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Butterfly Effect

What is Butterfly Effect?

The "Butterfly Effect" is the propensity of a system to be sensitive to initial conditions. Such systems over time become unpredictable, this idea gave rise to the notion of a butterfly flapping it's wings in one area of the world, causing a tornado or some such weather event to occur in another remote area of the world. It is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory. Small variations of the initial condition of a dynamical system may produce large variations in the long term behavior of the system. So this is sometimes presented as esoteric behavior, but can be exhibited by very simple systems: for example, a ball placed at the crest of a hill might roll into any of several valleys depending on slight differences in initial position

The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate or even prevent the occurrence of a tornado in a certain location. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale alterations of events. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different. Of course the butterfly cannot literally cause a tornado. The kinetic energy in a tornado is enormously larger than the energy in the turbulence of a butterfly. The kinetic energy of a tornado is ultimately provided by the sun and the butterfly can only influence certain details of weather events in a chaotic manner.

Who devised the idea?

1960, E Lorenz was doing weather prediction research at MIT. He managed to get funding to acquire a Royal McBee LGP-30 computer with 16 KB of memory that could do 60 multiplications per second. Lorenz set the new computer to solve a system of 12 differential equations that model a miniature atmosphere. To speed up the output, Lorenz altered the program to print only three significant digits of the solution trajectories, although the calculations themselves were carried out with a somewhat higher precision After seeing a particularly interesting run, he decided to repeat the calculation. As a shortcut on a number in the sequence, he entered the decimal .506 instead of entering the full .506127 the computer would hold and started the program. Lorenz went for a coffee break, and when he returned, he found that the results we completely different. At first he thought that some vacuum tubes in the computer were not working. Upon careful check, he realised that the discrepancies between the original and re-started calculations occurred gradually: First in the least significant decimal place and then eventually in the next, and so on

From the above mentioned event what Lorenz has discovered is that tiny differences in the starting conditions can have big effects on later on.

Why it is called Butterfly effect?

• Lorenz coined the phrase "butterfly effect" describing how a slight change in initial conditions can lead to drastically different outcomes. "The flap of a butterfly's wings in South America could be responsible for a tornado in Texas."

• He chose a butterfly because a three dimensional image of the trajectory mapped out by his equations looks like a butterfly.

Misconception

When the initial conditions change a bit, "does the flap of a butterfly's wings in Brazil set off a Tornado in Texas?"( Edward Lorenz Dec 1972, Talk given in Washington DC).There is a common misconception as with regards to the words "set off" (or cause in other formulations of the same idea). You cannot call uncle Eddie in Brazil and ask him to let his pet-butterflies flap their wings so that they cause a rain storm in Dhaka to soak your boy/girl-friend whom you are angry at. What is means is that you have to imagine two identical worlds. In one of the worlds you place a butterfly and let it flap its wings. In the other world you don't place the butterfly. Now you wait a while (a few months or more perhaps) and will see that the global weather patterns on your two worlds are completely different.

The butterfly effect refers to the exponential growth of any small perturbation. However, this exponential growth continues only so long as the disturbance remains very small compared to the size of the attractor. It then folds back onto the attractor. Unfortunately, most people miss this latter part and think that the small perturbation continues to grow until it is huge and has some large effect. The point of the effect is that it prevents us from making very detailed predictions at very small scales, but it does not have a significant effect at larger scales.

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