Talk:Markus-Lyapunov Fractals

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Response to Checklist

7/7/11 14:44 Overall, this page is in really great shape. I have a couple of minor comments in red. Great work!! AnnaP 7/10

References and footnotes

Click on any of the pictures to see author and original location. I include one reference at the bottom of the page, which is where I found the information on this page that isn't my own mathematics, "common sense" information, or information from discussion with Steve.


In the basic description, I relate it to population growth. In "why it's interesting," I relate it to fractals and the larger concept of fractal patterns in chaos. I also relate it to art. I wanted to explore the actual application for which Markus created this system, but was unable to find sufficient information.

Prose and Structure

I aimed for a logical progression of ideas, and tried to show this progression and remind readers of the connections between the ideas. Each of my sections starts with a sentence or two indicating the direction and point of the section. In the more mathematical section, I can't see any reasonable way to move the bulk of the math later than where it is.

Integration of Images

All of the images are either explicitly referenced in the text or are minor extensions or extra examples placed so that their correlation to the content is clear. Further, every image has a caption stating clearly how it relates to the content.


The content here is extensively related to content on Logistic Bifurcation, and this page links to that one in multiple key places. Such topics as chaos, summation notations, and modular arithmetic are also used and links to relevant pages are provided. In the section on fractal properties, the reader is directed to a large number of other examples of this property in related mathematical phenomena.


The mathematical section provides a derivation for the Lyapunov exponent and shows how and why it's used as it is for the logistic map. This section also provides a graph of Lyapunov exponents for logistic systems to show this application. The "forcing the rates of change" section includes an example of a fractal with a different period to show the impact of this mathematical idea.

I might actually include a couple of graphs to show how quickly the things converge when  \lambda <0 , and how quickly they diverge for  \lambda >0 . Just picking -1 and 1 and graphing the value of dxn/dxo as a function of n would be pretty easy
I added this in below the illustration of dx0 and dxn.

Accuracy and Precision

Terms that may be unfamiliar are all either explained in the text, included with explanatory balloons, or included as links to relevant helper pages. Where necessary, equations are provided to define terms. I stick to consistent, clear terms as often as possible.

In your bullet points in your basic description, please avoid the word "it" at least for the first one. I had to pause and think for a second "the logistic map? That doesn't make any sense if that's what 'it' means! OHH she means the exponent..."
Fixed this, I believe.


I've played with window size to make sure it doesn't do anything too dramatic to the page. Paragraphs are as short as I feel comfortable making them. Bubbles and links are used for almost all terms to define them. Text breaks are used to make sure images don't interfere with unrelated sections.

As is, the basic description looks like a wall of text. Are you completely, totally 100% sure that you can't break those up to add some needed white space? For example, I could see a paragraph break in between these sentences "...with those rates of change will behave. Markus then created a color scheme to represent different Lyapunov exponents..."
I added the break you suggested as well as one in the last paragraph.

General Comments

  • Kate 18:21, 6 July 2011 (UTC): I think this page is awesome, and probably just about ready for final review! :)

Here is the the Blue Fern page that I talked about. You might want to find a way to link to it somewhere on the page. Thanks! Done. AnnaP 6/16

Old comments:

Good things about the page:

  • This page completely blew me away. Your writing is incredibly clear, and it makes the whole page easy to understand. Keep up the good work!
  • Great use of pictures and good organization overall. I especially like the way you introduce the topic. It might be worthwhile for you to check out Chapter 6.5 of "Heart of Mathematics" (They have it on reserve in the library). The chapter is on the dynamics of change.
  • This is way way way way more clear and easy to follow than my first read through a few weeks ago. Good Job!Richard 6/30

Some suggestions:

  • Mouseovers:
  • In your lyaunov exponent bubble, I might precede the equation with a sentence like "The Lyapunov exponent will be denoted by lambda here, and is determined by the equation... where N is ___ and n is ____" so that people realize that they aren't already supposed to know that equation. It might help people from getting scared off. In fact, it might even be worthwhile to just give an short explanation in the bubble rather than giving the equation. I agree that the bubble is useful because otherwise people wont have any idea what the Lyapunov exponent is, but I'm not sure if they need to see the equation yet, and since the equation looks a bit intimidating, you might want to save it for later.
  • For the logistic equation mouseover, I think you might be able to do something similar. A one sentence explanation of why a logistic function is unique might be better than just giving an equation, especially because you don't need the equation to understand the section. The equation in this bubble is certainly less intimidating though, so you might be fine. Maybe ask Abram what he thinks about these suggestions too.

