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Wednesday, April 23, 2008

Chaos Theory: In Kitchen, Programming and Gambling

This is a guest post by Soham Das of Jump Up. His passion is analyzing markets and stocks. His blog stands out for his pragmatic style. He is also a regular writer for Seeking Alpha.

My mom always told, 'how lousy a cook, I will be'. Don’t ask me, why she told me, but yes, she had her reasons. I did things like, boiling a pan of water and just when it has started boiling, reduce the flame, and drop a droplet of ink. Funny eh!

Well, if you ask me, I enjoyed watching the intricate patterns that ink made while dissolving. It was awesome. Oh yeah! and it followed the image given in this post, for a while. [Drop a droplet of honey; you can see it forming better].

Ink dissolves in water really fast and hence it might be barely visible, so the next time when you are offering agarbattis to your Gods, observe the smoke. It will be whirling like that image.

So what does this has to do with me?, you might ask.

Well everything!

All these and much more are fantastic examples of chaotic system. What does chaos mean? In real life, we say chaos is randomness. But in maths, No!

Chaos Maths has an inherent thing going. You can find the end result of any chaotic system as long as you are not making a measurement error. The moment you make one, you are gone! This error will roll like a snowball, and keep on accumulating and getting scaled up, causing a total mismatch in the end results. Total mismatch? Think about, the statement,

Flapping of the butterfly in Brazil can cause typhoons in Mexico

That mismatch.

Pardon me, if I am dramatising, but in effect this is what is observed. Long long time back, one meteorologist, was studying and simulating weather models in his computer[wait a minute! computers existed those days? ]

What he found that, the same simulation running twice with different sets of initial condition, gave mind bogglingly different result. It was our dear ol’ friend Lorenz who talked first about Lorenz attractor and chaos theory. And in effect weather is a chaotic system.

Many of us think, that chaos is too much of a non-existent to even start bothering about, so all ye! Software engineers write a program to iterate this function: 2x2-1 with different initial conditions. Try 0.75 first and then try 0.7499. You will find some lovely reasons to think about!

So before the next time you gamble at a roulette table, forget that it’s a random system, and forget even predicting it, by Physics. It is most likely to turn wrong. No points for guessing why!

Till next time. Ciao!


sohamdas said...

Thanks for making me so bigger than life ;) :)

I have already started getting proposals from great looking girls from Wipro.. :)

Andrew said...

one interesting thing about chaos math is that it doesn't actually require measurement error. (if the system in question is iterative quadratic or some such blah blah, you can get "chaotic" results even with perfect accuracy. relevant distinction between "accuracy" and "precision".)

sohamdas said...

Andrew, can you shed some more light on it... I think suddenly I have hit upon something interesting here on a lazy Saturday afternoon...

Do you mean, that a lack in precision can lead to chaotic results?

Interesting... I am waiting

Andrew said...

Hi Soham, say you have an irrational number that you represent as a 32 bit number. All the bits of the "real" or actual number past 32 can't be coded in the number. So, even though you're not making an error with 0.33333..., the finite precision of the 32 bits basically introduces what is like an error or a flap of the butterfly's wings.

Makes it sound like computing is pointless for chaotic systems. But, in a weird twist, the prediction that you calculate, even though it's wrong for the 0.333 number, is right for some other input. (that's called the shadowing lemma I think.) so, while you can't predict the weather two days from now, you can run a bunch of simulations and in statistics/aggregate they still represent real weather patterns. isn't that cool as crap? don't ask any deeper questions on this though because I definitely do not know the answer ;-)

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