly complicated shape.

So again we see that simple systems, when iterated, when put in that feedback loop, are capable of producing outcome with surprising complexity.

In the next subunit we will see that not only iterated maps have strange attractors but differential equations can have strange attractors, too.

And the example we will use is the Lorenz equations.