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Principles of Chaos Theory

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Science is an art that depends heavily on predictability, organization, and painstaking accuracy. Since its earliest inception, scientists have worked to provide humanity with the most accurate and precise view of the world as possible. Using the one-two punch of mathematics and physics, mankind slowly progresses through every great phenomenon and mystery the world has to offer, attempting to explain it with simple, rational definitions and equations. Because of this, we are accustomed to having a linear view of the world - every question, no matter how complex, can have a one simple answer that completely addresses it. In many respects, this is a perfectly acceptable way of thinking. After all, we have a thorough understanding of many patterns found in nature. However, there are still many things left that linear mathematics is not able to explain. For example, how can we mathematically predict the shapes of clouds? Or the exact path of a lightning bolt? Or the pattern of a tree's bark? Questions like these are beyond the reach of conventional mathematics. To explain them, a different type of thinking must be used - something that can take science and mathematics to a different level and relate them to the inherent unpredictability in the world around us. This thinking is known as Chaos Theory.

Although the word "chaos" is traditionally associated with total disorder, it is misleading when describing Chaos Theory. In 1986, a group of scientists submitted a new definition for chaos: "Stochastic behavior occurring in a deterministic set." This definition was later changed to "Lawless behavior that is ruled by law."

Although Chaos Theory has only been around for about 50 years, its principles have been around for much longer than that. It wasn't until recently that scientific knowledge progressed to the point where the distinction between linear and chaotic science began to become apparent. In 1898, a French mathematician named Jacques Hadamard published a study describing the chaotic motion of free particles gliding on a negative curvature. Hadamard was able to show that their trajectories were unstable and unpredictable, diverging exponentially from one another. In the early 1900s, Henri Poincare, the great French mathematician/philosopher/physicist, declared the existence of non-periodic orbits. Both of these discoveries would play a major role in the formation of Chaos Theory 60 years later. As scientific and technological knowledge began to expand dramatically in the mid-1900s, it became more and more apparent that linear science could not explain certain observations. However, it wasn't until the invention of the computer that the stage was set for Chaos Theory to emerge. Early computers, such as ENIAC, were often used to run simple weather forecasts. One early user of these systems was Edward Lorenz, an American meteorologist. In 1961, Lorenz was using a basic computer to run a weather simulation. His goal was to use a sequence of data to simulate weather conditions at a future point in time. To do this, he composed a list of data variables corresponding to the different factors required to predict weather. As was a relatively common mathematical practice at the time, he decided to enter his numbers with three decimal places of accuracy. To his surprise, he got completely different results every time he ran the simulation. When he looked into it, he found that the simulation required at least 6 decimal places of accuracy to get any sort of consistency in simulation results. He was highly intrigued - How could a 0.000001 variation in any one of his factors be the difference between sunshine and rainstorm? It was a crucial realization that heralded the unofficial birth of modern Chaos Theory: Tremendously small changes in initial conditions produce large changes in the long-term. In 1963, Lorenz published his findings in a meteorological journal. Because he was not a mathematician or physicist, his discovery was largely ignored by the scientific community. His work would not be acknowledged until years later, when they were rediscovered by others. Nevertheless, Lorenz knew he had come across something very important, and he would dedicate his life to looking further into his new discovery.

Lorenz quickly went to work investigating chaotic motion. He discovered that the motion of particles described by certain systems "neither converged to a steady state nor diverged into infinity, but stayed in a bounded but chaotic region." He observed that particles that seem to move randomly still obey some sort of order in the long run. Lorenz tracked the location of a particle moving subject to atmospheric forces, and from that derived a series of differential equations. When he solved the system numerically, he found that his particle moved wildly and apparently randomly. After a while, though, he found that while the motion of the particle was chaotic, a general pattern began



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