Fractals and Chaos

Fractals deal with geometric patterns and quantitative ways of characterizing those patterns. Chaos, in contrast, deals with time evolution and its underlying or distinguishing characteristics. Fractals are a class of geometric forms; chaos is a class of dynamical behavior.
Fractals and chaos are closely intertwined and often occur together. For instance, most chaotic attractors have a fractal striated texture. Points on such attractors plot as a set of layers that look the same over a wide range of scales. The layered structure can be difficult to see because of noise and the small number of points in some datasets. The points then seem to plot on a single curve. Figure 1 is a chaotic attractor on which the same amount of detail appears as we repeatedly zoom in on smaller and smaller sections (and enlarge and refocus them for appropriate viewing). In general, "the chaotic attractors of flows or invertible maps typically are fractals; the chaotic attractors of noninvertible maps may or may not be fractals. The chaotic attractors of the logistic equation, for example, are not fractals" (Eubank & Farmer 1990). Other chaos-related geometric objects, such as the boundary between periodic and chaotic motions in phase space, also may have fractal properties (Moon 1992: 325). Because of those close relationships, fractals can help detect chaos.

Figure 1: Three different magnifications of part of the Hénon attractor. Entire middle diagram shows area outlined by the rectangle within upper diagram; entire lower diagram is region outlined by the rectangle within middle diagram. Computergenerated graphics by Sebastian Kuzminsky.