The logistic map a*x*(1-x) is a model problem showing simple chaotic behaviour. A bifurcation diagram of the logistic map is a graph showing the possible ranges of values of x against a bifurcation parameter, a. The values of x vary from 0 to 1, and are iterated using a discrete time step: x_{t+1}=a*x_t*(1-x_t). Stability can be seen for values of a=0:3, and from 3:4 period doubling and chaos occurs.
Writing a fast Matlab code to show this is not too difficult, Matlab optimizes vector iterations so this is the best path to take. Choosing a large number of points for each value of a and iterating a lot of times, then plotting the next 100 or so iterations seems to be the best way of seeing the long term behaviour of the system.
If you want to start learning about discrete dynamical systems the best place to start is in a book called chaos, by james Gleick.
[note to self, find good file hosting site]
