Tuesday, January 27, 2009

The Lorenz Attractor

2D mathematical systems (imagine the motion of a point in a flat plane) always move to a fixed point, a periodic/repeating orbit or to infinity. The motion thus stops, repeats itself or always keeps moving in the same direction. That seems logic to most and is stated (and proven) in the Poincare-Bendixson theorem. The funny thing is that this does not hold for 3D (imagine the motion of a point in 3D space) or higher dimensional systems. Not all motion in 3D moves to a fixed point, periodic motion or infinity. The 4th option is a chaotic motion. This motion is deterministic, stays in a bounded region, but does not follow a pattern. In 1963 Lorenz (not the more famous Dutch Lorentz), a meteorologist published a paper about convection in the atmosphere. He used the following model, This model created the following images when the "motion" is visualized in a 3D phase-space diagram, And the following graph when variable x is plotted versus time,

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