Chaos theory

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 ad 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: he ditterence between the sta rting values ot 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 utterfly’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 XO and Xl , 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 ariable d. To demonstrate the importance of infinite accuracy of initial conditions, we iterate T. Atter only one iteration, d, or the distance between T ), 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(XO) 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 ifferences, 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 completely random behaviour. When raphed the system, something surprising happened, the output stayed on a double spiral curve. Lorenz’s equations were certainly ordered, because they all had followed a spiral. The points never ended on a single point , but they weren’t periodic either, they never repeated the same thing.

He called his graphed equation the Lorenz attractor. In 1963 he published a paper describing his discovery and the unpredictability of the weather. This paper also included key information about the types of equations and what caused this behaviour. Since he was not a mathematician or a physicist he wasn’t cknowledged for his discoveries until year later, when there had already been rediscovered by others. Lorenz had to wait for someone to discover him, his discovery was revolutionary. Another example of sensitive dependance of initial conditions is flipping a coin. There are two variables.

One is how fast it is flipping, and the other is how fast it will hit the ground? Apparently, it should be likely to control how might the coin end up. In practice, it is impossible to control exactly how fast the coin flips and how high. There are similar problems like this in ecology. This occurs with the prediction of biological population. If the population rises continually, but with predators and limited food supply the equation is incorrect. next year’s population = r * this year’s population * (1 – this year’s population) Benoit Mandelbrot was a mathematician working at 18M, he was studying self- similarity.

One ot the areas ne studying was cotton prize tluctuation. He tried many times to analyze the data of the price for cotton, but the data did not go with the normal distribution. He decided to collect data from. Mandelbrot eventually gave up, until he decided to gather all the information dating back to 1900. He IBM’s computer and found a surprising fact: The numbers that produced aberrations from the point of view of normal distribution produced symmetry from the point of view of scaling. Each particular price change was random and changes unpredictable.

But the sequence of changes was independent on scale: curves for daily price changes and monthly price changes matched perfectly. Incredibly, analyzed Mandelbrot’s way, the degree of variation had remained constant over a tumultuous sixty-year period that saw the two World Wars and a depression. ” dames Gleick, Chaos – Making a New Science, pg. 86) Another example of the chaos theory is the human heart. The heart has a chaotic pattern. The time between the beats is not constant. It depends on how much actions a person is doing and there also among other things.