Consider the logistic map discussed in the text. To gauge the density of bifurcations, one uses a measure of distance between fixed points as follows. Define \(d=x^{*}-(1 / 2)\) as the distance between the fixed point \(1 / 2\) and the nearest fixed point to it - labeled \(x^{*}\). That is, you first find the \(r\) value corresponding to a convergence at value

\(x=1 / 2\), then you identify the closest fixed point \(x^{*}\) to \(1 / 2\) at this value of \(r\), and compute the distance \(d\). For example, at first period doubling, we have \(d_{1}=0.3090 \ldots\); then after the second, we have \(d_{2}=-0.1164 \ldots\) We then define the parameter \(\gamma\) as

\[\gamma=\lim _{n \rightarrow \infty}-\frac{d_{n}}{d_{n+1}}\]

Using numerical methods, compute \(\gamma\) and verify that it given by \(\gamma=2.502907 \ldots\)