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Chaos Theory: Better Modeling Through Nonlinear Patterns

01 Mar Posted by in Algebra, Science | Comments

Overview: Chaotic Patterns Aren’t Random, Just Nonlinear
Chaos theory is a system of mathematical modeling used for a number of systems found in mathematics, physics, biology, the social sciences, art, and philosophy. The mathematics involved is a simple way to describe relatively complex events. A small change on one variable can set off large oscillations that are not predictable to any degree.

The “Butterfly Effect” and Weather
The weather in a given location can only be predicted a week in advance, because so many small changes in air temperature, barometric pressure, humidity, and other things can cause changes in the weather. That is why there is only a chance of snow or a chance of rain. The predicted blizzard may miss the area entirely, or may hit even harder than was first thought. Scientists call that “the butterfly effect”. In a chaotic system, according to Lorenz, a butterfly’s wings flapping one part of the world could or could not set off a storm in another part of the world.

Convergence Points Form Strange Attractors
Some behaviors of people, animals, and systems when they are graphed, converge around two or three separate points, called “strange attractors”, and these curves are followed the more points are plotted. For example, convection currents in air follow an attractor pattern, as does the movement of a weight attached to a spring. Many concepts in areas as diverse as economics, psychology, mathematics, and physics will show attraction around separate points.

File:Lorenz attractor yb.svg

Chaos in Biology and Landscapes
Many natural objects follow the properties of chaos theory, such as clouds, coastlines, and earthquakes. Relatively simple calculations, repeated over and over with different values (called “iteration”) when graphed together will form many different and beautiful patterns, that differ only in scale. Some look like snowflakes, and others like swirling paisley prints. The Mandelbrot set, named after the mathematician who discovered it, shows a fractal image. Many of these images look like fine art.

Chaos in Medicine
Some of the most interesting applications of chaos theory have been in medicine. For example, an irregular heartbeat follows a chaotic pattern rather than the more typical pattern. Doctors can analyze the irregular pattern and find ways to introduce small changes to make the heartbeat more regular. Blood vessels follow fractal patterns, as they branch out into smaller and smaller capillaries. The EEG follows a chaotic pattern, as the underlying processes that occur in nerve cells converge in chaotic ways.

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