Consider the logistic map analyzed in the text. Divide the range \((0,1)\) into \(N\) equal intervals and numerically compute the probabilities \(p_{k}\) that the recursion lands in the \(k\) th interval. Then compute the entropy

\[S \equiv-\sum_{k=1}^{N} p_{k} \ln p_{k}\]
Do this so as to build up the function \(S(\alpha)\) for the range of \(2.8<\alpha<4\). Plot the function and correlate with the conclusions in the text.