Chaos theory and Strange attractor
Mandelbrot Set
Polish-born French mathematician Benoit Mandelbrot coined the term “fractal” to describe complex geometric shapes that, when magnified, continue to resemble the shape’s larger structure. This property, in which the pattern of the whole repeats itself on smaller and smaller scales, is called self similarity. The fractal shown here, called the Mandelbrot set, is the graphical representation of a mathematical function.
Chaos theory
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 (Earth’s 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 model’s equations and predictions.
Applications include the study of turbulent flow in fluids, irregularities in biological systems, population dynamics, chemical reactions, plasma physics, meteorology, the motions of groups and clusters of stars, transportation dynamics, and many other fields.
-------------------------------------------------------------------------------------------
Lorenz attractor (strange attractor)
(After Edward Lorenz, its discoverer) A region in the phase space of the solution to certain systems of (non-linear) differential equations. Under certain conditions, the motion of a particle described by such as system will neither converge to a steady state nor diverge to infinity, but will stay in a bounded but chaotically defined region. By chaotic, we mean that the particle's location, while definitely in the attractor, might as well be randomly placed there. That is, the particle appears to move randomly, and yet obeys a deeper order, since is never leaves the attractor.
Lorenz modelled the location of a particle moving subject to atmospheric forces and obtained a certain system of ordinary 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 momentary behaviour of the particle was chaotic, the general pattern of an attractor appeared. In his case, the pattern was the butterfly shaped attractor now known as the Lorenz attractor.
(1996-01-13)
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 butterfly’s 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, a Polish-born French mathematician who developed fractal geometry and coined the word fractal. The interesting relationship among fractals, chaos, and unstable phenomena such as turbulence is the subject of a field called nonlinear dynamics.
