This applet shows how the histograms of orbits of the logistic function f(x) = ax(1-x) change as the parameter a changes. In particular, we can observe the period doubling bifurcations and the occurence of ergodic orbits.

Begin by entering an initial value a_o for a and an initial value x_o for the orbit (for the latter, it doesn't really matter what number in [0,1] you choose, so the default value is good enough). Delta a is the increment of a at each step, and N is the number of points in each orbit that are computed at each iteration (more points give more detail in the histogram). Begin the applet by pressing the forward button or the backward button (to increase or decrease the value of a, respectively). You can speed up or slow down the steps, or pause the applet by pressing the required button.

Periodic orbits exhibit histograms that have a finite number of spikes; one spike occuring at each point of the orbit. As a increases from 2 to around 3.6, we see that the orbits bifurcate into orbits with longer and longer periods. That is, the orbits become more 'complicated'. (One can explain these bifurcations geometrically by looking at the shape of the graph of f^k as a changes; see the applet Logistic Bifurcation.) When a reaches approximately 3.6, the histograms become 'continuous'; they are no longer composed of a finite number of spikes, but are continuously distributed over regions of [0,1]. These are the ergodic orbits.

As a continues to increase however, we often see the histograms 'collapsing' back into a periodic one, and then passing through a cycle of period doubling bifurcations (but on a much 'faster' time scale) before becoming ergodic again.

The final state diagram is displayed on the top with a vertical line indicating the current value of a. Here we can see the 'periodic windows' among the ergodic orbits. The histograms will reflect these windows, but you will see many more that are not visible in the final state diagram. Especially if you choose the increments of a, Delta a, small. If you keep track of the parameter values of a for which a bifurcation occurs, then you can compute the Feigenbaum constant for these period doubling cascades.