Skip to main content
\( \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section2.2Graphical 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 1(a). Repeated application of this process represents repeated application of \(f\text{,}\) i.e. iteration. This is illustrated in Figure 1(b). Note that the orbit appears to be attractive.

<<SVG image is unavailable, or your browser cannot render it>>

Figure2.2.1Some cobweb plots

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 1(c); we notice that the fixed point now appears to be repelling. Finally, figure Figure 1(d) illustrates the fact that all hell can break loose.