A glance at the cobweb plots of $f(x)=x^2-2$ in Figure 2.6.1 and $g(x)=4x(1-x)$ in Figure 2.3.1 shows that they both exhibit very complicated behavior. In fact, they are chaotic in a perfectly quantitative sense. In this section, we'll introduce the doubling map, which is (in a sense) the prototypical chaotic map. After studying conjugacy in Section 2.9, we'll be able to extend this result to show that $f$ and $g$ are chaotic as well.