From Math Images
Sorry for taking so long to give feedback. The page is largely in great shape. The examples you use and the way you flush them out are very clear and well written.
A couple of content ideas:
- It might be worth pointing out that a chaotic system is deterministic, not random; it's just really hard/computationally impossible to make accurate predictions. I think a lot of people conflate the two ideas.
- It also might be worth mentioning that not every unpredictable system is chaotic, but that chaotic systems have properties that make them tractable in certain ways (this could go in the more mathematical section). Ignore this comment if I've just made a false assertion.
- Cool, but this sentence: While this is not true of all chaotic systems, it can be a powerful tool for analyzing many is a little too vague to be useful. Maybe drop this sentence and put the second sentence of the paragraph before the first sentence?
- I didn't want to scrap it, but I changed it
- Cool, looks good (10/16).
- Put a mouse-over on "dense", even though it's somewhat defined in the following sentence (I could go either way on this comment).
- Can you think of a short enough definition of dense to fit in a mouse over? I couldn't, which is why I didn't do it.
- What about the technical definition?
- As the person who's taken analysis, I'd rather have you put this in
- Done. I also changed the description below it, partly because the informal definition of dense was a little bit misleading, and partly because if I didn't change it, then the explanatory text was literally just going to repeat the mouse-overs (10/16).
A couple of slight wording things:
- Shouldn't the first sentence read "The definition of chaos is..." or "chaos is the term used to describe..." rather than "chaos is the definition of..."?
- There's something about your mouse-over definition of periodic points that I find not quite clear, though I would certainly concede that it's kind of annoying to define in a single sentence.
- Any alternate phrasing ideas would be appreciated.
- How about "A periodic point represents a state of a system that may change over time but eventually returns to its starting point"
- That isn't actually true... returning to a starting point can actually have nothing to do with periodic behavior. I tried something else.
- I'm still not sure what the problem is with what I said, but ok, the new thing is fine (10/16).
Alan, I'd really like to see you address some of the information in my comments below in your page. If you want to meet with me about it, we could work on it Wednesday after your interview.
My previous response was hidden in your comments; I hadn't italicized it before. I haven't incorporated your suggestions because I am hesitant about writing things I don't understand. let's talk more about this on Wednesday.
For a rigorous definition, see R. Devaney, "An Introduction to Chaotic Dynamical Systems" It's in Cornell.
This is guy who pretty much established the mathematical definition of chaos used in analysis. I hope this helps!
I've been thinking about this... and it might be easiest for you to put this page on hold, because I can definitely either fill in the mathematical holes, or go over them with you. I think that that would be a lot faster and easier than you reading a book.
The shorter version is that there are three characteristics:
1 Sensitive dependence on initial conditions (two initial values close to each other eventually go apart)
2) A property called transitivity. Basically, if you take any interval in your space, and you iterate it enough times, it will eventually intersect any other interval in your space (your space can be restricted to the points where the dynamics are chaotic. Many systems are only chaotic on certain subsets. For example, Julia Sets are precisely the points where a function behaves chaotically).
3) Dense periodic points. This means that if you pick any point in your space, there is a periodic point within a distance epsilon of it for any epsilon greater than zero.
One of the nifty things is that 2 and 3 imply 1, so you really only need the second two (but, the first one gives us a more intuitive property of a chaotic system).
I read the section on chaos from the book several weeks ago, and had trouble following it. I'll put it on hold and work on it with you when you get back.
- Alan 6/17
Anyways, we can talk about this more when I'm back. I just thought I'd give you the short version to think about in the mean time.