Talk:Logistic Bifurcation

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Response to Checklist

Chris 7.17.11

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?

General Comments

  • 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.

Section-specific Comments

Basic Description

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."

Logistic Map

Old comments:

  • (How) does the number between 0 and 1 relate to an actual population? If there's 400 zebras in some area, is there a way to connect that number with the logistic function?

  • Is the function for a generic population, or is it customizable to specific populations?

  • I think this section could make the difference between the logistic equation and the logistic map clearer. What are their specific purposes and uses?

    • I removed the logistic equation section and spent more time talking about what actually makes up the logistic map -- is it clearer now?

Yeah, I think it does make more sense this way.
  • xd 20:44, 12 June 2011 (UTC) regarding Kate's comments, the x values are not exactly the population per ce. it is actually the ratio between two periods which is actually pretty short. I think wikipedia has a good explanation.

Bifurcation and Chaos

Old comments:

  • I know I had to look up "fecundity" - perhaps this is a good place to use a green bubble? done.

  • While both the initial population size and the fecundity rate of that population are variables in the logistic map, the rate of change is more mathematically powerful…
fecundity rate = rate of (population?) change??? Or rate of change is more powerful than either initial pop. and fecundity rate? unclear.

    • In conjunction with everything I added to the "Basic Description," is this clear?

Yup, I think it's clear.

  • instead of telling me not to think of continuous motion, I would have found it more helpful to be told not to think of one single community. But the rest of the explanation in that paragraph was very clear.

    • I added this in, but didn't really integrate that view in -- is what I stuck in sufficient?

Yeah, I think this is one of those "for me personally, this explanation would be a tiny bit better" type things, but the way you have it now isn't confusing.

  • The second paragraph in this section is perhaps a bit long - can it be split anywhere? done.

  • xd 20:46, 12 June 2011 (UTC) I think this section is pretty good already. With the picture, the explanation is very clear. As said before, you can have a little section on how matlab plotted those pictures.

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))

Old comments:

Kate 17:53, 9 June 2011 (UTC):

  • I know the section heading says it, but I still would have appreciated an initial sentence telling me that we were about to begin deriving the LM. good point. done.

  • In this way, the overall rate of change, R, is higher when xn is lower and lower when xn is higher.
Why do we want it to be higher when xn is lower & v.v.? How do we know that this specific way of changing the rate is accurate to what happens in nature? Something like R=r cos(xn) would meet the stated goal of providing an R that changes, so why is r(1-xn) better?

    • Along with everything else I've added and the extra sentence here, does this make more sense?

Yup! The pond pictures helped address that very well up top, and then you referred back to it in the derivation.

*Re: Eq. 4 - I'd like to see an intermediate step, so I'm not trying to do all the differentiating in my head. Also, I'm having a hard time reading the bit in the denominator - is that an x-sub-D or an x-sub-0?

  • At this point, I'm really feeling like I'd like to have a bit more background on the logistic equation and where it comes from.

    • Ok, I've just removed all mention of the logistic equation. Is the article clear without it?

Yeah, I think so. Might wanna get the opinion of someone who knows more of the math, but I think it reads fine.

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:

Kate 14:04, 17 June 2011 (UTC): I think I'm understanding the web diagrams now, but I still have a couple questions:

