Help with brute force method of producing bifurcation diagrams of discrete-time systems

Question

I have a homework question concerning a brute force method of creating bifurcation diagrams. This seems really abstract for me and would like a clearer description of how the method works. Can someone help?

Answer 1

Essentially the way these bifurcation diagrams are produced is to slowly step through a range of values for your bifurcation parameter and for each value, choose a bunch of random points and iterate your map on them a large number of times. By doing this, these points should tend toward whatever stable points you have. Throw away the early values of where the point is as these aren't particularly interesting, but plot all the later points against the value of your bifurcation parameter and that way you'll clearly see these stable sets and how they change.