From Math Images
Response to Checklist
Reading over the comments by others such as Anna and Rebecca who have much more math and physics experience, I understand that these may very well be the people for whom an article like this is written. That said, I spent a long time on this trying to make sense out of it as a "layman" when it comes to this topic, and so I offer my feedback with that perspective. If it's helpful, I'm glad, since I spent a long time on it (worth it just for the sake of making sense of such an interesting topic). If it's just way too much given your time constraints, I totally understand. I have two key points, one about something I loved about your page and one where I needed help.
1. It's clear the incredible amount of work you put into this page. The highlight for me was the web systems, which were fascinating. I loved that section, particularly because the examples you chose illustrated the concepts so well. The graphs were great.
2. I couldn't make sense of logistic bifurcation without really mastering the logistic map. It would be very helpful to me to get that before the article gets far into the bifurcation.
The first sentence in Deriving the Logistic Map "The logistic map is a function that defines the amount of change a system goes through in exactly one time interval" still does not make sense to me. I don't understand the system and how a map connects with exactly one time interval. When I read the word "map," I think of a full scale image with lots of detail. I got stuck on that point for a long time and it wasn't until I looked at Wikipedia's Logistic Map that I really got it. It also helped me understand the y-axis, which is a ratio between the existing population and the maximum population.
Question: Isn't one of the vertical lines graphed in "Image 2" (does it need a number?) a logistic map? Doesn't that define the change over a large time interval?
Recommendation: I know that you are trying to separate out the "More Mathematical Explanation" to spare the reader, but much of what's in there is critical to helping the reader make sense out of the concept. As such, I would put "Deriving the Logistic Map" before "Logistic Bifurcation and Chaos." Before you show the bifurcation, it would be really helpful for the reader to understand why it focuses on 2.5 to 4. You can do that by giving the examples for r between 0 and 1, then between 1 and 2, etc.
Final Note: The top image is dark and it's very hard to tell that there is a line that is bifurcating.
17:00, 7/7/11 My comments are up AnnaP 7/10
Messages to the Future It might be kind of fun to include a hidden section at the end about how to plot the logistic map--maybe put in a request for a computer science oriented person to explain how to do it? You can also take a look at my mathematica code to plot it--you can download that by clicking [here]
References and Footnotes
All of the images are cited, if you click on them. Most of them were created by me, anyway. I emailed the creators of the applet at the bottom of the page to get their permission, and it's very clearly cited on the page. No footnotes were necessary for the information on this page.
The basic description is very clear about how logistic systems connect to biology and populations. It also shows how various oscillations would manifest, graphically in a population in a way that is easy to understand. "Why it's interesting" includes further discussion of real-world use and an interactive applet demonstrating self-similarity.
Quality of Prose
Because of the length of the page, I carefully used "signpost" sentences and paragraphs throughout the article, along with clear logical links from one paragraph to the next. Each section is clear about its purpose and direction. All of the more mathematical section is pretty heavy, and it's in the only order that makes sense, so I can't move the detailed math any further down.
- your first sentence "A section of a bifurcation diagram showing how logistic systems change as fecundity changes." might scare away readers by using "bifurcation" and "fecundity" Perhaps use a slightly less accurate, more generic first sentence?
- You don't need the word "recursively" here: "or apply it over and over recursively" Again, it's a word that will confuse some readers
- Rebecca 02:24, 16 July 2011 (UTC) I would say "far from scattered" instead of "anything but scattered" in the bifurcation and chaos section.
- "Using a web diagram, we confirmed that, for the example of r = 2.9, the point we can find analytically is in fact the point the system approaches." I don't think you need the comma after "that."
Integration of Images
The images are meticulously numbered, referenced in the text, linked between image and text, and explained in both the text and the captions.
This page links extensively to chaos and iterations, including a thorough discussion of how iteration can be applied to this topic. The idea of fractals is also discussed.
The logistic map is derived with clear explanations of steps. The mathematical discussion of bifurcation uses extensive examples in multiple forms -- diagrams, graphs, and analytical discussions. All new ideas are defined, discussed, or linked out to helper pages.
Accuracy and Precision
All terms are defined in text, bubbled out, or linked out. Mathematical ideas are expressed both in equations and pictorially, where possible. In the mathematical bifurcation section, because it is so long, great pains are taken to keep the reader on track and return consistently to the main ideas.
