Under certain conditions, the motion of a particle described by certain systems will neither converge to a steady state nor diverge to infinity, but will stay in a bounded but chaotically defined region. By chaotic, I mean that the particle’s location, while definitely in the attractor (the 3-D region), might as well be randomly placed there. That is, the particle appears to move randomly, and yet obeys a deeper order, since it never leaves the attractor.
Edward Lorenz modeled 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 behavior of the particle was chaotic, the general recursive pattern of an attractor appeared. In his case, the pattern was the butterfly shaped attractor now known as the Lorenz Attractor.
Notice how the sequence of values, despite being non-periodic and unpredictable, never jump off the 3-D space that is seen above. This signals the presence of an ‘attractor’. This is the Lorenz Attractor.