Chaotic motion

Lorenz looked for a way to visually represent chaotic motion; this is possible by creating a phase diagram (any type of motion can be represented by a phase diagram). A phase diagram has each axis represent one dimension (x,y,z) and time is implied. Examples are a system that is at rest will be represented as a point and a periodic motion system will be plotted as a closed curve. The motion of the system depends on the initial state of the system and the set of parameters, but phase diagrams frequently show the motion as identical for all initial states in a region around the motion. The system is almost “attracted” to the motion; hence this motion for the system is called an “attractor” and is common. Attractors usually are simple (points and curves) and formed by most types of motion. The Lorenz system of three differential equations gave rise to the Lorenz Attractor, which is an example of a “strange attractor,” because it was formed from a chaotic system diagram. All strange attractors have a fractal structure and are usually detailed and complex. The Lorenz Attractor and all strange attractors show order but never repeat. “The Lorenz Attractor always has the same butterfly shape, no matter how random each variable may appear to be on its own, the combination of the three always produces the same picture” (www.gweet.net/-rocko).

Inherent in chaos is randomness; the understanding of chaos theory has “implied fundamental limits on the ability to make predictions” (Crutchfield, Farmer, Packard and Shaw, 56) yet there is underlying order in chaos, represented by the strange attractors or specifically the butterfly-shaped Lorenz Attractor. In nature we see chaos everywhere, beneficially employed, “through amplification of small fluctuations it can provide natural systems with access to novelty” (Crutchfield, Farmer, Packard and Shaw, 56).