  • I think I figured out what k is, but you never address it, although it shows up in the legends of your graphs.
    • I've addressed it in the caption of the first image that uses it; is this sufficient, or should I talk about it in the text?
Kate 20:31, 30 June 2011 (UTC): I think the caption's good enough, that's where people will look if they're confused about the picture.
  • It would probably be a good idea to mention which equation goes with the parabola and which goes with the straight line the first time you reference one of those graphs, even though the reader can probably figure it out.
    • Done, for all.
  • Be more explicit about how/why the web diagram verifies that that point is a fixed point. It's because it spirals in towards it, right? I think it would be clearer if you said that.
    • I've attempted to do this. Did I succeed?
Kate 20:31, 30 June 2011 (UTC): about to do another read-through for clarity, if I don't leave a new comment about this, assume it's ok.
  • It would also be clearer if you state why we need the web diagram - how come looking at the picture and noting the intersection of the line and the parabola isn't good enough to find the fixed point?
    • The web diagrams show where the system goes, whereas systems of equations just show intersections -- the web diagrams just show that what I'm saying is true. Is it clear now that this is why I'm using them?
Kate 20:31, 30 June 2011 (UTC): about to do another read-through for clarity, if I don't leave a new comment about this, assume it's ok.
  • I need more explanation about how you can have a fixed point but not converge to a single value, and more explicit explanation about how "the web diagram for r = 3.4 on the right" shows that.
    • Done? I think I did this...
Kate 20:31, 30 June 2011 (UTC): about to do another read-through for clarity, if I don't leave a new comment about this, assume it's ok.
  • Are you sure Eq. 7 is correct? What does the un-subscripted x mean?
    • I think I fixed the issues here. I showed where I got it.
Kate 20:31, 30 June 2011 (UTC): I think it's good.
  • The first time you point us to a graph with k=2 (right after Eq. 7), it'd be helpful to be explicit about which intersections we're supposed to be looking at. The first time I read through, I was looking at the intersections of the parabola and the 4th degree polynomial, instead of the 4th degree polynomial and the line. Also, I think it is unhelpful/confusing to have unlabeled parabolas drawn in in all of these pictures - either give it a k=1 label like you do in the second to last one on the right or don't graph it.
    • Ok, I haven't actually labeled all the curves, but I explain in the captions what you're seeing. Is that sufficient?
Kate 20:31, 30 June 2011 (UTC): about to do another read-through for clarity, if I don't leave a new comment about this, assume it's ok.
  • This sentence confused me:
We can see by the graph that, while both x(n+1) = xn and x(n+2) = xn have solutions, the two solutions are equal, so the system does not generate any new values.
This sentence confused me so much the first five or so times I read it. I think it would be clearer if you first pointed out that the curve for k=2 only intersects the line once, and then point out that this solution is the same as the one provided by k=1.
    • I think I've made this more clear.
Kate 20:31, 30 June 2011 (UTC): about to do another read-through for clarity, if I don't leave a new comment about this, assume it's ok.
  • There are too many unlabeled lines in the graph at the end on the bottom left (with r=3.54). Also, I don't see how you get four period oscillations out of that picture. When I look at it, it looks like the red polynomial intersects the line in 7 places, which I thought would correspond to oscillation between seven values.
    • Again, please let me know if I've clarified this sufficiently. I think I have...
Kate 20:31, 30 June 2011 (UTC): about to do another read-through for clarity, if I don't leave a new comment about this, assume it's ok.

Two overall comments for this section:

  • Referring to your pictures only by location is confusing, especially for those of us who often mix up right and left. It would be so much easier if they were labeled.
    • Done.
  • I still don't understand what the web diagrams add to our understanding here - why are they better than just looking at the intersections of the lines and the polynomials?
    • Mentioned above, I think I've made the use of both diagram clearer.
Kate 20:31, 30 June 2011 (UTC): about to do another read-through for clarity, if I don't leave a new comment about this, assume it's ok.

  • xd 20:51, 12 June 2011 (UTC) I don't quite get this part. You are not done with this section are you?

Kate 17:53, 9 June 2011 (UTC):

  • I still don't really get what's going on in the two pictures near where you introduce "fixed point" - where did the parabola and the line come from? What's different about what's going on in the two pictures?

  • xd 20:16, 22 June 2011 (UTC) In your fig 2, what does that red label "k-1" mean? In addition, why is the y label x_n+k instead of x_n+1?

Special Cases

Old comments:

Kate 14:04, 17 June 2011 (UTC):

  • The web diagram to the left should help illustrate why: no matter what x0 begins the system, the values inevitably move toward zero.
Neither the web diagram nor this sentence helped illustrate that to me. Where is x0 in the web diagram? What parabola is being graphed there? Where is the zero it's moving towards? How do I know it's for every x0 even though only one blue line is graphed? What is x0 and why am I allowed to pick different values for it? I am exaggerating my confusion a little, but I know that you can explain it much better than this.
    • I think the animation helps here.
Kate 20:34, 30 June 2011 (UTC): Animation is so helpful.
  • None of the equations under 0<r<1 make any sense to me. Where do they come from? What does the random bar in Eq. 8 mean? I assume Eq. 8 came from the top of the page somewhere, but I don't really remember what it is. And how on earth did you get from the second equation in this section to the third?
    • I put in something more like a full proof here. Does it make sense?
Kate 20:34, 30 June 2011 (UTC): about to do another read-through for clarity, if I don't leave a new comment about this, assume it's ok.

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,