Now what was all that about a butterfly controlling the future??? "Chaos" began Edward Lorenz, a meteorologist from MIT began simulating whether patterns in his computer. He devised three equations to simulate weather patterns (as is obvious this was as simplified as whether patterns get.) He would do this at a stretch and "play" with his weather, where he was god and weather his toy. He once decided to start from somewhere in the middle instead of beginning the simulation from the simulation from the beginning that is time, t=0. Using the values he got from the previous print out .......... but something happened. He ended up with something absolutely different from the previous trial.......something absolutely different. 

                    Why?.....? Lorenz had hardly any idea, he even contemplated the possibility of instrument failure. The value which was referred to from the printout was accurate only up to three decimal places where as the values in the memory of the computer had six decimal places. An approximation of such small proportions had catapulted into completely changing the weather in Lorenz's computer. 

                   This discovery by Lorenz completely contradicted the earlier belief that rough approximations at the start lead to roughly approximate results. The concept of a deterministic universe was shattered. Determination of all the parameters involved, infinitely accurately is something possible only in dreams and hence even a so called "Very Accurate" values of the contributing parameters reduces itself into a gross approximation at some scale, thereby making long term predictions of weather like systems not difficult but an impossibility.

                   But how exactly can a discrepancy of something like say 0.0001 change the result not by degree but by type? 

The answer lies in the type of systems we are dealing with. A microscopic difference in the motion in wind currents say around your hand changes the behavior of the layer if air around it and this process continues. It is an iterative system, where the behavior of a section is dependent on the behavior of the previous section. This gives rise to a system that is capable of not just transportation of the initial discrepancy but also provides for the manifestation of the initially "microscopic" discrepancy in a significant manner.

                    Hence Lorenz once half jokingly asked "Can a butterfly flapping its wings in Tokyo cause a tornado in Texas?" 

                    Still not convinced about the -' Sensitivity to initial conditions' of such a system? then you ought to perform this simple mathematical calculation.

Take a simple iterative function f(x)=2*x*x-1 ,

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 Hold on a minute, what on earth is an iterative function ?

An iterative function is one where a series of calculations are carried forward  repetitively or iteratively on the     equation, the result obtained from the first calculation is used for the next one and that for the next....... this keeps on going.

But what has this got to with the weather?

Read the paragraph explaining (I hope) how minor discrepancies grow. Since the behavior of one affects the next the property of repetitiveness is spawned.

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                                                OK then back to the function mentioned above. If x is given a value =1, irrespective of the number of iterations, the value of the function remains equal to 1. Well then what if you take x=1.00000001, what difference do you think would an increment of .0000001 make. 

                    With a simple calculator, keep performing this calculation, use the value obtained after the first iteration for the second iteration and so on.

The values obtained from this process are given below. 

Where the +e64 means that there are 64 more 0's following that number.

                We carried out only 21 iterations we arrived at a 128 digit number where as for initial conditions of x=1, it would still have been f(x)=1......................get the picture? (At least a fuzzy one.)

                The point is that the flap of the butterfly's wing does not provide the energy required for the tornado that is absurd, but it initiates a sequence of events ,which eventually build up to such massive changes. It is to say that minor changes at the start cause unexplainably large changes after sometime. 

                So does this mean that when you clicked the mouse to view this page, you changed the weather for all time? ......................................  You possibly did.

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