When you click on any point in the graph window, you see the graphical iterates of that x value. The righthand scrollbar changes the parameter, c . Below the picture is an explanation. The dynamics of f is understanding what happens to points after successive iterates.
Graphical iteration is a visual technique to see where points go under iteraton.
Repeating this process gives all the iterates of x . As c decreases, the fixed point ceases to be attracting and a period two orbit forms. This is called a period doubling bifurcation . Move the scroll bar to see this bifurcation occur. As c decreases further, the period two orbit ceases to attract and there is an attracting period four orbit.