- The paragraph that begins with this sentence: "Earlier, we found that period-one fixed points exist in all logistic systems, whether or not they approach only one point. Now, we will see that the condition x(n+2) = xn also has a solution for logistic systems with r < 3. " Could use a bit of work. It'd be better to explain that a fixed point will certainly come back to itself when you iterate it. If you have f(x)=x, then f(f(x))=f(x)=x. So anything that is a solution to the first equation is also a solution to f(f(x))=x. Similarly, all two two-cycles will show up when you look for solutions to the period four equation f(f(f(f(x))))=x. An explanation in those terms in place of, or addition to, what you already have will make that section make more sense.
Rebecca 01:42, 16 July 2011 (UTC) I don't know if you can just say "(hence the name)" when you're talking about bifurcation. I had to look up the fact that "furcate" is to divide into branches. I think mentioning that in the parentheses would be helpful. My vocabulary is pretty weak, but i think other people might have the same issue.
Layout I made the paragraphs as short as I could, put in breaks to make sure pictures didn't overlap with other sections, and tried to place image location to create the least possible interference with readability at any window size. Definitions are generally bubbled or linked out, with some in boldface.
- Why don't you add an extra break between each paragraph in your basic description? That will help break up the text a bit better.
- The paragraphs between images 3 and 4 and 5 and 6 could also use some extra spaces.
- Image 5 is a bit distracting where it is, since you talk about it much lower than the image itself. Try to move it around.
- Can you make the blurb on web diagrams it's own subsection, or make the phrase Web Diagrams at the beginning of that area bold to stand out?
- Kate 14:58, 1 July 2011 (UTC): Much clearer than before. Be careful with your "So"s - you begin a lot of sentences with "So". None of them are things I would object to individually, but the fact that they stood out to me probably means you're overusing them.
- AnnaP 6/16 - Diana, have you seen the Iterated Functions Page? That page uses the logistic map for examples, and you might find that it's useful to link to that page (or lift some of the images) for examples.
- I had already linked to this, bit it was pretty subtle, so I've added it in more prominently.
Kate 14:55, 1 July 2011 (UTC):
Typo: into multiple values. in the case of logistic bifurcation…
- The branching behavior of bifurcation occurs over the range of multiple logistic systems as their fecundities or maximum rates of change increase.
- I don't understand this sentence at all.
But to read this diagram, do not think of the branching action as a continuous motion. Instead consider a single vertical line through the image; it captures exactly one
- I think you should get rid of the "But" and the semi-colon should be a colon.
The last picture in this section is helpful but kind of giant - what do you think about hiding it?
Kate 17:53, 9 June 2011 (UTC):
Out of curiosity, are there any populations whose size is chaotic?
- Planning to put this in "Why It's Interesting."
Bifurcation and Chaos
A More Mathematical Explanation
Deriving the Logistic Map
Kate 14:55, 1 July 2011 (UTC):
- exactly one time interval - this phrasing seemed odd to me, I think it was the contrast of "exactly" with some unspecified length of time, like how "He's exactly an unknown height tall" would be weird.
- I couldn't figure out a great way to fix this. I realize it's strange, but I don't know a graceful way to specify that the "exactly" modifies "one" and not "interval." (Grammatically, "one" is technically what "exactly" modifies here, but it does read ambiguously.)
- Okey dokey. It's not a big deal, the sentence is still perfectly understandable. (Kate 20:14, 6 July 2011 (UTC))
A Mathematical View of Bifurcation
Kate 14:55, 1 July 2011 (UTC):
- The web diagram explanation is much clearer, yay!
I don't understand the unlabeled equation that I'd probably call equation eight and a half…
xd 7/6 One typo "Earlier, we found that period-one fixed points exist in all logistic systems, whether on not they approach only one point." I fixed it for you. Old comments:
Why It's Interesting
Kate 14:55, 1 July 2011 (UTC):
- This might be a personal preference, but I've been hiding applets, because the way they flicker when I scroll drives me crazy.
- Actually, that drives me crazy, too, but I'm leaving this one unhidden because it's at the bottom of the page, and so there's not much space in which the reader is scrolling. And it lets me direct the reader to the applet's actual position.
- Reasonable. (Kate 20:15, 6 July 2011 (UTC))
The bit about chaos in nature was quite interesting. I think it might be a good idea to link to it from the top of the basic description - people may be interested in the pictures of fish in ponds and also the biology at the bottom but not the math in the middle. Typo: no matter how fari it is from x0,