Period Doubling Bifurcation for ga(x) = ax2sin(Pi*x)



Here we have an example of a unimodal function, and by varying the parameter a we can see why it goes through a period-doubling bifurcations just like the logistic family does. You can see for yourself the period-doubling bifurcations of this function using the VB program Time Series.

The first two graphs are of ga for two different values of a;



The next three graphs are of the second iterate g2 for different values of a. We see that there is a period doubling bifurcation (period 1 to period 2) at a = 2 (2 period two points bifurcated out from the the fixed point on the right).