| CHAOS THEORY and THE LORENZ ATTRACTOR | |
| Edward N. Lorenz, a research
meteorologist, made his discovery in the 60's by observing weather
phenomena. He took
various mathematical models of fluid convection and simplified them
into a system of differential equations. Minute variations in the initial
values of variables in his primitive computer weather model (c. 1960)
would result in grossly divergent weather patterns. He found that the
length of time spent around an attractor could not be predicted, the
function would alter, seemingly randomly between the two sides of the
attractor. He also found that even slightly different starting points
could have drastically different results. At first other scientists
and mathematicians would not believe that his results were correct
because they thought it would take very complex equations to get such
chaotic results, but eventually they came around to believe that his
results were accurate. The butterfly shape formed by his function
has become rather famous and is known as the Butterfly effect.
Lorenz explored the underlying mathematics and published his conclusions
in a seminal work in the
annals of chaos theory, Deterministic Nonperiodic Flow, in
which he described a relatively simple system of equations that resulted
in
a pattern of infinite complexity; The Lorenz Attractor Named after Edward Lorenz, the Lorenz attractor is an example of a strange attractor showing chaotic motion from a simple three-dimensional model of atmospheric convection. In the study of dynamical systems, an attractor is a 'set', 'curve', or 'space' to which that a system irreversibly evolves, if left undisturbed. It is other-wise known as a 'limit set'. There are three types of attractors; point attractors, periodic attractors and strange attractors. Attractors are the pinnacle and origin of chaos theory. A strange attractor is a non-periodic attractor. This is the most common type of attractor. It is characterized by a set of coupled nonlinear ordinary differential equations. The first strange attractor discovered was the Lorenz attractor. Chaos Theory In mathematics and physics, chaos theory deals with the behavior of certain nonlinear dynamical systems that (under certain conditions) exhibit the phenomenon known as chaos, most famously characterized by sensitivity to initial conditions. Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economies, and population growth. Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder. The Butterfly Effect The butterfly effect, used to describe many chaotic phenomena, was first described as such in reference to weather: that the beating of a butterfly's wings in Brazil might set off a tornado in Texas months later. Chaos theory posits that complex systems such as the weather, or the stock market, are difficult to predict due to their sensitivity to small changes. The cumulative effect of these small changes, and their timing, makes it very difficult or impossible to predict future conditions with a high degree of certainty. Edward Lorenz, in a paper in 1963 given to the New York Academy of Sciences, said: "One meteorologist remarked that if the theory were correct, one flap of a seagull's wings would be enough to alter the course of the weather forever." Later speeches and papers by Lorenz used the more poetic butterfly. |
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