# chaos theory

**Topics:**Chaos theory, Butterfly effect, Fractal

**Pages:**7 (1228 words)

**Published:**December 18, 2013

What exactly is the chaos theory? Some believe the chaos theory is one of the

many theories that will be recognized in the centuries to come. The chaos theory embodies many

conditions of science, such as physics, engineering, economics, philosophy, mathematics, music,

and even psychology. The chaos theory is only beginning. The chaos theory is a theory used in

different categories of science that a seemingly possible phenomena has an underlying meaning.

When was chaos first discovered? Edward Lorenz was the first true experimenter in chaos,

he was a meteorologist. In 1960 Edward Lorenz was working on a weather prediction problem, he

had a computer set up to model the weather with twelve equations. His computer program did not

predict the weather, but theoretically predicted what the weather might be.

In 1961 Edward Lorenz wanted to see a specific sequence again, to save time he began in

the middle of the sequence. He entered his printout number and let it run. An hour later the

sequence had changed differently. The pattern had diverged, ending up being extremely different.

His computer had saved the numbers to a six decimal place, he printed it out for three decimal

places to save paper. The original sequence was 0.506127 he had it as 0.506. Lorenz's experiment: the difference between the starting values of these curves is only .000127. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)

This is how the butterfly effect became, because of the number of differences of the two

curves starting points was that of a butterfly's wings flapping. Unpredictability is one of the most

important element is a complicated system. Lorenz calls this unpredictability “sensitivity to initial

conditions,” which is also known to be the butterfly effect. This idea means with a non-linear,

complex systems, starting conditions will effect in extremely dissimilar outputs. The effect of the

butterfly's movements, to predict the weather. An example is if a butterfly flaps it's wings in

Tokyo, it could predict a storm in Texas in several weeks time.

The dependance on initial conditions is extreme. There is a rule for complicated systems

that one cannot create a model that will predict outcomes accurately. The idea initial conditions

on sensitive dependance mathematical roots are powerful. If you have a circle with the points X0

and X1, this represents the starting value for a variable. “We assume that the difference between

there two numbers is represented by the distance between the points on the circle, given by the

variable d. To demonstrate the importance of infinite accuracy of initial conditions, we iterate T.

After only one iteration, d, or the distance between T(X0) and T(X1), has doubled. Iterating again,

we find that the distance between the two points, already twice its initial size, doubles again. In

this pattern, we find that the distance between the two points, Tn(X0) and Tn(X1), is 2nd. Clearly,

d is expanding quite rapidly, leading the model further and further astray. After only ten iterations,

the distance between the two points has grown to a whopping 210d = 1024d.”

This example determines that to close conditions begin, after only a few minor differences,

and iterations. The exact point on the circle can only be describes with an infinite amount of decimal

places, the other remaining decimal places are discarded. There will always be a decimal error even if

you enter the initial numbers into the computer with precision. Chaos is deterministic, sensitive to

initial conditions, and orderly. Chaotic systems do have a sense of order, non chaotic systems are

random. In a chaotic system even a minor in the starting point can lead to different outcomes.

Equations for this system appear to show an increase to...

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