This makes sense. I tried to make them both into word-based explanation bubbles, but now I'm not sure if maybe I should just put the sentences in the bubbles directly into the body of the article? -Diana 16:25 5-23

I see why you would say this, and you definitely could move them. I actually think I like it how it is currently though. The explanation of the Lyaunov exponent in the bubble might be a bit difficult for some people to understand, and you can still understand the section without it... so I say leave it in a bubble. Rebecca 22:04, 25 May 2011 (UTC)
  • I'm having trouble understanding what the difference is between your first bullet point and your second one. It seems like the population reaching a fixed point and staying there and the population becoming stable are the same thing.

I added a bubble for the word "stable" to explain why stability doesn't imply "fixed." -Diana 16:25 5-23

  • Maybe link to the general fractals page in the sentence "Self-similarity is that trait that makes fractals what they are – zooming in on the image reveals smaller and smaller parts that resemble the whole." done. -Diana 16:25 5-23
  • I would consider moving the black and white picture in basic explanation section to a Why It's Interesting section along with the paragraph beginning with "The movements from light to dark..." This is a great discussion, but you don't necessarily need it in the basic description because it's not part to your explanation of population growth or anything. You can actually zoom in on the picture and show self-similarity, and maybe expand a little on what you say in that paragraph.

I've moved it down to "why it's interesting," but now I'm not sure whether I need something the basic description to explain why it says "fractal" in the page heading, because now it's not clear... Also, either way, I need to figure out why chaos creates self-similarity (because I don't understand it myself right now), and add an explanation. -Diana 16:25 5-23


  • I think you've got something weird with your html - there's some odd stuff happening in the upper left.done.
  • Yay, his name is Aleksandr!
  • Missing a period after the chaotic bubble.done.
  • Typo: "the population is change is neutral"done.
  • Neutral makes more sense now, though
  • Glad you added zero in to the color explanation
  • Equations 2 and 3 don't look consistent to me - is it 1-x or 1+x?ah. that was a typo; done.
  • The explanation of bifurcation doesn't really make sense to me, I don't understand how it connects to the graph.

I've changed it, but this is a recurring comment I've gotten. I hope it's actually intelligible now.

  • The Lyapunov Exponent explanation is really good.
  • Little green bubbles are all working well and are useful!
-Kate 19:11, 19 May 2011 (UTC)

actual population, Verhulst constructed


For pronounciation, see: [1]

:I think the problem with the image stretching is because you wrote a lot inside "Image Created By". The basic description is really easy to follow, imo. One thing I'd like to see added is a first name for Lyapunov the first time you mention him. Something that wasn't immediately clear was superstable, but it looks like you're gonna put a bubble there. Also, near the end, you've written "floating," where I think you should have "floating", but that's not a big deal. Also, what color is the blue/yellow image when it's zero? -Kate, 5/18done.

xd 19:46, 25 May 2011 (UTC) Under The Logistic Forumula

1. Explain what are x_n and x_n+1. I am assuming that they represent the population at time n and time n+1.

2. If that is the case, then I wonder if 1-x_n is always negative. In addition, what does r mean? is it just some parameter that varies? actually, i went to wikipedia Logistic map and got a better idea.

3. I am trying to understand your explanation of bifurcation. The picture is too small to discern the values on the axis. In addition, I am trying to use Matlab to plot the picture and get a better understanding of its significance. I will talk to you about this.

