chaos theory

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chaos theory

Postby papernuke » Tue Apr 03, 2007 12:21 am

waht in detail is the chaos theory? (please explain in a way i can understand) i read about it on the wiki and places, but its confusing :oops:
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Postby pat » Tue Apr 03, 2007 4:49 am

Say that you're trying to model the behaviour of some complicated system like the way convection currents move in a pot of water or the weather or something.

Suppose, you put in an input value of 0.3999293929113 for one of the inputs to your model. Now, say that you want to re-run the simulation again tomorrow, but instead of 0.3999293929113, you accidentally type 0.3999293929118. You might think, "Well, my results might be off a tiny bit, but the inputs agree to 12 decimal places so I should still be in the same ballpark."

If your system is chaotic, you may well not be in the same ballpark at all any longer.

Chaos theory is the study of what kinds of systems show this behaviour, what kind of way can we measure how bad the problem is, and are there overarching things we can still say about the system even if the details are hopelessly dependent on the exact initial conditions.
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Postby papernuke » Wed Apr 04, 2007 2:15 am

oh, ok, so you mean if something VERY small is off, (in math presumably) the result from earlier is wayway different?
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Postby pat » Wed Apr 04, 2007 3:35 am

Yes, or if you fail to account for something very small...
The phrase *butterfly effect* comes from chaos theory. The idea is that if you are trying to predict the weather, and you fail to take into account a butterfly flapping its wings in Tokyo, your results are going to be way off base in a few days.
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Postby papernuke » Wed Apr 04, 2007 4:15 am

pat wrote:If your system is chaotic, you may well not be in the same ballpark at all any longer.


what does it mean if its chaotic, and how can it be chaotic?
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Postby pat » Wed Apr 04, 2007 3:05 pm

It is called chaotic if these small differences matter.

Generally, three interacting differential equations is enough. For example, trying to solve Newton's gravitational equations for a system containing three objects.

Also, many iterated functions exhibit chaotic behaviour, too. For example, using Newton's Method to try to find the complex zeros of y = x<sup>4</sup>. There are lots of places where you zoom into a different zero if you move even the slightest bit...
http://archive.ncsa.uiuc.edu/Classes/MATH198/alexrein/semester.html
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Postby Nick » Sun Apr 08, 2007 11:27 am

Chaotic systems are so complicated, they are nearly impossible to predict. Drop dye into a glass of water and try to predict the movements of the dye, for instance. ;)
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