We were introduced to the idea of real iteration back in Subsection 1.1.2 but there's no reason the function we iterate needs to come from Newton's method. Figure 1 illustrates iteration using a geometric cool called a cobweb plot. In part (a) we see the iteration of \(N(x)=x/2+1/x\text{,}\) that arises in Newton's method. In part (b) we the iteration of \(f(x)=3.1x(1-x)\text{.}\) This is an example of a logistic function and really has nothing directly to do with Newton's method.

The logistic function above is a member of a family of functions - the logistic family. These functions have the form \(f_r(x) = rx(1-x)\text{.}\) When studying a family of functions like this we are often interested in the various types of behavior that might arise. We can explore these possibilites with a tool called a bifurcation diagram. The bifurcation diagram for the logistic family is shown in Figure 2.