Section 2.3 Graphical analysis
There is an efficient geometric tool to visualize functional iteration. The basic idea is simple: Suppose we graph the function \(f\) together with the line \(y=x\text{.}\) If those two graphs intersect; that point of intersection is a fixed point. Now, suppose we're on the line at the point \((x_i,x_i)\text{.}\) If we move vertically to the graph of the function, we preserve the \(x\) coordinate but change the \(y\) coordinate to \(f(x_i)\text{.}\) Thus, we arrive at the point \((x_i,f(x_i)) = (x_i,x_{i+1})\text{.}\) If we then move horizontally back to the line \(y=x\) we now preserve the \(y\) coordinate but change the \(x\) coordinate so that the \(x\) and \(y\) coordinates are the same. Thus, we arrive at the point \((x_{i+1},x_{i+1})\text{.}\)
In summary: The process of moving vertically from a point on the line \(y=x\) to the graph of \(f\) and back to the line horizontally is a geometric representation of one application of the function \(f\text{.}\) This step is illustrated in Figure 2.3.1(a). Repeated application of this process represents repeated application of \(f\text{,}\) i.e. iteration. This is illustrated in Figure 2.3.1(b). Note that the orbit appears to be attractive.
It turns out that the process is quite sensitive to the slope of the function at the point of intersection. A slightly steeper function is shown in Figure 2.3.1(c); we notice that the fixed point now appears to be repelling. Finally, figure Figure 2.3.1(d) illustrates the fact that all hell can break loose.
Playing with cobweb plots is a great way to get a feel for the possibilites that arise in discrete dynamical systems. Here's an interactive tool that we'll use for this purpose:
https://marksmath.org/visualization/cobwebs/