Chaos-order Theory (summarized from several sources)

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1. Introduction to Chaos

What exactly is chaos? The word chaos is sometimes taken to mean the opposite of cosmos, in that the later term has connotations of order. The dictionary definition of chaos is complete disorder or confusion, turmoil, turbulence, primordial abyss, and undesired randomness. The other definitions are: an order without periodicity, reply acts in a simple deterministic system, a qualitative study about instable no periodic act in a deterministic nonlinear dynamic system. Ian Stewart defined that chaos is a simple model ability, which does not obtain disarranged particles, to produces a very irregular and random act. There are many definitions of chaos, but put simply, it is the idea that is possible to get completely random results from normal equations. Chaos also covers the reverse: finding the order in what appears to be completely random data.

2. Early Chaos

When was chaos first discovered? The word chaos might have first appeared by Hesiod, a Greece, in his poetry, Theogeny (700 B.C), in part I: “At the beginning there was chaos, nothing but void, formless matter, infinite space”. Later in Milton’s Paradise Lost: “In the beginning, how the heaven and earth rose out of chaos”. Both Shakespeare (Othello) and Henry Miller (Black Spring) refer to chaos. Ilya Prigogine showed that complex structures could come from simpler ones. This is like order coming from chaos. Henry Adams previously described this with his quote; “Chaos often breeds life, when order breeds habit”. Henri Poincare, who discovered the planet Neptune, was really the “Father of Chaos (Theory)”, however. The planet Neptune was discovered in 1846 and had been predicted from the observation of deviation in Uranus’ orbit.

During the 1960’s, Edward Lorenz, a meteorologist at MIT (Massachusetts Institute of Technology) was working on a project to simulate weather patterns on a computer. He had a computer set up, with a set of twelve equations to model the weather. It did not predict the weather itself. However, this computer programme did theoretically predict what the weather might be. One day, he wanted to se a particular sequence again. To save time, he started in the middle of the sequence, instead of the beginning. He entered the number off his printout and left to let it run. When he came back, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up wildly from the original. Eventually he figured out what happened. The computer stored the number to six decimal in its memory. To save paper, he only had print out three decimal places. In the original sequence, the number was .506127 and he had only typed the first three digits, .506. By all conventional ideas of the time, it should have worked. He should have gotten a sequence very close to the original one. A scientist considers himself lucky if he can get measurements with accuracy to three decimal places. Surely the fourth and fifth, impossible to measure using reasonable methods, cannot have a huge effect on the outcome of the experiment. Lorenz proved this idea wrong.

This effect came to be known as the butterfly effect. The amount of difference in the starting points of the two curves is so small that is comparable to a butterfly flapping its wings. The flapping of single butterfly’s wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month’s time, a tornado that would have done devastated the Indonesian coast does not happened. Or maybe that was not going to happen, does. This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behavior of a system

3. Chaos Theory

Chaos theory describes complex motion and the dynamics of sensitive systems. Chaotic systems are mathematically deterministic but nearly impossible to predict. Chaos is more evident in long-term systems than in short-systems. Behavior in chaotic systems is a periodic, meaning that no variable describing the state of the system undergoes a regular repetition of values. A chaotic system can actually evolve in a way that appears to be smooth and ordered, however. Chaos refers to the issue of whether or not it is possible to make accurate long-term predictions of any system if the initial conditions are known to an accurate degree.

 

(this essay summarized from several sources. I wrote it about 5 years ago, and I miss the sources’ list. I’ll find it asap)

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