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Despite advances made in systems analysis, many systems remain beyond the reach of current mathematics. Chaos theory, a relatively new area of mathematics, concerns the analysis of unpredictable systems that are extremely sensitive to initial conditions. One important example of a chaotic system is climate. Global climate modeling is an area of mathematical research that seeks to develop models for predicting the weather, given accurate data from weather satellites orbiting Earth. The problem in developing such models arises not from lack of data but from the difficulty of modeling such a complex system (Earths atmosphere) with a small number of equations. In such models even a thousand equations may be considered small. The solution of these equations is very sensitive to changes in the initial conditions. The term initial conditions refers to all the measurements at the starting time. A tiny inaccuracy in a single measurement of a chaotic system–such as a temperature variation of a fraction of a degree–can produce large errors in solutions to the models equations and predictions.
Meteorologist Edward Lorenz tried to model climate in a series of equations during the 1960s. In doing so, he produced a chaotic system of three related differential equations, now known as a Lorenz attractor, or strange attractor. Through his models he discovered the sensitivity of chaotic systems to initial conditions, which he phrased in the question “Does the flap of a butterflys wings in Brazil set off a tornado in Texas?”
The Lorenz attractor is an example of a fractal, a pattern produced by applying a function repeatedly, much like pushing a button on a calculator over and over. The sequence {x, f(x), f(f(x)), f(f(f(x))), }, when graphed in two dimensions, gives rise to beautiful, complex geometric images such as the Mandelbrot set pictured in this article. These fractal images are named after Benoit Mandelbrot,