Chapter 6: Starting from Randomness

Section 7: The Notion of Attractors

Attractors in systems based on numbers

Particularly for systems based on ordinary differential equations (see page 922) a geometrical classification of possible attractors exists. There are fixed points, limit cycles and so-called strange attractors. (The first two of these were identified around the end of the 1800s; the last with clarity only in the 1960s.) Fixed points correspond to zero-dimensional subsets of the space of possible states, limit cycles to one-dimensional subsets (circles, solenoids, etc.). Strange attractors often have a nested structure with non-integer fractal dimension. But even in cases where the behavior obtained with a particular random initial condition is very complicated the structure of the attractor is almost invariably quite simple.

From Stephen Wolfram: A New Kind of Science [citation]