Graphical Method

 

In addition to doing a calculation, there is a graphical procedure for finding the itinerary of an orbit. This graphical method is illustrated in Figure 2.4 for the same parameter value and initial condition used in the previous example

  
Figure 2.4: Graphical method for iterating the quadratic map. (Generated by the Quadratic Map program.)

and is based on the following observation. To find from we note that ; graphically, to get we start at on the horizontal axis and move vertically until we hit the graph y = f(x). Now this current value of y must be transferred from the vertical axis back to the horizontal axis so that it can be used as the next seed for the quadratic map. The simplest way to transfer the y axis to the x axis is by folding the x-y plane through the diagonal line y = x since points on the vertical axis are identical to points on the horizontal axis on this line. This insight suggests the following graphical recipe for finding the orbit for some initial condition  :

1. Start at on the horizontal axis.
2. Move vertically up until you hit the graph f(x).
3. Move horizontally until you hit the diagonal line y = x.
4. Move vertically--up or down--until you hit the graph f(x).
5. Repeat steps 3 and 4 to generate new points.

For the example in Figure 2.4 it is clear that the orbit is converging to the point 1/2. This same graphical technique is also illustrated in Figure 2.5 for the more complicated orbit that arises when .

  
Figure 2.5: Graphical iteration of the quadratic map for a chaotic orbit. (Generated by the Quadratic Map program.)

We can also ask again about the fate of a whole collection of initial conditions, instead of just a single orbit. In particular we can consider the transformation of all initial conditions on the unit interval ,

 

subject to the quadratic map, equation (2.5). As shown in Figure 2.6, the quadratic map for can be viewed as transforming the unit interval in two steps.

  
Figure 2.6: Stretching and folding in the quadratic map ( ).

The first step is a stretch , which takes the interval to twice its length, . The second step is a fold, which takes the lower half of the interval to the whole unit interval, and the upper half of the interval also to the whole unit interval with its direction reversed, and . These two operations of stretching and folding are the key geometric constructions leading to the complex behavior found in nonlinear systems. The stretching operation tends to quickly separate nearby points, while the folding operation ensures that all points will remain bounded in some region of phase space.

Another way to visualize this stretching and folding process is presented in Figure 2.7. Imagine taking a rubber sheet  and dividing it into two sections by slicing it down the middle.

  
Figure 2.7: Rubber sheet model of stretching and folding in the quadratic map.

The sheet separates into two branches at the gap (the upper part of the sheet where the slice begins); the left branch is flat, while the right branch has a half-twist in it. These two branches are rejoined, or glued together again, at the branch line seen at the bottom of the diagram. Notice that the left branch passes behind the right branch. Next, imagine that there is a simple rule, indicated by the wide arrows in the diagram, that smoothly carries points at the top of the sheet to points at the bottom. In particular, the unit interval at the top of the sheet gets stretched and folded so that it ends up as the bent line segment indicated at the bottom of the sheet. At this point Figure 2.7 is just a visual aid illustrating how stretching and folding can occur in a dynamical system. However, observe that the resulting folded line segment resembles a horseshoe. These horseshoes were first identified and analyzed as recurring elements in nonlinear systems by the mathematician Steve Smale . We will say much more about these horseshoes, and make more extensive use of such diagrams, in section 4.8 and Chapter 5 when we try to unravel the topological organization of strange sets.