What do the functions f(x), f(f(x)), and f(f(f(x))) actually look like? Graphed for the logistic map.
On this page we take a closer look at what the logistic function itself looks like when it is iterated. Below left is the graph of f(x) itself; middle is the graph of f(f(x)) (f composed with itself, plotted against x); on the right is f(f(f(x))). Drag the slider to adjust the value of a and watch the graphs change. You are already familiar with the first one, but the next two may surprise you.
What this experiment shows is that the iterated functions start to get very complicated. And each time you iterate, it gets more so! So perhaps it is not so surprising that the orbits are also quite complicated. Imagine how elaborate the function gets after it has been iterated (composed with itself) one hundred times.