# Previous Henon Attractor Discussion

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## Checklist

```  1. Are the words that you define bold? YES
2. Do you have references and other interesting links? YES
3. Did you cite your pictures, or say if you created them? YES
4. Have you considered all of the comments on the discussion page? YES
5. Have you looked over everyone else’s pages, and linked to the relevant ones? YES
```

#### Abram 7/9

Really nice edits. OK, last comments, I promise.

I tried to do some more research on the history/application of Henon Attractors, and could only find a bit about the origin of the attractor and Michel Henon's intent in modeling celestial orbits and developing the attractor - nothing on other applications.

Huh, weird. Do you have any idea what particular problem he was studying, or what physical system the equations were supposed to represent? I'm guessing not...

Actually, the points are not appearing one at a time, because I created the animation in Matlab and had to set the number of points plotted per frame to increase exponentially for each frame so that the animation wouldn't take too long. I've asked the Drexel students to make a better animation with smaller points and better timing. Hopefully, that will do the trick. Oh, gotcha. It makes sense that you had to do that. I still think it would be nice, then, to describe in the text what's "wrong" with the animation, but I could go either way on this, and it would certainly take some time.

Yeah, I think I'm running out of time, so I've put up a correction to my animation suggestion for the Drexel Students. I'll be in contact with them to make sure that the animation is done correctly. This way, I don't think I'll write an explanation of my current animation on the page, because it'll be replaced soon enough.

I just looked more closely at the solving of the systems of equations, and saw a few small things to change.

• Immediatley after $x_{n+1} = x_{n}\$, and $y_{n+1} = y_{n}\$, write something like "we can just refer these values as x and y, respectively."
Ok, that's a good idea.
• Between the statement of the system of equations and the next equation you arrive at via substitution, write something like "substituting the solution for x from the first equation into the second equation gives..." or maybe you can come up with better wording?
Sure.
• I worry about this x_1,2 notation. I'm worried it will throw people off because they are used to seeing the subscripts refer to an iteration number. I think you could drop the subscripts if you wanted.
Ok, that makes sense. Thanks! Ryang1 (7/10)

#### Abram 7/9

Nice edits. The page is really well laid out now. There are just a few details that are a bit unclear. They are as follows:

More important

Basic Description:

• You do a good job summarizing how the mathematical process works. Could you somewhere include a description of what kinds of objects are modeled using Henon Attractors, or more generally why they are interesting to mathematicians? I feel like that's the one thing I need to be drawn in a bit more.
I tried to do some more research on the history/application of Henon Attractors, and could only find a bit about the origin of the attractor and Michel Henon's intent in modeling celestial orbits and developing the attractor - nothing on other applications.
• The animation is great, but I still feel like it's hard to tell that points are appearing one at a time. Can you explain this in the text?
Actually, the points are not appearing one at a time, because I created the animation in Matlab and had to set the number of points plotted per frame to increase exponentially for each frame so that the animation wouldn't take too long. I've asked the Drexel students to make a better animation with smaller points and better timing. Hopefully, that will do the trick.

Changing "a" and "b":

• Although the original Henon Attractor... By the "original", do you mean the one from the main image in this page or Michael Henon's original attractor, or something else?
I mean Michel Henon's "original" attractor...which is also the fractal in the main image. I'll try to be more explicit.
• As we can see below, a and b are limited to the range of values outside of which the fractal ceases to resemble the Henon Attractor. It's not at all clear from simply looking at the examples that there are values of a and b that don't allow the creation of a Henon Attractor. Instead of saying, "as we can see below," what about just informing us that only certain values of a and b will "work."
Ok, that's a good idea.

Fixed points:

• $x_{n+1} = x\$, and $y_{n+1} = y\$ should read $x_{n+1} = x_{n}\$, and $y_{n+1} = y_{n}\$
Yes, you are right.
• I'm curious -- is it possible to actually arrive at a fixed point from outside a fixed point. i.e, for the fractal to become "stagnant" as you put it (and it would be cool to answer this question not just to me personally, but to put the answer on the page).
I'm still looking into this...
• What does "within the bounds of the attractor" mean?
Hmm, I think this phrase refers to the attractor's "basin of attraction", which is something that I just came across recently and did not have time to research (it's in the Further Directions section). I'll re-word it, so that if someone comes along and adds the section on "basin of attraction", they can explain it better in that new section.

Less important

Basic Description:

• It is plotted in an irregularly matter... The manner of plotting is totally regular. It's the sequence of points that seems chaotic. I would just drop this phrase. You explain the irregularity later.
Good point (and whoops, horrible grammar - I must have re-worded that phrase a few times).
• This image results from an iterated function, meaning that the equations that describe it can be applied to itself an infinite amount of times. You've already made this clear. I say drop it.
Ok. Ryang1 (7/9)

#### Abram 7/8

I just read this page for the first time, and I like it a lot. It's really interesting material, and you've presented it in a way that is clear and accessible to readers at many levels.

I second all of Anna's questions. I have a couple of others as well.

The main thing I'm missing is an informal description at the beginning of the page of what the Henon Attractor represents.

