Butterfly Effect (Chaos Theory)

[TheShinje]

[quote][font=Courier New]The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141) [/font][/quote]
Can the single flap of a butterflys wings cause a storm in China? Or set off a tornado in Texas?

The theory is quite intriguing to me. It insists that the simple flap of a butterfly's wings can set off a chain of events that would have not occured if the butterfly hadn't been there ( not existed), It's a miniscule detail that, when added to the equation, mutates itself into something more.


[quote]


....One day in 1961, he wanted to see a particular sequence again. To save time, he started in the middle of the sequence, instead of the beginning. He entered the number off his printout and left to let it run.

When he came back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up wildly different from the original. (See figure 1.) Eventually he figured out what happened. The computer stored the numbers to six decimal places in its memory. To save paper, he only had it print out three decimal places. In the original sequence, the number was .506127, and he had only typed the first three digits, .506. [url="http://www.imho.com/grae/chaos/images/chaosfig1.gif"][img]http://www.imho.com/grae/chaos/images/chaosfig1.gif[/img][/url]
By all conventional ideas of the time, it should have worked. He should have gotten a sequence very close to the original sequence. A scientist considers himself lucky if he can get measurements with accuracy to three decimal places. Surely the fourth and fifth, impossible to measure using reasonable methods, can't have a huge effect on the outcome of the experiment. Lorenz proved this idea wrong. [/quote]So, in the experiment above, somethign so small as leaving a few decimal places out has a profound impact on the outcome of the sequence, this is the[b] Butterfly Effect.[/b]

Chaos theory, anyone? What are your views on it?

[outlawstar69]

If I could say that one field of math was interesting, at least from a perspective of one who watches and not participates, I'd have to say that chaos theory has some really cool aspects to it. I believe that the concept was touched upon in the book Jurassic Park, and that's what introduced me to the whole idea. (I still think that malcom was the best character in the book, but anyway..) In response, yes I do think that something small can have a profound effect on the enviroment. I mean, there's really no way to tell if it doesn't have such effects, because there's no way to recreate conditions exactly for any scientific experiment, let alone one with such a large margin of error. That is, unless we discver time travel, do the experiment, go back, change something, then see the effects...

[Guest ScirosDarkblade]

Well, that .000127 example seems bogus. Sure, when calculating many quantities, that amount won't have too much of an effect, especially when it is only the difference between two much larger numbers (in comparison). However, there are plenty of situations where it'll make a huge difference, including when you use these kinds of numbers to calculate trajectories or orbits of heavenly bodies or something of that sort. It's not that difficult to have a small value in an equation get very large by the time calculations are complete, as long as there are enough relations where that value is the square root or cubic root (or whatever) of some dependent variable.

My point is that with enough (not much, often) analysis one would be able to predict what magnitude of change in the outcome a small change in a variable would create (when dealing with concrete equations). So in such cases it would be possible to tell when chaos theory applies and when it does not. Making it sort of worthless.

The flapping of a butterfly's wing creating a typhoon is also a bit hard to buy, because at the same time as a butterfly flaps its wings a couple of people (a couple of billion) breathe and some birds fly around or something. Isolating some key instigator in this case, if one even exists, is flat-out impossible.

So chaos theory is fun to talk about because it has fun stuff like the "alternate reality" implications, but ultimately seems hardly applicable.

[outlawstar69]

[QUOTE=ScirosDarkblade]The flapping of a butterfly's wing creating a typhoon is also a bit hard to buy, because at the same time as a butterfly flaps its wings a couple of people (a couple of billion) breathe and some birds fly around or something. Isolating some key instigator in this case, if one even exists, is flat-out impossible.

So chaos theory is fun to talk about because it has fun stuff like the "alternate reality" implications, but ultimately seems hardly applicable.[/QUOTE]

That's just one example that people use to try and explain the concept to people who haven't studied it for 10 years and recieved a doctorate. There are other applications, or examples. For instance, the way a droplet of water rolls along on a propeller surface. Or, the way that roots grow in the soil... chaos theory deals with the sort of situations where it's not cut and dry, not the type wherein 2 + 2 always equals 4.

[JuliasPeach]

[COLOR=DarkOrange]yeah the whole concept of the "chaos theory" is confuing at times but i think its pretty true. i my self got the whole introdution to this from the Jurassic Park book and thats some indeapth stuff! i think its also pretty scarry that a buterfly can cause a tornado but i shouldn'y worry :D but the chaos theory should be taken seriosly or who knows might happen? :eek: [/COLOR]

[Guest ScirosDarkblade]

[quote name='outlawstar69']That's just one example that people use to try and explain the concept to people who haven't studied it for 10 years and recieved a doctorate. There are other applications, or examples. For instance, the way a droplet of water rolls along on a propeller surface. Or, the way that roots grow in the soil... chaos theory deals with the sort of situations where it's not cut and dry, not the type wherein 2 + 2 always equals 4.[/quote]

Well, from what I've read on chaos theory, it's never seemed like a specific science. It's basically the study of nonlinear, aperiodic systems (i.e. things that are modeled by something which does not ever repeat, although it may have overall trends). Sure, it has its applications, but only in the sense that when certain systems are studied, they are found to behave chaotically but within some framework. There is no real "body of knowledge" that chaos theory encompasses. There's not much to derive from it.

Now, if chaos theory somehow got the point where there turned out to be a recognizeable characteristic among chaotic systems, meaning that systems could be predicted to be chaotic (or not), then we'd have something. At this point, the study of chaotic systems has led to some interesting revelations here and there, but nothing really substantial as far as I know.