I made a fancier version of my previous logistic-map diagram:

This version has the old representation on the left, the more traditional cobweb diagram in the middle, and a cumulative distribution of all the x values on the right, along with giving you the ability to change the number of iterations to go up to 256. Rather than trying to explain everything in detain in text, I made a 10-minute YouTube video (which you can probably get away with watching at 2x speed, as I was perhaps being too deliberate with my statements…):