Zones of Popularity on the Attractor

The fourth orderly structure of chaos is zones of relatively greater popularity on each chaotic attractor. These are zones a chaotic system is more likely to visit during its evolution. On the graphs of Figures 1 and 2, areas the trajectory visits the most are proportionally darker. The darkest zones are most likely to be visited, for a given k. In contrast, the trajectory never goes to white zones at all. The zones of greater popularity could provide some guidance in devising a statistical theory for accurately predicting the likelihood of x_t taking on a particular value (Jensen 1987). A fifth orderly feature of chaos is the fractal structure.
All this regularity indicates that order is a basic ingredient in chaos. That's why some people look upon chaos as order disguised as disorder. Some authors believe that pure chaos sets in only when the control parameter is at a maximum (such as k=4 in the logistic equation).

Figure 1: Chaotic domain of logistic equation. Computergenerated graphics by Sebastian Kuzminsky.

Figure 2: Window consisting of a period-three cycle, occurring at about k = 1.76 for the quadratic equation (after Grebogi et al. 1987). Computer-generated graphics by Sebastian Kuzminsky.