# Talk:Markus-Lyapunov Fractals

## Contents

# 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.

**Context**

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.

**Connections**

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.

**Examples**

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 , and how quickly they diverge for . 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 d
*x*_{0}and d*x*_{n}.

- I added this in below the illustration of d

**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.

**Layout**

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:

# Section-specific Comments

## Intro

~~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.

- 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

~~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,~~Done.**it**reaches a fixed point and remains there.

Old comments:

## 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 "d
*x*" notion and used the word "difference" instead of 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 "d

## 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:

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