Chaos Theory

[Jonahtan]

I was readin the thread on "new math formula" and the part about reducing pi to two decimal places and it reminded me of something i read while researching chaos theory. However i do not really understand chaos theory and am having trouble understanding it and its practical mathematical application (Its use in the world). It seems more like an interesting observation that a useful application. The article i read was http://www.imho.com/grae/chaos/chaos.html

i think i would understand it with more mathematical background however i am yet to begin grade 11 math, i have only done physics this year.

i am very interested in this theory and a more comprehensive and perhaps simplified explanation would be greatly appreciated.

[Tormod]

Welcome to our forums, Jonathan. We have people here who are more apt to answer your questions than I am, so I'll just say "hi" for now.

[gpdone]

This is an example of the fundamental concept of CHAOS and non linear dynamics.

"It has been said that "Chaos is a name for any order that produces confusion in our minds." (George Santayana, thanks to Fred Klingener for finding this). However, the mathematical definition is, roughly speaking,

Chaos: effectively unpredictable long time behavior arising in a deterministic dynamical system because of sensitivity to initial conditions.
It must be emphasized that a deterministic dynamical system is perfectly predictable given perfect knowledge of the initial condition, and is in practice always predictable in the short term. The key to long-term unpredictability is a property known as sensitivity to (or sensitive dependence on) initial conditions"

Source; http://amath.colorado.edu/faculty/jdm/faq-[2].html

Hope this helps get you started.

Lee

[pgrmdave]

so, what you're saying is that some systems are so complex, that unless everything was known about it, little can be predicted?

[Tim_Lou]

does this have something to do with uncertainty principle?

the thing about uncertainty principle is that when observation is made, photons strike the samples and must thus transfer momentum to that object (velocity in a way), if you decrease the energy of the photon to diminish this affect, the wavelength of the photons increases, then the position of the object would not be precise enough.

(hehe, finally i have a full understanding about the uncertainty principle cheer!)

[gpdone]

Originally Posted by pgrmdave

so, what you're saying is that some systems are so complex, that unless everything was known about it, little can be predicted?

Yes. Happily for us there are many cases where knowledge of only a few initial conditions will allow for high levels of predictability.

And yes to Tim_lou, Goodonya mate. Tis a good example of the situation we are in regarding ALL of our observations; they are all after the fact.

Lee

[Bo]

hmm purely mathematicly speaking, i dont agree that chaos theory and the uncertainty principle have anything to do with each other, from a more philosophical point, there of course is an overlap.

As for chaos theory: the basic idea is that most systems are so complex, and so highly sensitive to a small change in input, that effectively we lose all our predictive power. The standard example is the weather. If you take a weathermodel and give it some initial data, and you compare this to the same model, same data, but 1 parameter changed only a very tiny little bit, then for a while the 2 models give the same predictions, but after a while, there is no coherence whatsoever. From this comes the observence that "a butterfly in china can make a hurricane in amsterdam".

Bo

[TINNY]

a recent movie was titled The Butterfly Effect. anyone seen it?
shows that small changes in the past brings about drastic differences in the future.
ahhh.. inconsequential. just jumpin' in.

[lindagarrette]

Pretty good movie, actually, with a moral, to boot. " Be careful what you ask for. " Seriously, though,it does illustrate in a superficial way how the universe is determinmistic.

[TINNY]

Seriously, though,it does illustrate in a superficial way how the universe is determinmistic.

see? the determinism vs freewill battle pops up everywhere. someone's got to start a thread on it. no effense
it also shows that humans can make a big difference when they utilize their free will properly.