Section-specific Comments


  • You should make that blurb a complete sentence. It sounds unfinished to me.Richard 6/30
Kate 18:21, 6 July 2011 (UTC): Seconded. Good point. Done.

Basic Description

  • Kate 18:21, 6 July 2011 (UTC): This is a useful indicator because, for the logistic map,
I think the first comma is unnecessary.
    • I see what you're saying. It's definitely a bit choppy. I'm leaving it in for now, because I want it to be very clear that these bullet points are not true for every dynamic system. I tried to figure out a more graceful way to maintain the emphasis created by that comma without having the slightly awkward punctuation, and I couldn't find one. I'm happy to talk about this more, though.

  • In the first bullet, it might be good to clarify what "it" refers to...the rate of change in population?Richard 6/30
If it is zero, the population change is neutral; at some point in time, it reaches a fixed point and remains there. Done.

Old comments:

Kate 15:29, 13 June 2011 (UTC):

  • As much as I love it, I'm not sure the semicolon is the right punctuation mark to use in the middle of your first sentence. You probably know the actual rules about it better than I do, but it keeps sticking out to me in a way that is mildly disruptive if not wrong.

  • Second sentence is really really good. Awesome one-sentence description of the image.

  • In your bubble on stable, I think you want a lowercase "a" and a colon instead of a semicolon.

  • I think you can say "any point (x,y)" without commas around it.

  • Overall, I don't see how the clarity or organization of this section could be improved upon. I think it's excellent.
  • You have a mouseover for stable, but what about chaos? What does that mean for a population? You may have it on your other page, but it might be worth a sentence description here.Richard 6/30

Gene 17:30, 6 July 2011 (UTC) Neat Basic Descript but is "The logistic map is one of the simplest mathematical representations of population growth." really fair? It don't look to simple to lotsa folks. Maybe something else instead of "simple"?

*I described it as "concise" instead. Valid?

'Tis a way cool ending to this section, too, but it rests on a lot of data/info not available to your reader. Is it available somewhere so you could say "Based on all the junk found at XXX, if you take a point (x,y) and it comes out blue, this means that ...". I think you've oversimplified a wee bit--but with nice results.

*We talked about this, and that sort of data isn't available. Instead, I provided loosely numerical examples in the final paragraph. Does this help?

A More Mathematical Explanation

  • Kate 18:21, 6 July 2011 (UTC): I think it might be a good idea to set it up so that there's that little note saying that understanding of this section requires knowledge of the logistic map, and link to that page, because I'd probably be quite confused here if I hadn't read that page. Done.

The Lyapunov Exponent

  • When you first introduce summation notation it might be cool to change it to thisRichard 6/30
"Generalizing this for all n, we consider every step of iteration using summation notation: "
Kate 18:21, 6 July 2011 (UTC): Please do this! My poor helper page isn't linked to from anywhere and is currently useless. Love it. Done.
  • You still refer to "change" in the paragraphs at the end of this section, but are you referring to the rate of change or the population's change?
    • Had a conversation about this issue with Richard. We decided that it would be much clearer if I provided a visual representation of the "dx" notion and used the word "difference" instead of change.

Why It's Interesting

  • Kate 18:21, 6 July 2011 (UTC): Your second image in this section interferes with the Teaching Materials heading when the window's large.Taken care of.
  • Self-similarity would be a good word to bold in this section.Richard 6/30 Done.
  • Is there a picture of it on a t-shirt or something?Richard 6/30
    • Um... No. I couldn't find one. Thing is, this fractal is used in graphics all the time; I've seen it. But in those cases, it's not usually called by it's name, so it's sort of impossible to find online.

Old comments:

  • Rebecca 12:33, 31 May 2011 (UTC)For the first picture in the Why It's Interesting section, it would be good to show the main image again I think. I don't think people will be able to see the self-similarity if they have to keep scrolling back and forth. Something similar to the picture you show for bifurcation would be great! Good point. Done.
  • 12:33, 31 May 2011 (UTC)I love the artistic extensions section!