It's hard to tell from your otherwise great animation that the picture is starting with just one point, and that one point at a time is being added to the picture. Can you reassure the reader that this is the case? The Drexel animation you suggest (which sounds great) would address this issue, but maybe some text would make a good stopgap measure.

Ok, I see what you mean. I re-worded the Basic Description section to try to explain the process of creating the Henon Attractor more clearly.

"Chaotic system" section:

• I can't tell if those equations are the equations for any Henon Attractor or just for your example.
• Can you clarify this, and maybe rename the section "Equations for a Henon Attractor" or something else that matches the content of the section?
Anna brought this up too. I decided to switch out the featured picture with just the original Henon Attractor and added a new section for Henon Attractors with different a and b values.
• Somewhere in this section, explain that x_n represents the x-value after the nth iteration.
Good idea.

``` The first fixed point (0.6314, 0.1894), labeled "1" on the image, is located within the bounds of the attractor and is unstable.
This means that if the system gets close to the point, it will exponentially move away from the fixed point to continue
chaotically. The second fixed point, labeled "2", is considered stable, and it is located outside of the bounds of the attractor.
```
• When you say "it will exponentially move away from point 1", do you mean just for a brief period of time?
• You should explain what it means that point 2 is stable, similar to the way you explain what it means that point 1 is unstable.
• I'm a bit confused on one point here. I thought, based on your page on attractors, that once points get near attractors, they stay near attractors, but you are saying that point 1, which is within the attractor is an unstable fixed point, while point 2, which is not within the attractor, is stable. Can you help explain this?
I worked out the points and found out that my "point 1" and "point 2" were switched. Hopefully, it makes more sense now.
• In the equations "x_n+1 = x and y_n+1 = y", it's unclear what x_n+1 and y_n+1 mean. I think you are suggesting that you have gone through some iterative sequence to arrive at a fixed point, but you don't say that that's what you've done. Can you clean this up a bit? Maybe something like, "If an iterative sequence arrives at a fixed point after a certain number iterations, that means that for every iteration after that arrival, the value of x_n no longer changes, so x_n = x_n+1..."
Ok.

I know this is a lot of comments, but your writing is really good, and I'm guessing you'll do a great job with them.

#### Anna 7/6

I still think that the terms "chaotic attractor" and "basin of attraction" in that one paragraph need a bit more explanation.

I also just noticed some inconsistencies in the values. You say that "The Henon Attractor uses the values a = 1.4 and b = 0.3 and begin with a starting point (1,1)." but later "The artistic image of the Henon Attractor that is the subject of this page instead uses the values a = 1 and b = 0.542."

Can you clarify what's going on a bit? Maybe say which pictures use which values, or which Henon first used?

I also like what you did to the paragraph that I didn't like before. What you've got now works much better.

I see what you mean. I'll try to make the distinction between the two a,b values and the images they produce. I think I'll actually put a picture of the original Henon Attractor (a=1.4, b=0.3) as the main image and have another separate section about what happens when you change a and b where I'll insert the current featured picture. And I'll also put in the mouseovers of the "chaotic attractor" - I forgot last time. I was mistaken about the "basin of attraction" which is the set of initial points that lead to the chaotic attractor and not the same as the chaotic attractor. The "basin of attraction" should belong in its own section, but I won't have time to research it, so I can put it in the Further Ideas section.
Ryang1 (7/8)

#### Anna 7/4

Can you have mouse overs on "chaotic attractor" and "basin of attraction"?

I'm also not totally into this paragraph:

"The shape of the Henon Attractor is often described as a smooth fractal in one direction and as a Cantor Set in another direction. If we zoom into the Henon Attractor near the doubled-tip of the fractal (as seen in the animation), we can see that the points form layers of lines that appear to resemble the Canter Set. If we follow the attractor backwards from the doubled-tip, we can see that the fractal is more smooth and contains less bundles of lines. "

But... I don't know how to rephrase it. It takes a lot of thinking to understand what you're getting at.

That's all I've got for you!

Yes, I had problems trying to understand and explain that concept, so I added a suggestion at the bottom of the page. Ryang1

#### Anna 6/25

this looks really good overall. I just have a few small comments.

Under the fractal section, can you say which direction has which behavior? It's really not clear.

I'll try to explain it in words and add an image.

Your mouse over for "system of equation" is sort of unclear. Can you actually do the steps of solving there?

That's a good idea.

Can you bold definitions, instead of just italicizing?

Sure Ryang1 (7/2)

#### David 6/26

This is a really great concept and it would totally captivate me except I find the initial image difficult to see. Can it be brightened up a bit?

One other thing is that you write in the self-similarity section about "bundles of lines" are they lines, line segments, or points? I ask because the iterations plot points not lines.

Thanks David, I tried to increase the contrast in the picture, so hopefully it's easier to see.
Yes, you bring up a good point. The fractal does consist of points, but I was just generalizing the pattern into a pattern of lines - I'll explain this on the page.
Ryang1 (6/26)

#### Anna (6/10)

I really like the structure of this page, and the way you've broken down and organized the information.

Ryang1 (6/10)

"The Henon system can be described as chaotic and random- its function does not plot points that go to infinity not does it plot points in a repetitive pattern. "

--Mhershey1 13:12, 22 June 2009 (EDT)

Thanks for catching my mistake, Ryang1