Chaos and Fractals

A glance at the cobweb plots of \(f(x)=x^2-2\) and \(g(x)=4x(1-x)\) 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 seeing why it's chaotic, we'll show that it's conjugate to \(f(x)=x^2-2\text{,}\) implying that it too is chaotic.