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What is chaos anyways?

We think of a lot of systems as "chaotic". The weather, the stock market, the motion of a double pendulum - all these systems have structures which to the human eye do not have an obvious pattern. However, the lack of a discernable pattern does not mean that a system is necessarily chaotic. Consider, for example, a gas of non-interacting molecules trapped in a cubical box. At any moment in time, a snapshot of the molecules will show them all moving around in different directions and at different positions. Hence there will be no obvious pattern. However, if I start each molecule at \(t=0\) with a known initial position and velocity to within some experimental accuracy, then I can predict the position of each molecule at a later time \(t=T\) to the same accuracy. Hence the system is deterministic, even though it might appear random to our eyes. The randomness comes from the variation in the initial conditions, not from an inherent randomness in the evolution of the system itself.

In contrast, a chaotic system possesses a randomness in the evolution of the system itself. This set of problems will help illuminate the difference between a deterministic system and a chaotic one.

Note by David Mattingly
2 years, 11 months ago

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I'm quite intrigued by chaos theory and things like the butterfly effect. Very interesting! Vishnuram Leonardodavinci · 2 years, 11 months ago

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A simple but deeply interwoven theory... Its implications would be amazing to see. Tejas Menon · 2 years, 11 months ago

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Absolutely stunning.... theory... What ultimately translates is that there is predictability in a chaotic situation. What about unexpected results despite knowing the exact parameters, !!! Samarth Shukla · 2 years, 9 months ago

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