Strange attractor fractal structure

Figure 4.12 shows sample a vs y plots obtained in this manner for a few elementary CA rules. Note that the patterns for nonlinear rules such as R18, R22, and 122 appear to possess a characteristic fractal-like structure reminiscent of the strange attractors appearing in continuous systems shown earlier. We will comment on the nature of this similarity a bit later on in this chapter. 

Our goals in this chapter are modest. We want to become familiar with the simplest fractals and to understand the various notions of fractal dimension. These ideas will be used in Chapter 12 to clarify the geometric structure of strange attractors. 

In a striking series of plots, Henon provided the first direct visualization of the fractal structure of a strange attractor. He set a = 1.4, b = 0.3 and generated the at-... 

One approach is to use equations with chaotic solutions. Such solutions are unpredictable over the long term yet exhibit interesting structures as they move about on a strange attractor, a fractal object with noninteger dimension. Our books show many examples of computer art produced by these and related methods. In this chapter, we propose another simple method for producing fractal art. 

There are many ways to display the solution to our general maps. One way is to plot successive values of x and y as dots on the screen. Many solutions will move toward a point and remain there. Others will settle into a periodic orbit or will move off toward infinity. The interesting cases are the chaotic ones that remain confined to a limited region but whose orbits produce a strange attractor with intricate fractal structure. You can choose the starting values of x and y arbitrarily within the basin of attraction, but you should discard the first few iterates because they probably lie off the attractor. 

In a three-dimensional differential equation the system is going to collapse on a structure that is strictly less than three in dimension, but strictly more than two. Such a strange observation leads us immediately to the concept of fractional dimensionality and the attractor associated with such a fractional dimensionality is called a strange attractor, due to its unusual nature. The fractional dimensionality is also desaibed as a fractal. ... 

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