Does it have mathematicalalgoriths yet? ...We understand the concept of the Butterfly Effect from movies (of that title, or as in Run, Lola Run) ; but isn't there much more to it? Analysis of Simplexity and Complexity? ...Of convergent patterns arising out of apparent stochastic processes?
Please answer by enriching the above rather amorphous and inchoate ideas.....Thanks and may the Best Answer win !
Samandriel
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Although not an example of chaos, a ladder teetering on one of its edges gives the basic idea of what chaotic motion is about. When it is teetering, you have no idea which way it will fall because tiny differences in position determine whether it falls one way or the other.
Mathematically chaotic systems are always in a "teetering" state like this, where tiny changes in its position and/or velocity lead to very different trajectories. The double pendulum is probably one of the simplest examples of such a system:
http://video.mit.edu/watch/double-pendulum-6392/
That is the main idea underlying chaos in the mathematical sense of the word.
Lex Lodge
Chaos Theory is primarily mathematic and deals with the sensitivity of large systems to change made in minutiae detail. Often a cross-over impact to theoretical physics dealing with subatomics all the way to structure of the universe. The very nature of space, time, and what is real can get questioned. At the heart of Chaos Theory is an observation called Entrophy; which says everything heads toward a condition less-organized. Many modern considerations give consideration to a "recovery cycle," which maybe inheirant to Multiverse or Time Cycle concepts.
DinDjinn
But are ALL systems extraordinarily sensitive to initial conditions?